Research Article Identification of Skeletal Muscle...

Hindawi Publishing CorporationISRN RehabilitationVolume 2013 Article ID 610709 10 pageshttpdxdoiorg1011552013610709

Research ArticleIn Vivo Identification of Skeletal Muscle Dynamics withNonlinear Kalman Filter Comparison between EKF and SPKF

Mitsuhiro Hayashibe David Guiraud and Philippe Poignet

DEMAR Project INRIA Sophia-Antipolis and LIRMM CNRS University of Montpellier 34095 Montpellier France

Correspondence should be addressed to Mitsuhiro Hayashibe mitsuhirohayashibeinriafr

Received 24 March 2013 Accepted 20 April 2013

Academic Editors A Cappozzo and S Park

Copyright copy 2013 Mitsuhiro Hayashibe et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Skeletal muscle system has nonlinear dynamics and subject-specific characteristics Thus it is essential to identify the unknownparameters from noisy biomedical signals to improve the modeling accuracy in neuroprosthetic control The objective of this workis to develop an experimental identificationmethod for subject-specific biomechanical parameters of a physiological muscle modelwhich can be employed to predict the nonlinear force properties of stimulated muscle Our previously proposed muscle modelwhich can describe multiscale physiological system based on the Hill and Huxley models was used for the identification Theidentification protocols were performed on two rabbit experiments where the medial gastrocnemius was attached to a motorizedlever system to record the force by the nerve stimulation The muscle model was identified using nonlinear Kalman filters sigma-point and extended Kalman filter The identified model was evaluated by comparison with experimental measurements in thecross-validation mannerThe feasibility could be demonstrated by comparison between the estimated parameter and the measuredvalue The estimates with SPKF showed 57 and 29 error in each experiment with 7 different initial conditions It reveals thatSPKF has great advantage especially for the identification of multiscale muscle model which accounts for the high nonlinearity anddiscontinuous states between muscle contraction and relaxation process

1 Introduction

11 Muscle Modeling and FES Functional electrical stimula-tion (FES) is an effective technique to evoke artificial con-tractions of paralyzed skeletal muscles It has been employedas a general method in modern rehabilitation to partiallyrestore motor function for patients with upper neural lesions[1 2] Recently the rapid progress in microprocessor tech-nology provided the means for computer-controlled FESsystems [3ndash5] A fundamental problem concerning FES ishow to handle the high complexity and nonlinearity ofthe neuromusculoskeletal system [6 7] In addition thereis a large variety of patient situations depending on thetype of neurological disorder To improve the performanceof motor neuroprosthetics beyond the current limited useof such system subject-specific modeling is essential Theuse of a mathematical model can improve the developmentof neuroprosthetics by optimizing their functionality forindividual patients

A mathematical model makes it possible to describethe relevant characteristics of the patientrsquos skeletal muscleand to accurately predict the force as a function of thestimulation parameters Indeed synthesis of stimulationsequences or control strategies to achieve movement can beefficiently computed and optimized using numerical models[8] Therefore it can contribute to enhancing the design andfunction of controls applied to FES Over the years a greatvariety of muscle models have been proposed differing intheir intended application mathematical complexity levelof structure considered and fidelity to the biological factsSome of them have attempted to exhibit the microscopic ormacroscopic functional behavior for instanceHuxley [9] andHill [10] The distribution-moment model [11] constitutes abridge between the microscopic and macroscopic levels Itis a model for sarcomeres or whole muscle which has beenextracted via a formal mathematical approximation fromHuxley crossbridge models Models integrating geometryof the tendon and other macroscopic consideration can be

2 ISRN Rehabilitation

found in [12] A study by Bestel and Sorine [13] based onboth microscopic Huxley and macroscopic Hill type modelproposed an explanation of how the beating of cardiacmuscle is achieved through a chemical control input Itintegrated the calciumdynamics inmuscle cells that stimulatethe contractile element of the model Starting with thisconcept we adapted it for striated muscle [14] We proposeda musculotendinous model which considered the muscularmasses and viscous frictions in muscle-tendon complexThismodel is represented by differential equations where theoutputs are the musclersquos active stiffness and force The modelinput represents the actual electrical signal as provided by thestimulator in FES

12 In Vivo Identification for Subject-Specific ParametersIn actual FES system the appropriate tuning is achievedempirically by intensively stimulating the patientrsquos musclefor each task If this adjustment could be calculated in thesimulation and if we could find the best signal patternusing virtual skeletal muscle such method would be veryhelpful for movement synthesis for spinal cord injured (SCI)patients However in order to perform the simulation anaccurate skeletal muscle model is required to reproduce awell-predicted force for each muscle corresponding to thepatient-specific characteristics

For any biological systems identification is a difficultproblemdue to the fact that (i)measurementsmust be as non-invasive particularly on humans (ii) some entities cannot bedirectly measured (iii) an experimental setup and protocolhave to be designed and certified (iv) intersubject variationscan be large and (v) the large nonlinearity and complex-ity of the models cause some optimization algorithms tofail Thus few papers address biomechanical parametersidentification in FES context and they used macroscopicmodel for global force production [15] Consequently wedescribed an approach for coupling the model with in-vivomeasurements that is using a multiscale muscle model in anestimation procedure in order to perform the identificationof the parameters hence giving access to physiologicallymeaningful parameters of the muscle Preliminary result wasreported in [16]

The skeletal muscle dynamics are in particular highlynonlinear and we need to identify many unknown phys-iological parameters if a multiscale model is applied Themain objective of this paper is to develop an experimen-tal computational method to identify unknown internalparameters from the limited information For such identi-fication of nonlinear system extended Kalman filter (EKF)has often been used for skeletal muscle [17] and cardiacmuscle identification [18] It can work when the model isnot complicated and not highly nonlinear In this paper asigma-point Kalman filter (SPKF) was applied to the in vivoexperimental data to identify internal states in the nonlineardynamics of multiscale skeletal muscle model SPKF hashigher accuracy and consistency for nonlinear estimationthan EKF theoretically The feasibility of both identificationsis verified by comparisonThe computational performance isdiscussed

The outline of the paper is as follows The next sec-tion presents the formulation of the skeletal muscle modelcontrolled by FES The following section is devoted tothe experimental identification of the model for isometriccontraction including the identification protocolThe experi-mentalmeasurementwas performed in-vivo on rabbitsThenwe present detailed results of the parameter estimation thecomparison with EKF and cross-validation which illustratesthe pertinence of the identification Finally we present somediscussions conclusions and perspectives in the last sections

2 Skeletal Muscle Model

Muscle modeling is complex in particular when the modelis based on biomechanics and physiological realities Mostof the muscle models have been based on phenomenologicalmodels derived from Hillrsquos classic work [10] and well sum-marized by Zajac [12] Hill macroscopic model is a standardmuscle model for practical use Recent work [19] performedthe validation of Hill model during functionally relevantconditions They concluded that model errors are large fordifferent firing frequencies and largest at the low motor unitfiring rates relevant to normal movement They pointed outthat the reason may come from the Hill model assump-tion to consider muscle activation force-length and force-velocity properties independently It was suggested that morephysiological coupling between activation and force-velocityproperties can be demonstrated in microscopic crossbridgemodels incorporating a dependence between physiologicallybased activation and crossbridge attachment [19] Thus ourapproach is to provide a multiscale physiology-based modelon the both micro- andmacroscale fact to obtain meaningfulinternal parameters

Our muscle model is composed of two elements ofdifferent nature (i) an activation model which describes howan electrical stimulus generates an action potential (AP) andinitiates the contraction and (ii) a mechanical model whichdescribes the dynamics in force (Figure 1) For details of themuscle model refer to the previously published article [14]Here a brief summary of the model and the informationnecessary for the identification are given

21 Activation Model The activation model describes theelectrical activity of muscle which represents the excitation-contraction phenomena In muscle physiology it is knownthat two input elements dominate muscle contraction fiberrecruitment rates and temporal activation [20] The recruit-ment rates determine the percentage of recruitedmotor unitsIn Hill model it has only one input which corresponds to thisrecruitment rate Hill model is mainly applied for voluntarycontraction where temporal activation is not dominantHowever in FES since all muscle fibers are synchronouslyactivated this temporal activation which occurs by everystimulation pulse is important The recruitment rates can bedetermined on the pulse width (PW) and pulse amplitude ofsignal 119868 generated by the stimulator [21]The recruitment rate120572(PW 119868) can be assumed to be a constant valuewhenPWand119868 remain constant

ISRN Rehabilitation 3

Muscle model

PW

stimulation signal

Recruitment rateActivation model

Temporal activation

Mechanical model

Forceactive stiffness

strainu

InputOutput

Identification

T = 1f

I

120572

Figure 1 Outline of skeletal muscle model and its identification

The temporal activation can be considered as the under-lying physiological processes which describes the chemicalinput signal 119906 that brings muscle cells into contraction asshown in Figure 2 Muscle contraction is initiated by an APalong the muscle fiber membrane which goes deep into thecell through the T-tubules It causes calcium releases thatinduce the contraction process when its concentration risesabove a threshold and is sustained till the concentration dropsback below this threshold once again Hatze [22] gives anexample of calcium dynamics (Ca2+) modeling Since wefocus on the mechanical response in this paper we choosea simple model that renders the main characteristics of thedynamics The contraction-relaxation cycle is triggered bythe (Ca2+) associated with two phases (i) contraction and(ii) relaxation as in Figure 2 We use a delayed (120591) modelto take into account the propagation time of the AP andan average time delay due to the calcium dynamics Thefrequency of the chemical input 119906 can be defined from thestimulation frequency The time delay and the contractiontime can be obtained from a twitch test by single stimulationpulse A contraction takes place with a kinetics 119880

119888then if

no other AP has been received in the mean time an activerelaxation follows indefinitely with a kinetics 119880

119903 119880119888is linked

to the rate of actine-myosine cycle whereas 119880119903is related to

the rate of crossbridge breakage Finally 119906 can be written asfollowswhereΠ

119888(119905) is a trapezoidal switching functionwhich

connects relaxation and contraction state from 0 to 1

119906 = Π119888(119905) 119880119888+ (1 minus Π

119888(119905)) 119880

119903 (1)

22 Mechanical Model The model is based on the macro-scopicHill-Maxwell typemodel and themicroscopic descrip-tion of Huxley [9] For crossbridge modeling Huxley [9]proposed an explanation of the crossbridge interaction in asarcomere A sarcomere model can be used to represent awhole muscle which is assumed to be a homogeneous assem-bly of identical sarcomeres The distribution-moment modelof Zahalak [11] is a model for sarcomeres or whole musclewhich is extracted via a formal mathematical approximationfrom Huxley crossbridge models This model constitutesa bridge between the microscopic and macroscopic levelsBased on Huxley and Hill-type models Bestel and Sorine[13] proposed an explanation of how the beating of cardiacmuscle may be performed through a chemical control inputconnected to the calcium dynamics in muscle cell thatstimulates the contractile element of the model

Stim PW

Contraction Relaxation Contraction Relaxation

T = 1f[Ca+2]

uUc

Ur

C

120591

I

Figure 2 Chemical control input 119906 for temporal activation

The model is composed of macroscopic passive elementsand a contractile element 119864

119888controlled by input commands

the chemical input 119906 as suggested by Bestel and Sorine[13] for the cardiac muscle on the sarcomere scale and therecruitment rate 120572 on the fiber scale as shown in Figure 3 Inorder to express isometric contractions whereas the skeletonis not actuated our muscle model is introduced with masses119898 (kg) and linear viscous dampers 120582

1199041 1205821199042(Nsm) to ensure

energy dissipation On both sides of 119864119888 there are elastic

springs 1198961199041 1198961199042(Nm) and viscous dampers to express the

viscoelasticity of the muscle-tendon complex The parallelelement119870

119901 mainly represents surrounding tissues but it can

be omitted in isometric contraction mode We assume thesymmetric form with the two masses and passive elementsidentical Here 120576

119904= 1205761199041= 1205761199042 119896119904= 1198961199041= 1198961199042 and 120582 = 120582

1199041=

1205821199042 1198711198880and 119871

1199040(m) are the lengths of 119864

119888and 119896

119904in the rest

state Initially we can define the relative length variation 120576 aspositive when the length increases as in (2) In particular inthe case of isometric contraction the following relationshipexists (3)

120576119904=119871119904minus 1198711199040

1198711199040

120576119888=119871119888minus 1198711198880

1198711198880

(2)

21198711199040120576119904+ 1198711198880120576119888= 0 (3)

The dynamical equation of one of the masses is given by (4)119865119888and 119896119888express the force and the stiffness of119864

119888 respectively

The force 119865119890at the output of this muscle model is the sum


mm

120582s2Ec

Commandu 120572

120582s1

Kp

Ls1 Ls2

Lc

Fc

Fs

Fd

ks2ks1

Figure 3 Macroscopic mechanical configuration of the musclemodel

of the spring force 119865119904and the damping force 119865

119889 When

we measure the tension of skeletal muscle under in vivoconditions the experimentally measured force is equivalentto119865119890Whenwe take the ratio of119865

119888and119865119890 1198711199040is offset and can

be written as in (6) (7) is the relational equation in Laplacetransform From this relationship the differential equation(8) can be obtained

1198981198711199040

120576119904= 119865119888minus 1198961199041198711199040120576119904minus 1205821198711199040

120576119904 (4)

119865119890= 119865119904+ 119865119889= 1198961199041198711199040120576119904+ 1205821198711199040

120576119904 (5)

119865119888

119865119890

=119898 120576119904+ 120582 120576119904+ 119896119904120576119904

120582 120576119904+ 119896119904120576119904

(6)

L [119865119888]

L [119865119890]=1198981199042+ 120582119904 + 119896

119904

120582119904 + 119896119904

(7)

119898119890+ 120582119890+ 119896119904119865119890= 120582119888+ 119896119904119865119888 (8)

Finally in isometric contraction the differential equations ofthis model can be described as follows

119888= minus119896119888119906 + 120572119896

119898Π119888119880119888minus 119896119888

1003816100381610038161003816120576119888

1003816100381610038161003816(9)

119888= minus119865119888119906 + 120572119865

119898Π119888119880119888minus 119865119888

1003816100381610038161003816120576119888

1003816100381610038161003816 + 1198711198880119896119888120576119888 (10)

119890= minus

120582

119898119890minus119896119904

119898119865119890+

120582

119898119888+119896119904

119898119865119888 (11)

120576119888= minus

2119865119888

1198981198711198880

minus119896119904

119898120576119888minus

120582

119898120576119888 (12)

The dynamics of the contractile element correspond to (9)and (10) The terms 119896

119898and 119865

119898are the maximum values

for 119896119888and 119865

119888 respectively From (3) and (4) the differential

equation for 120576119888is obtained as in (12) The internal state vector

of this system should be set as x = [119896119888119865119888119865119890119890120576119888

120576119888]

3 Experimental Identification

In this study we have developed a method to identify theparameters in the mechanical part of a skeletal musclemodel The input controls of the model are set as the staticrecruitment rate 120572 and the chemical control input 119906 from theactivation model These two controls are computed from theFES input signal Note that the identification was performedwith constant FES parameters for pulse width and intensity of

electrical stimulation so that the recruitment rate is constantThe amplitude and pulse width were selected to recruit 90of the maximum muscular force then 120572 can be set as 09for the fiber recruitment In addition the calcium dynamicsin our model induce a time delay and an onoff controlwhich represents contractionrelaxation so that correct dataprocessing can avoid the detailed calcium dynamics Thetrigger for signal 119906 can be calculated from the timing ofthe electrical stimulation 119880

119888and 119880

119903are set as 20 and 15

respectively as in [23]In isometric contraction the differential equations of

skeletal muscle dynamics are straightly given in (9)ndash(12)In this case 119870

119888 119865119888 119865119890 and 120576

119888are unknown time-varying

values and 119898 120582119904 and 119871

1198880are unknown static parameters

to be estimated For the identification of this model it is anonlinear state space model and many state variables arenot measurable Moreover in-vivo experimental data includesome noise Hence we need an efficient recursive filter thatestimates the state of a dynamic system from a series of noisymeasurements

31 Sigma-Point Kalman Filter For this kind of nonlinearidentification extended Kalman filter (EKF) is the well-known standard method In EKF the nonlinear equationshould be linearized to first order with partial derivatives(Jacobian matrix) around a mean of the state The optimalKalman filtering is then applied to the linearized systemWhen the model is highly nonlinear EKF may give particu-larly poor performance and diverge easily In skeletal muscledynamics its state-space is dramatically changed betweenthe contraction and relaxation phases At this time partialderivatives would be incorrect due to the discontinuityTherefore we introduced the sigma-point Kalman filter(SPKF)The initial idea was proposed by Julier and Uhlmann[24] and has been well described by Merwe and Wan [25]SPKF uses a deterministic sampling technique known as theunscented transform to pick a minimal set of sample points(called sigma points) around the mean These sigma pointsare propagated through the true nonlinearity This approachresults in approximations that are accurate to at least secondorder in a Taylor series expansion In contrast EKF results inonly first-order accuracy

Anoutline of the SPKF algorithm is described For detailsthe reader should refer to [25 26] The general Kalmanframework involves estimation of the state of a discrete-timenonlinear dynamic system

x119896+1

= f (x119896 k119896)

y119896= h (x

119896n119896)

(13)

where x119896represents the internal state of the system to be

estimated and y119896is the only observed signal The process

noise k119896drives the dynamic system and the observation

noise is given by n119896 The filter starts by augmenting the state

vector to 119871 dimensions where 119871 is the sum of dimensions inthe original state model noise and measurement noise Thecorresponding covariance matrix is similarly augmented to


an 119871 by 119871matrix In this form the augmented state vector x119886119896

and covariance matrix P119886119896can be defined as

x119886119896= 119864 [x119886

119896] = [x119879

119896k119879119896

n119879119896]119879

P119886119896= 119864 [(x119886

119896minus x119886119896) (x119886119896minus x119886119896)119879

]

= [

[

Px119896 0 0

0 Qk1198960

0 0 Rn119896

]

]

(14)

where Px is the state covariance Qk is the process noisecovariance and Rn is the observation noise covariance

In the process update the 2119871 + 1 sigma points arecomputed based on a square root decomposition of the priorcovariance as in (15) where 120574 = radic119871 + 120582 and 120582 is found using120582 = 120572

2(119871 + 120581) minus 119871 120572 is chosen in 0 lt 120572 lt 1 which determines

the spread of the sigma points around the prior meanand 120581 is usually chosen equal to 0 The augmented sigma-point matrix is formed by the concatenation of the statesigma-point matrix the process noise sigma-point matrixand the measurement noise sigma-point matrix such thatX119886 = [(X119909)

119879(XV

)119879

(X119899)119879]119879

The sigma-point weights tobe used for mean and covariance estimates are defined as in(16) The optimal value of 2 is usually assigned to 120573

X119886

119896minus1= [x119886119896minus1

x119886119896minus1

+ 120574radicP119886119896minus1

x119886119896minus1

minus 120574radicP119886119896minus1

] (15)

120596119898

0=

120582

(119871 + 120582)

120596119888

0= 120596119898

0+ (1 minus 120572

2+ 120573)

120596119888

119894= 120596119898

119894=

1

(2 (119871 + 120582))(for 119894 = 1 2119871)

(16)

And these sigma points are propagated through the nonlinearfunction Predicted mean and covariance are computed asin (18) and (19) and predicted observation is calculated asfollows

X119909

119896|119896minus1= f (X119909

119896minus1X

V119896minus1

) (17)

xminus119896=

2119871

sum

119894=0

120596119898

119894X119909

119894119896|119896minus1 (18)

Pminusx119896 =2119871

sum

119894=0

120596119888

119894(X119909

119894119896|119896minus1minus xminus119896) (X119909

119894119896|119896minus1minus xminus119896)119879

(19)

Y119896|119896minus1

= h (X119909119896|119896minus1

X119899

119896minus1) (20)

yminus119896=

2119871

sum

119894=0

120596119898

119894Y119894119896|119896minus1

(21)

The predictions are then updated with new measurementsby first calculating the measurement covariance and state-measurement cross-correlationmatrices which are then usedto determine the Kalman gain Finally the updated estimate

and covariance are determined from this Kalman gain asbelow

Py119896 =2119871

sum

119894=0

120596119888

119894(Y119894119896|119896minus1

minus yminus119896) (Y119894119896|119896minus1

minus yminus119896)119879

Px119896y119896 =2119871

sum

119894=0

120596119888

119894(X119909

119894119896|119896minus1minus xminus119896) (Y119894119896|119896minus1


K119896= Px119896y119896P

minus1

y119896

x119896= xminus119896+ K119896(y119896minus yminus119896)

Px119896 = Pminusx119896 minus K119896Py119896K119879

119896

(22)

These process updates and measurement updates should berecursively calculated in 119896 = 1 infin until the end point ofthe measurement

32 Experimental Measurement for Identification Stimula-tion experiments were performed on two New Zealandwhite rabbits at Aarhus University Hospital in AalborgDenmark as depicted in Figure 4 Anesthesia was inducedand maintained with periodic intramuscular doses of acocktail of 015mgkg Midazolam (DormicumR AlpharmaAS) 003mgkg Fetanyl and 1mgkg Fluranison (combinedin HypnormR Janssen Pharmaceutica) [14]

The left leg of the rabbit was anchored at the kneeand ankle joints to a fixed mechanical frame using bonepins placed through the distal epiphyses of the femur andtibia A tendon of the medial gastrocnemius (MG) musclewas attached to the arm of a motorized lever system (Dualmode system 310B Aurora Scientific Inc) The position andforce of the lever arm were recorded An initial muscle-tendon length was established by flexing the ankle to 90∘ Abipolar cuff electrode was implanted around the sciatic nerveallowing the MG muscle to be stimulated Data acquisitionwas performed at a 48 kHz sampling rate The muscleforce against the electrical stimulation was measured underisometric conditions

4 Result of Identification

In order to facilitate the convergence of the identificationthe estimation process has been split into two steps In thefirst step we only estimate geometrical parameter 119871

1198880 In the

second step we estimate the dynamic parameters 119898 and 120582The biomechanicalmusclemodel to be identified is presentedas (9)ndash(12) We define the state vectors for geometric anddynamic estimations in SPKF as follows

xg = [119896119888119865119888119865119890119890120576119888

1205761198881198711198880]

xd = [119896119888119865119888119865119890119890120576119888

120576119888119898 120582]

(23)

41 Parameter Estimation Parameter estimation was per-formed using two rabbit experimental data The stimulationsignal input used for the estimation is composed of two


Sciatic nerve

Stimulator

Rabbit

Figure 4 Overview of the rabbit experiment

successive pulses (doublet) at 16Hz and 20Hzwith amplitude105 120583A and pulse width 300 120583s The stiffness 119896

119904has been

estimated separately using the experiments performed onthe isolated muscle The isolated muscle was pulled with themotorized lever The stiffness is taken as equal to the slope ofthe straight line of the passive length versus force relationshipin the experimental measurement as shown in Figure 5 The119896119904obtained was 4500Nm 119865

119898and 119896

119898can be obtained

knowing the force response of muscle to a stimulationpattern with maximum signal parameter values (frequencyamplitude pulse and width) In this case 119896

119898= 1000Nm

and 119865119898= 15N were obtained from the measurement

The experimental muscle force against the doublet stim-ulation (20Hz) was used for parameter estimation duringmeasurement updates for 119865

119890in SPKF The covariance matrix

has been initialized as in Table 1 for both geometric anddynamic parameter identifications The evolutions of esti-mated internal states for 119896

119888and 120576

119888are obtained as shown

in Figure 6 From these behaviors we can confirm that thecontractile element of the model is successfully trackingthe contraction represented by the dynamics of differentialequations under the estimation process Figure 7 shows theestimated parameter 119871

1198880and the error covariance The fact

that the error covariace is being decreased during identifica-tion process shows stable convergence of the estimation Evenwith the different initial values setting parameter estimationresults converged into a particular value in SPKF as shown inFigure 7(a) We tested the estimation from 7 different initialvalues with an interval of 5mm from 55mm to 85mm Thecolor is changed from magenta to cyan in the plot

As can be seen from the resultant computational behavioron the graphs the internal state vectors of the muscle modelconverged well to stationary values even with different initialvalues After the complete estimation process for geometricand dynamic parameters the estimated values are summa-rized in Table 2 Normalized RMS errors between measuredand estimated force were also computed for each case More-over the estimated length of the contractile element witheach stimulation showed values close to the actual measuredlengths of the extracted muscle The border between thecontractile element and tendon is not so clear visually butit was approximately 7 cm for both rabbit GM muscles

Slope = 4500Nm

24

21

18

15

12

9

6

3

Non linear region

00

0001 0002 0003 0004 0005 0006 0007 0008 0009Relative length (L minus L0) (m)

Forc

e (N

)

Toe region

Line

ar reg

ion

Figure 5 Force measurement for muscle passive stiffness

050

100150200250300350

Stiff

ness

(Nm

)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0

002

004

006

0 005 01 015 02 025 03 035 04 045 05 Time (s)

minus012

minus01

minus008

minus006

minus004

minus002

Inte

rnal

stat

e120576c

(b)

Figure 6 Estimated state of the stiffness119870119888(a) and the strain 120576

119888(b)

for the contractile element with SPKF

The sizes of the two rabbits were similar in particular theestimated length of 119871

1198880had good correspondence with the

measured length for both rabbits andwith different frequencystimulations Physical parameters are impossible to makevisual verification but the estimated intrasubject propertyis maintained among the parameters obtained by differentfrequency stimulations from Table 2


002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)

Figure 7 Estimated parameter for the original length of thecontractile element 119871

1198880(a) and its error covariance (b) with SPKF

The plot color is changed from magenta to cyan with 7 differentinitial values from 55mm to 85mm

Table 1 The covariance matrix initialization

Initial covariance matrix (geometric)Px = diag([1 01 01 1 1 10 10

minus2])

Qk = diag([25 01 01 25 25 10 10minus3] times 10

minus4)

Rn = 25 times 10minus3

Initial covariance matrix for (dynamic)Px = diag([1 01 01 1 1 10 10

minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)


Table 2 Parameter estimation

Parameters Rabbit 1 Rabbit 216Hz 20Hz 16Hz 20Hz

1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082

Measured length ofcontractile element 7 cm 7 cm

Body weight 45 kg 42 kg

42 Comparison with the Extended Kalman Filter Theextended Kalman filter is generally chosen for nonlinearsystem identification However in EKF first-order partialderivatives are used for the computation which means thata matrix of partial derivatives (Jacobian) is computed aroundthe estimate for each stepThe detail of EKF is summarized inthe appendix When the process and measurement functionsf and h are highly nonlinear EKF can give particularlypoor performance and diverge easily [25] The estimate canhave a bias due to the linear approximation especially fordiscontinuous systems

Using the same computational conditions such as ini-tial values for states parameters and covariance matricesparameter estimation was also performed with EKF to com-pare the estimation quality for this nonlinear system Theestimation results with EKF are shown in Figures 8 and 9The numerical comparison for SPKF and EKF in 7 differentinitial conditions is summarized in Table 3 both for rabbit1and rabbit2With the observation noise covariance119877 = 25 times

10minus3 the same as with SPKF the result with EKF converged

computationally as shown at the error covariance of EKF1 inthe table However the matched result in force level does notmean directly that the estimated internal state and parameterare correct The internal state 120576

119888in EKF estimation was

not well estimated as the contracted strain variation Thedifference can be observed by comparing the plots of Figures9 and 6 for the EKF and SPKF estimates respectively InFigure 6 the estimated state of 120576

119888has two sharp peaks which

correspond to the strain of the contractile element induced bymuscle contraction by doublet stimulationThus this internalstate reflects the expected phenomenon in muscle dynamicsHowever the estimated state in Figure 9 does not show sharpdeformation induced by stimulation pulse

The linear approximation in the EKF transformationmatrix could be the cause of the lower quality of thestate estimation Finally it resulted in a bias for parameterestimation Thus the converged value for 119871

1198880under these

conditions is not realistic The EKF estimation was alsoperformed with the observation noise covariance 119877 = 25 times

10minus2 giving more uncertainty to the measurement update In

this case the estimated value became more realistic as theSPKF estimate did However the convergence became pooras in Figure 8(b) and it was highly dependent on the initialvalues as in Figure 8(a)

43 Model Cross-Validation A cross-validation of the iden-tified model was carried out to confirm the validity of thismethod The resultant muscle force was calculated using theidentified models with the control input generated by thestimulation frequency Figure 10 shows the measured forceresponse of the MG muscle of the rabbit and the simulatedforce with the identified model The stimulation was threesuccessive pulses in 119868 = 105 120583A PW = 300 120583s and Freq =3125Hz The red line indicates the measured muscle forcethe blue and green dotted lines are the plots by the identifiedmodel with SPKF and EKF1 respectively The model identi-fied by SPKF could predict the nonlinear force properties ofstimulatedmuscle quitewell in this cross-validation Figure 11


Table 3 Quantitative comparison on identification (SPKF versusEKF) in 7 different initial conditions

Rabbit 1 SPKF EKF1 EKF2119875 for 119865

119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75

Deviation 1198711198880(cm) 612 times 10

minus2767 times 10

minus3034

Average errors () 57 336 69Rabbit 2 SPKF EKF1 EKF2119875 for 119865

119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2

Average errors () 29 449 479EKF1 and EKF2 represent EKF estimation with the observation noisecovariance 119877 = 25 times 10minus3 and 119877 = 25 times 10minus2 respectively 119875 representssteady-state error covariance

is the result in the case of stimulation trains with six pulses in20Hz There is a good agreement between the measured andthe predicted force however when muscle fatigue appearsthe experimental force is lower than that from the simulationParticualrly the difference between the measured and theestimated forces at the end can be considered as the errorcoming from modeling without considering muscle fatigueApart from the error coming from muscle fatigue we couldconfirm the feasibility of the FES muscle modeling andthe effectiveness of the identification Normally the muscleproperty varies greatly being dependent on the type ofmuscle and the force response is highly subject-specificTherefore the muscle force prediction by cross-validation isquite difficult even for the approximate prediction especiallywhen force production is predicted for each stimulationpulse The identification based on experimental responsewould contribute strongly to realistic force prediction inelectrical stimulation and FES controller development Fur-ther investigation is required for the expression of fatiguephenomena [27] for its reproducibility and identification

5 Discussion

The skeletal muscle model used in this identification protocolis based on both macroscopic and microscopic physiologywhich is unlike the black-box or other approaches usingsimple Hill muscle model The structured model requiresmore parameters in highly nonlinear dynamics which haveto be estimated by experiment However the great advantageof our model is the insight which it can give to a musclersquosbiomechanical and physiological connections where theparameters have a physical significance such as length andmass This paper describes an identification method whichuses experimental response and a nonlinear physiologicalmuscle model to obtain subject-specific parameters Theadvantage of animal experiment is to have direct access toextracted muscle for confirmation of the obtained parameterafter the experiment

002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

R = 25 times 10minus2


(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)

Figure 8 Estimated parameter for the original length of thecontractile element119871

1198880(a) and its error covariance (b) with EKFThe

plot color is changed from magenta to cyan with 7 different initialvalues from 55mm to 85mmThe results with two different settingsfor the observation noise covariance are shown

002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)

minus006minus004minus002

Iden

tifica

tion

of p

aram

eter

120576 c

Figure 9 Estimated state of the strain of the contractile element 120576119888

with EKF

This work was performed for application of FES stim-ulated muscles however since many living systems havenonlinear dynamics and subject specificity this kind ofidentification approach itself could be applied to other organmodels and clinical situations In this work both SPKFand EKF algorithms were applied for muscle dynamicsidentification EKF showed a high dependency on initialsettings of the computation and it was difficult to find theeffective range of initial states and covariances In SPKFthe obtained result was more consistent and robust withrespect to various initial conditions Moreover for SPKF


0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured

Simulated by identified model with SPKFSimulated by identified model with EKF1

Figure 10 Measured and simulated isometric muscle force by theidentified models with three successive stimulation pulses (in 119868 =

105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Figure 11 Measured (red) and simulated isometric muscle force bythe identifiedmodel (blue) with six successive stimulation pulses (in119868 = 105 120583A PW = 300120583s and Freq = 20Hz)

the computation of a Jacobian is not necessary so it caneasily be applied even to complex dynamics SPKF resultsin approximations that are accurate to at least second orderin Taylor series expansion In contrast EKF results in first-order accuracy Further the identification accuracy is clealyimproved especially for nonlinear systems but the compu-tation cost still remains the same as for EKF Advanced androbust system identification including designing experimen-tal protocols has a very important role to improve the controlissues in neuroprosthetics In addition the main difficultyin understanding human systems is caused by their time-varying properties The systems are not static and changeover time The function of a human being is not alwaysthe same for example muscle fatigue can easily change theexpected force response In order to deal with the time-varying characteristics of a human system robust biosignalprocessing and model-based control which corresponded tononlinearity and time variancewould provide a breakthroughin the development of neuroprosthetics [28]

6 Conclusion

We have developed an experimental identification methodfor subject-specific biomechanical parameters of a skele-tal muscle model which can be employed to predict thenonlinear force properties of stimulated muscle The math-ematical muscle model accounts for the multiscale majoreffects occurring during electrical stimulation Thus theidentificationmethodwas required to deal with the high non-linearlity and discontinous states betweenmuscle contractionand relaxation phase The identified model was evaluatedby comparison with experimental measurements in cross-validation There was a good agreement between measuredand simulated muscle force outputs The result showedthe performance which can contribute to the prediction ofthe nonlinear force of stimulated muscle under FES Thefeasibility of the identification could be demonstrated bycomparison between the estimated parameter and the mea-sured value In this study the identification was performedby sigma-point Kalman filter and extended Kalman filterThe performance was compared and summarized underthe same computational conditions SPKF gives more stableperformance than EKF The internal state in EKF estimationwas not well estimated as the contracted muscle strainIn SPKF estimation keeping the realistic state transitionand independence from initial conditions it could realizeconverged solution for each identification trial

SPKF is a Bayesian estimation algorithm which recur-sively updates the posterior density of the system state asnew observations arrive online This framework can allowus to calculate any optimal estimate of the state using newlyarriving information We believe that the proposed identi-fication method has also the advantage for human muscleidentification while it provides not only better accuracy innonlinear dynamics but also adaptability to time-varyingsystems The preliminary result is reported as in [29]

Appendix

Extended Kalman Filter

The extended Kalman filter [30] extends the scope of Kalmanfilter to nonlinear optimal filtering problems by forming aGaussian approximation of the distribution of state x andmeasurements y using a Taylor series-based transformationIt is based on linear approximations to the transformationmatrix Assuming the same nonlinear system as in (13)EKF approximates a process with nonlinear difference andmeasurement relationships as follows

x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)

Note that the difference between EKF and KF is that thematricesA

119896andC

119896in KF are replaced with Jacobianmatrices

which are partial derivatives around the estimate in EKF


In order to avoid numerical problems it is necessary toproceed the algorithm by using a square root factorizationas a UD decomposition which ensures that the covariancematrix remains a positive definite matrix

References

[1] A Kralj and T Bajd Functional Electrical Stimulation Standingand Walking After Spinal Cord Injury CRC Press Boca RatonFla USA 1989

[2] R Kobetic R J Triolo and E B Marsolais ldquoMuscle selectionand walking performance of multichannel FES systems forambulation in paraplegiardquo IEEE Transactions on RehabilitationEngineering vol 5 no 1 pp 23ndash29 1997

[3] N D N Donaldson T A Perkins and A C M WorleyldquoLumbar root stimulation for restoring leg function stimulatorand measurement of muscle actionsrdquo Artificial Organs vol 21no 3 pp 247ndash249 1997

[4] R Kobetic R J Triolo J P Uhlir et al ldquoImplanted functionalelectrical stimulation system for mobility in paraplegia afollow-up case reportrdquo IEEE Transactions on RehabilitationEngineering vol 7 no 4 pp 390ndash398 1999

[5] DGuiraud T Stieglitz K P Koch J Divoux andP RabischongldquoAn implantable neuroprosthesis for standing and walking inparaplegia 5-year patient follow-uprdquo Journal of Neural Engi-neering vol 3 no 4 pp 268ndash275 2006

[6] W K Durfee ldquoControl of standing and gait using electricalstimulation influence of muscle model complexity on controlstrategyrdquo Progress in Brain Research vol 97 pp 369ndash381 1993

[7] R Riener ldquoModel-based development of neuroprostheses forparaplegic patientsrdquo Philosophical Transactions of the RoyalSociety B vol 354 no 1385 pp 877ndash894 1999

[8] D Popovic and T Sinkjaer Control of Movement for the Phys-ically Disabled Control for Rehabilitation Technology SpingerLondon UK 2000

[9] A F Huxley ldquoMuscle structure and theories of contractionrdquoProgress in Biophysics and Biophysical Chemistry vol 7 pp 255ndash318 1957

[10] A V Hill ldquoThe heat of shortening and the dynamic constants inmusclerdquo Proceedings of the Royal Society of London B vol 126no 843 pp 136ndash195 1938

[11] G I Zahalak ldquoA distribution-moment approximation forkinetic theories of muscular contractionrdquo Mathematical Bio-sciences vol 55 no 1-2 pp 89ndash114 1981

[12] F E Zajac ldquoMuscle and tendon properties models scalingand application to biomechanics and motor controlrdquo CriticalReviews in Biomedical Engineering vol 17 no 4 pp 359ndash4111989

[13] J Bestel and M Sorine ldquoA differential model of musclecontraction and applicationsrdquo in Schloessmann Seminar onMathematical Models in Biology Chemistry and Physics MaxPlank Society Bad Lausick Germany 2000

[14] H El Makssoud D Guiraud P Poignet et al ldquoMultiscale mod-eling of skeletal muscle properties and experimental validationsin isometric conditionsrdquo Biological Cybernetics vol 105 no 2pp 121ndash138 2011

[15] R Riener M Ferrarin E E Pavan and C A Frigo ldquoPatient-driven control of FES-supported standing up and sitting downexperimental resultsrdquo IEEE Transactions on Rehabilitation Engi-neering vol 8 no 4 pp 523ndash529 2000

[16] M Hayashibe P Poignet D Guiraud and H El MakssoudldquoNonlinear identification of skeletal muscle dynamics withsigma-pointKalmanFilter formodel-based FESrdquo inProceedingsof the IEEE International Conference on Robotics and Automa-tion (ICRA rsquo08) pp 2049ndash2054 Pasadena Calif USA May2008

[17] T Schauer N Negard F Previdi et al ldquoOnline identificationand nonlinear control of the electrically stimulated quadricepsmusclerdquo Control Engineering Practice vol 13 no 9 pp 1207ndash1219 2005

[18] O Sayadi andM B Shamsollahi ldquoECGdenoising and compres-sion using a modified extended Kalman filter structurerdquo IEEETransactions on Biomedical Engineering vol 55 no 9 pp 2240ndash2248 2008

[19] E J Perreault C J Heckman and T G Sandercock ldquoHillmuscle model errors during movement are greatest within thephysiologically relevant range ofmotor unit firing ratesrdquo Journalof Biomechanics vol 36 no 2 pp 211ndash218 2003

[20] R Riener and J Quintern ldquoA physiologically based model ofmuscle activation verified by electrical stimulationrdquo Bioelectro-chemistry and Bioenergetics vol 43 no 2 pp 257ndash264 1997

[21] W K Durfee and K E MacLean ldquoMethods for estimatingisometric recruitment curves of electrically stimulated musclerdquoIEEE Transactions on Biomedical Engineering vol 36 no 7 pp654ndash667 1989

[22] H Hatze ldquoA general mycocybernetic control model of skeletalmusclerdquo Biological Cybernetics vol 28 no 3 pp 143ndash157 1978

[23] H El Makssoud Modelisation et identification des musclessquelettiques sous stimulation electrique fonctionnelle [PhDthesis] University of Montpellier II 2005

[24] S J Julier and J K Uhlmann ldquoA new extension of the Kalmanfilter to nonlinear systemsrdquo in Proceedings of the 11th Interna-tional Symposim on Aerospace Defence Sensing Simulation andControls (AeroSence rsquo97) 1997

[25] R Merwe and E Wan ldquoSigma-point Kalman filters for proba-bilistic inference in dynamic state-spacemodelsrdquo in Proceedingsof the Workshop on Advances in Machine Learning MontrealCanada June 2003

[26] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[27] E Rabischong andD Guiraud ldquoDetermination of fatigue in theelectrically stimulated quadriceps muscle and relative effect ofischaemiardquo Journal of Biomedical Engineering vol 15 no 6 pp443ndash450 1993

[28] M Hayashibe Q Zhang D Guiraud and C Fattal ldquoEvokedEMG-based torque prediction under muscle fatigue inimplanted neural stimulationrdquo Journal of Neural Engineeringvol 8 no 6 Article ID 064001 2011

[29] M Hayashibe M Benoussaad D Guiraud P Poignet andC Fattal ldquoNonlinear identification method corresponding tomuscle property variation in FESmdashexperiments in paraplegicpatientsrdquo in Proceedings of the 3rd IEEE RAS and EMBS Interna-tional Conference on Biomedical Robotics and Biomechatronics(BioRob rsquo10) pp 401ndash406 Tokyo Japan September 2010

[30] P Maybeck Stochastic Models Estimation and Control vol 2Academic Press New York NY USA 1982

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


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Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


found in [12] A study by Bestel and Sorine [13] based onboth microscopic Huxley and macroscopic Hill type modelproposed an explanation of how the beating of cardiacmuscle is achieved through a chemical control input Itintegrated the calciumdynamics inmuscle cells that stimulatethe contractile element of the model Starting with thisconcept we adapted it for striated muscle [14] We proposeda musculotendinous model which considered the muscularmasses and viscous frictions in muscle-tendon complexThismodel is represented by differential equations where theoutputs are the musclersquos active stiffness and force The modelinput represents the actual electrical signal as provided by thestimulator in FES

12 In Vivo Identification for Subject-Specific ParametersIn actual FES system the appropriate tuning is achievedempirically by intensively stimulating the patientrsquos musclefor each task If this adjustment could be calculated in thesimulation and if we could find the best signal patternusing virtual skeletal muscle such method would be veryhelpful for movement synthesis for spinal cord injured (SCI)patients However in order to perform the simulation anaccurate skeletal muscle model is required to reproduce awell-predicted force for each muscle corresponding to thepatient-specific characteristics

For any biological systems identification is a difficultproblemdue to the fact that (i)measurementsmust be as non-invasive particularly on humans (ii) some entities cannot bedirectly measured (iii) an experimental setup and protocolhave to be designed and certified (iv) intersubject variationscan be large and (v) the large nonlinearity and complex-ity of the models cause some optimization algorithms tofail Thus few papers address biomechanical parametersidentification in FES context and they used macroscopicmodel for global force production [15] Consequently wedescribed an approach for coupling the model with in-vivomeasurements that is using a multiscale muscle model in anestimation procedure in order to perform the identificationof the parameters hence giving access to physiologicallymeaningful parameters of the muscle Preliminary result wasreported in [16]

The skeletal muscle dynamics are in particular highlynonlinear and we need to identify many unknown phys-iological parameters if a multiscale model is applied Themain objective of this paper is to develop an experimen-tal computational method to identify unknown internalparameters from the limited information For such identi-fication of nonlinear system extended Kalman filter (EKF)has often been used for skeletal muscle [17] and cardiacmuscle identification [18] It can work when the model isnot complicated and not highly nonlinear In this paper asigma-point Kalman filter (SPKF) was applied to the in vivoexperimental data to identify internal states in the nonlineardynamics of multiscale skeletal muscle model SPKF hashigher accuracy and consistency for nonlinear estimationthan EKF theoretically The feasibility of both identificationsis verified by comparisonThe computational performance isdiscussed

The outline of the paper is as follows The next sec-tion presents the formulation of the skeletal muscle modelcontrolled by FES The following section is devoted tothe experimental identification of the model for isometriccontraction including the identification protocolThe experi-mentalmeasurementwas performed in-vivo on rabbitsThenwe present detailed results of the parameter estimation thecomparison with EKF and cross-validation which illustratesthe pertinence of the identification Finally we present somediscussions conclusions and perspectives in the last sections

2 Skeletal Muscle Model

Muscle modeling is complex in particular when the modelis based on biomechanics and physiological realities Mostof the muscle models have been based on phenomenologicalmodels derived from Hillrsquos classic work [10] and well sum-marized by Zajac [12] Hill macroscopic model is a standardmuscle model for practical use Recent work [19] performedthe validation of Hill model during functionally relevantconditions They concluded that model errors are large fordifferent firing frequencies and largest at the low motor unitfiring rates relevant to normal movement They pointed outthat the reason may come from the Hill model assump-tion to consider muscle activation force-length and force-velocity properties independently It was suggested that morephysiological coupling between activation and force-velocityproperties can be demonstrated in microscopic crossbridgemodels incorporating a dependence between physiologicallybased activation and crossbridge attachment [19] Thus ourapproach is to provide a multiscale physiology-based modelon the both micro- andmacroscale fact to obtain meaningfulinternal parameters

Our muscle model is composed of two elements ofdifferent nature (i) an activation model which describes howan electrical stimulus generates an action potential (AP) andinitiates the contraction and (ii) a mechanical model whichdescribes the dynamics in force (Figure 1) For details of themuscle model refer to the previously published article [14]Here a brief summary of the model and the informationnecessary for the identification are given

21 Activation Model The activation model describes theelectrical activity of muscle which represents the excitation-contraction phenomena In muscle physiology it is knownthat two input elements dominate muscle contraction fiberrecruitment rates and temporal activation [20] The recruit-ment rates determine the percentage of recruitedmotor unitsIn Hill model it has only one input which corresponds to thisrecruitment rate Hill model is mainly applied for voluntarycontraction where temporal activation is not dominantHowever in FES since all muscle fibers are synchronouslyactivated this temporal activation which occurs by everystimulation pulse is important The recruitment rates can bedetermined on the pulse width (PW) and pulse amplitude ofsignal 119868 generated by the stimulator [21]The recruitment rate120572(PW 119868) can be assumed to be a constant valuewhenPWand119868 remain constant


Muscle model

PW

stimulation signal


Temporal activation

Mechanical model


strainu

InputOutput

Identification

T = 1f

I

120572



119888then if


119903 119880119888is linked





119906 = Π119888(119905) 119880119888+ (1 minus Π

119888(119905)) 119880

119903 (1)


Stim PW


T = 1f[Ca+2]

uUc

Ur

C

120591

I





1199041 1205821199042(Nsm) to ensure






119904= 1205761199041= 1205761199042 119896119904= 1198961199041= 1198961199042 and 120582 = 120582

1199041=

1205821199042 1198711198880and 119871


119888and 119896

119904in the rest


120576119904=119871119904minus 1198711199040

1198711199040

120576119888=119871119888minus 1198711198880

1198711198880

(2)

21198711199040120576119904+ 1198711198880120576119888= 0 (3)


119888 respectively



mm

120582s2Ec

Commandu 120572

120582s1

Kp

Ls1 Ls2

Lc

Fc

Fs

Fd

ks2ks1



119889 When




1198981198711199040

120576119904= 119865119888minus 1198961199041198711199040120576119904minus 1205821198711199040

120576119904 (4)

119865119890= 119865119904+ 119865119889= 1198961199041198711199040120576119904+ 1205821198711199040

120576119904 (5)

119865119888

119865119890

=119898 120576119904+ 120582 120576119904+ 119896119904120576119904

120582 120576119904+ 119896119904120576119904

(6)

L [119865119888]

L [119865119890]=1198981199042+ 120582119904 + 119896

119904

120582119904 + 119896119904

(7)

119898119890+ 120582119890+ 119896119904119865119890= 120582119888+ 119896119904119865119888 (8)


119888= minus119896119888119906 + 120572119896

119898Π119888119880119888minus 119896119888

1003816100381610038161003816120576119888

1003816100381610038161003816(9)

119888= minus119865119888119906 + 120572119865

119898Π119888119880119888minus 119865119888

1003816100381610038161003816120576119888

1003816100381610038161003816 + 1198711198880119896119888120576119888 (10)

119890= minus

120582

119898119890minus119896119904

119898119865119890+

120582

119898119888+119896119904

119898119865119888 (11)

120576119888= minus

2119865119888

1198981198711198880

minus119896119904

119898120576119888minus

120582

119898120576119888 (12)


119898and 119865


for 119896119888and 119865




120576119888]




119888and 119880




119888 119865119888 119865119890 and 120576


values and 119898 120582119904 and 119871





x119896+1

= f (x119896 k119896)

y119896= h (x

119896n119896)

(13)









x119886119896= 119864 [x119886

119896] = [x119879

119896k119879119896

n119879119896]119879

P119886119896= 119864 [(x119886

119896minus x119886119896) (x119886119896minus x119886119896)119879

]

= [

[

Px119896 0 0

0 Qk1198960

0 0 Rn119896

]

]

(14)





119879(XV

)119879

(X119899)119879]119879


X119886

119896minus1= [x119886119896minus1

x119886119896minus1

+ 120574radicP119886119896minus1

x119886119896minus1


] (15)

120596119898

0=

120582

(119871 + 120582)

120596119888

0= 120596119898

0+ (1 minus 120572

2+ 120573)

120596119888

119894= 120596119898

119894=

1

(2 (119871 + 120582))(for 119894 = 1 2119871)

(16)


X119909

119896|119896minus1= f (X119909

119896minus1X

V119896minus1

) (17)

xminus119896=

2119871

sum

119894=0

120596119898

119894X119909

119894119896|119896minus1 (18)

Pminusx119896 =2119871

sum

119894=0

120596119888

119894(X119909



(19)

Y119896|119896minus1

= h (X119909119896|119896minus1

X119899

119896minus1) (20)

yminus119896=

2119871

sum

119894=0

120596119898

119894Y119894119896|119896minus1

(21)



Py119896 =2119871

sum

119894=0

120596119888

119894(Y119894119896|119896minus1



Px119896y119896 =2119871

sum

119894=0

120596119888

119894(X119909



K119896= Px119896y119896P

minus1

y119896



119896

(22)






1198880 In the


xg = [119896119888119865119888119865119890119890120576119888

1205761198881198711198880]

xd = [119896119888119865119888119865119890119890120576119888

120576119888119898 120582]

(23)



Sciatic nerve

Stimulator

Rabbit



119904has been


119898and 119896



119898= 1000Nm





119888and 120576






Slope = 4500Nm

24

21

18

15

12

9

6

3

Non linear region

00


Forc

e (N

)

Toe region

Line

ar reg

ion


050

100150200250300350

Stiff

ness

(Nm

)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0

002

004

006

0 005 01 015 02 025 03 035 04 045 05 Time (s)

minus012

minus01

minus008

minus006

minus004

minus002

Inte

rnal

stat

e120576c

(b)


119888(b)






002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)






minus2])


minus4)



minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)




1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082












1198880under these








119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Muscle model

PW

stimulation signal


Temporal activation

Mechanical model


strainu

InputOutput

Identification

T = 1f

I

120572



119888then if


119903 119880119888is linked





119906 = Π119888(119905) 119880119888+ (1 minus Π

119888(119905)) 119880

119903 (1)


Stim PW


T = 1f[Ca+2]

uUc

Ur

C

120591

I





1199041 1205821199042(Nsm) to ensure






119904= 1205761199041= 1205761199042 119896119904= 1198961199041= 1198961199042 and 120582 = 120582

1199041=

1205821199042 1198711198880and 119871


119888and 119896

119904in the rest


120576119904=119871119904minus 1198711199040

1198711199040

120576119888=119871119888minus 1198711198880

1198711198880

(2)

21198711199040120576119904+ 1198711198880120576119888= 0 (3)


119888 respectively



mm

120582s2Ec

Commandu 120572

120582s1

Kp

Ls1 Ls2

Lc

Fc

Fs

Fd

ks2ks1



119889 When




1198981198711199040

120576119904= 119865119888minus 1198961199041198711199040120576119904minus 1205821198711199040

120576119904 (4)

119865119890= 119865119904+ 119865119889= 1198961199041198711199040120576119904+ 1205821198711199040

120576119904 (5)

119865119888

119865119890

=119898 120576119904+ 120582 120576119904+ 119896119904120576119904

120582 120576119904+ 119896119904120576119904

(6)

L [119865119888]

L [119865119890]=1198981199042+ 120582119904 + 119896

119904

120582119904 + 119896119904

(7)

119898119890+ 120582119890+ 119896119904119865119890= 120582119888+ 119896119904119865119888 (8)


119888= minus119896119888119906 + 120572119896

119898Π119888119880119888minus 119896119888

1003816100381610038161003816120576119888

1003816100381610038161003816(9)

119888= minus119865119888119906 + 120572119865

119898Π119888119880119888minus 119865119888

1003816100381610038161003816120576119888

1003816100381610038161003816 + 1198711198880119896119888120576119888 (10)

119890= minus

120582

119898119890minus119896119904

119898119865119890+

120582

119898119888+119896119904

119898119865119888 (11)

120576119888= minus

2119865119888

1198981198711198880

minus119896119904

119898120576119888minus

120582

119898120576119888 (12)


119898and 119865


for 119896119888and 119865




120576119888]




119888and 119880




119888 119865119888 119865119890 and 120576


values and 119898 120582119904 and 119871





x119896+1

= f (x119896 k119896)

y119896= h (x

119896n119896)

(13)









x119886119896= 119864 [x119886

119896] = [x119879

119896k119879119896

n119879119896]119879

P119886119896= 119864 [(x119886

119896minus x119886119896) (x119886119896minus x119886119896)119879

]

= [

[

Px119896 0 0

0 Qk1198960

0 0 Rn119896

]

]

(14)





119879(XV

)119879

(X119899)119879]119879


X119886

119896minus1= [x119886119896minus1

x119886119896minus1

+ 120574radicP119886119896minus1

x119886119896minus1


] (15)

120596119898

0=

120582

(119871 + 120582)

120596119888

0= 120596119898

0+ (1 minus 120572

2+ 120573)

120596119888

119894= 120596119898

119894=

1

(2 (119871 + 120582))(for 119894 = 1 2119871)

(16)


X119909

119896|119896minus1= f (X119909

119896minus1X

V119896minus1

) (17)

xminus119896=

2119871

sum

119894=0

120596119898

119894X119909

119894119896|119896minus1 (18)

Pminusx119896 =2119871

sum

119894=0

120596119888

119894(X119909



(19)

Y119896|119896minus1

= h (X119909119896|119896minus1

X119899

119896minus1) (20)

yminus119896=

2119871

sum

119894=0

120596119898

119894Y119894119896|119896minus1

(21)



Py119896 =2119871

sum

119894=0

120596119888

119894(Y119894119896|119896minus1



Px119896y119896 =2119871

sum

119894=0

120596119888

119894(X119909



K119896= Px119896y119896P

minus1

y119896



119896

(22)






1198880 In the


xg = [119896119888119865119888119865119890119890120576119888

1205761198881198711198880]

xd = [119896119888119865119888119865119890119890120576119888

120576119888119898 120582]

(23)



Sciatic nerve

Stimulator

Rabbit



119904has been


119898and 119896



119898= 1000Nm





119888and 120576






Slope = 4500Nm

24

21

18

15

12

9

6

3

Non linear region

00


Forc

e (N

)

Toe region

Line

ar reg

ion


050

100150200250300350

Stiff

ness

(Nm

)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0

002

004

006

0 005 01 015 02 025 03 035 04 045 05 Time (s)

minus012

minus01

minus008

minus006

minus004

minus002

Inte

rnal

stat

e120576c

(b)


119888(b)






002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)






minus2])


minus4)



minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)




1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082












1198880under these








119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















mm

120582s2Ec

Commandu 120572

120582s1

Kp

Ls1 Ls2

Lc

Fc

Fs

Fd

ks2ks1



119889 When




1198981198711199040

120576119904= 119865119888minus 1198961199041198711199040120576119904minus 1205821198711199040

120576119904 (4)

119865119890= 119865119904+ 119865119889= 1198961199041198711199040120576119904+ 1205821198711199040

120576119904 (5)

119865119888

119865119890

=119898 120576119904+ 120582 120576119904+ 119896119904120576119904

120582 120576119904+ 119896119904120576119904

(6)

L [119865119888]

L [119865119890]=1198981199042+ 120582119904 + 119896

119904

120582119904 + 119896119904

(7)

119898119890+ 120582119890+ 119896119904119865119890= 120582119888+ 119896119904119865119888 (8)


119888= minus119896119888119906 + 120572119896

119898Π119888119880119888minus 119896119888

1003816100381610038161003816120576119888

1003816100381610038161003816(9)

119888= minus119865119888119906 + 120572119865

119898Π119888119880119888minus 119865119888

1003816100381610038161003816120576119888

1003816100381610038161003816 + 1198711198880119896119888120576119888 (10)

119890= minus

120582

119898119890minus119896119904

119898119865119890+

120582

119898119888+119896119904

119898119865119888 (11)

120576119888= minus

2119865119888

1198981198711198880

minus119896119904

119898120576119888minus

120582

119898120576119888 (12)


119898and 119865


for 119896119888and 119865




120576119888]




119888and 119880




119888 119865119888 119865119890 and 120576


values and 119898 120582119904 and 119871





x119896+1

= f (x119896 k119896)

y119896= h (x

119896n119896)

(13)









x119886119896= 119864 [x119886

119896] = [x119879

119896k119879119896

n119879119896]119879

P119886119896= 119864 [(x119886

119896minus x119886119896) (x119886119896minus x119886119896)119879

]

= [

[

Px119896 0 0

0 Qk1198960

0 0 Rn119896

]

]

(14)





119879(XV

)119879

(X119899)119879]119879


X119886

119896minus1= [x119886119896minus1

x119886119896minus1

+ 120574radicP119886119896minus1

x119886119896minus1


] (15)

120596119898

0=

120582

(119871 + 120582)

120596119888

0= 120596119898

0+ (1 minus 120572

2+ 120573)

120596119888

119894= 120596119898

119894=

1

(2 (119871 + 120582))(for 119894 = 1 2119871)

(16)


X119909

119896|119896minus1= f (X119909

119896minus1X

V119896minus1

) (17)

xminus119896=

2119871

sum

119894=0

120596119898

119894X119909

119894119896|119896minus1 (18)

Pminusx119896 =2119871

sum

119894=0

120596119888

119894(X119909



(19)

Y119896|119896minus1

= h (X119909119896|119896minus1

X119899

119896minus1) (20)

yminus119896=

2119871

sum

119894=0

120596119898

119894Y119894119896|119896minus1

(21)



Py119896 =2119871

sum

119894=0

120596119888

119894(Y119894119896|119896minus1



Px119896y119896 =2119871

sum

119894=0

120596119888

119894(X119909



K119896= Px119896y119896P

minus1

y119896



119896

(22)






1198880 In the


xg = [119896119888119865119888119865119890119890120576119888

1205761198881198711198880]

xd = [119896119888119865119888119865119890119890120576119888

120576119888119898 120582]

(23)



Sciatic nerve

Stimulator

Rabbit



119904has been


119898and 119896



119898= 1000Nm





119888and 120576






Slope = 4500Nm

24

21

18

15

12

9

6

3

Non linear region

00


Forc

e (N

)

Toe region

Line

ar reg

ion


050

100150200250300350

Stiff

ness

(Nm

)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0

002

004

006

0 005 01 015 02 025 03 035 04 045 05 Time (s)

minus012

minus01

minus008

minus006

minus004

minus002

Inte

rnal

stat

e120576c

(b)


119888(b)






002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)






minus2])


minus4)



minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)




1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082












1198880under these








119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















x119886119896= 119864 [x119886

119896] = [x119879

119896k119879119896

n119879119896]119879

P119886119896= 119864 [(x119886

119896minus x119886119896) (x119886119896minus x119886119896)119879

]

= [

[

Px119896 0 0

0 Qk1198960

0 0 Rn119896

]

]

(14)





119879(XV

)119879

(X119899)119879]119879


X119886

119896minus1= [x119886119896minus1

x119886119896minus1

+ 120574radicP119886119896minus1

x119886119896minus1


] (15)

120596119898

0=

120582

(119871 + 120582)

120596119888

0= 120596119898

0+ (1 minus 120572

2+ 120573)

120596119888

119894= 120596119898

119894=

1

(2 (119871 + 120582))(for 119894 = 1 2119871)

(16)


X119909

119896|119896minus1= f (X119909

119896minus1X

V119896minus1

) (17)

xminus119896=

2119871

sum

119894=0

120596119898

119894X119909

119894119896|119896minus1 (18)

Pminusx119896 =2119871

sum

119894=0

120596119888

119894(X119909



(19)

Y119896|119896minus1

= h (X119909119896|119896minus1

X119899

119896minus1) (20)

yminus119896=

2119871

sum

119894=0

120596119898

119894Y119894119896|119896minus1

(21)



Py119896 =2119871

sum

119894=0

120596119888

119894(Y119894119896|119896minus1



Px119896y119896 =2119871

sum

119894=0

120596119888

119894(X119909



K119896= Px119896y119896P

minus1

y119896



119896

(22)






1198880 In the


xg = [119896119888119865119888119865119890119890120576119888

1205761198881198711198880]

xd = [119896119888119865119888119865119890119890120576119888

120576119888119898 120582]

(23)



Sciatic nerve

Stimulator

Rabbit



119904has been


119898and 119896



119898= 1000Nm





119888and 120576






Slope = 4500Nm

24

21

18

15

12

9

6

3

Non linear region

00


Forc

e (N

)

Toe region

Line

ar reg

ion


050

100150200250300350

Stiff

ness

(Nm

)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0

002

004

006

0 005 01 015 02 025 03 035 04 045 05 Time (s)

minus012

minus01

minus008

minus006

minus004

minus002

Inte

rnal

stat

e120576c

(b)


119888(b)






002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)






minus2])


minus4)



minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)




1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082












1198880under these








119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Sciatic nerve

Stimulator

Rabbit



119904has been


119898and 119896



119898= 1000Nm





119888and 120576






Slope = 4500Nm

24

21

18

15

12

9

6

3

Non linear region

00


Forc

e (N

)

Toe region

Line

ar reg

ion


050

100150200250300350

Stiff

ness

(Nm

)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0

002

004

006

0 005 01 015 02 025 03 035 04 045 05 Time (s)

minus012

minus01

minus008

minus006

minus004

minus002

Inte

rnal

stat

e120576c

(b)


119888(b)






002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)






minus2])


minus4)



minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)




1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082












1198880under these








119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















002

003

004

005

006

007

008

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)

(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0

(b)






minus2])


minus4)



minus410])

Qk = diag([25 01 01 25 25 10 10minus5

10minus3] times 10

minus4)




1198711198880(cm) 71 74 74 68

119898 (g) 165 198 365 398

120582 (Nsm) 197 194 197 192

NRMS () 060 046 102 082












1198880under these








119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119890161 times 10

minus4161 times 10

minus4451 times 10

minus4

119875 for 1198711198880

181 times 10minus4

247 times 10minus4

29 times 10minus3

Mean 1198711198880(cm) 74 47 75


minus2767 times 10

minus3034


119890172 times 10

minus4161 times 10

minus4453 times 10

minus4

119875 for 1198711198880

897 times 10minus4

393 times 10minus4

25 times 10minus3

Mean 1198711198880(cm) 68 39 104


minus2338 times 10

minus2649 times 10

minus2



5 Discussion


002003004005006007008

01101

009

Iden

tifica

tion

of p

aram

eter

Lc0

(m)

0 005 01 015 02 025 03 035 04 045 05 Time (s)



(a)

0 005 01 015 02 025 03 035 04 045 05 0

000100020003000400050006000700080009

001

Time (s)Erro

r cov

aria

nce o

f par

amet

erLc0



(b)




002004006

0

0

005 01 015 02 025 03 035 04 045 05 Time (s)


Iden

tifica

tion

of p

aram

eter

120576 c


with EKF



0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 005 01 015 02 025 03 035 04 0450

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)

Measured



105 120583A PW = 300 120583s and Freq = 3125Hz)

0 01 02 03 04 05 060

1

2

3

4

5

6

7

8

Time (s)

Forc

e (N

)



6 Conclusion



Appendix



x119896+1

asymp f (x119896 0) + A

119896(x119896minus x119896)

y119896asymp h (x

119896 0) + C

119896(x119896minus x119896)

A119896=

120597f120597x

10038161003816100381610038161003816100381610038161003816x=x119896 C

119896=

120597h120597x

10038161003816100381610038161003816100381610038161003816x=x119896

(A1)


119896andC





References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















References




































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Research Article Identification of Skeletal Muscle...

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