Skeletal muscle system has nonlinear dynamics and subjectspecific characteristics. Thus, it is essential to identify the unknown parameters from noisy biomedical signals to improve the modeling accuracy in neuroprosthetic control. The objective of this work is to develop an experimental identification method for subjectspecific biomechanical parameters of a physiological muscle model which can be employed to predict the nonlinear force properties of stimulated muscle. Our previously proposed muscle model, which can describe multiscale physiological system based on the Hill and Huxley models, was used for the identification. The identification protocols were performed on two rabbit experiments, where the medial gastrocnemius was attached to a motorized lever system to record the force by the nerve stimulation. The muscle model was identified using nonlinear Kalman filters: sigmapoint and extended Kalman filter. The identified model was evaluated by comparison with experimental measurements in the crossvalidation manner. The feasibility could be demonstrated by comparison between the estimated parameter and the measured value. The estimates with SPKF showed 5.7% and 2.9% error in each experiment with 7 different initial conditions. It reveals that SPKF has great advantage especially for the identification of multiscale muscle model which accounts for the high nonlinearity and discontinuous states between muscle contraction and relaxation process.
Functional electrical stimulation (FES) is an effective technique to evoke artificial contractions of paralyzed skeletal muscles. It has been employed as a general method in modern rehabilitation to partially restore motor function for patients with upper neural lesions [
A mathematical model makes it possible to describe the relevant characteristics of the patient’s skeletal muscle and to accurately predict the force as a function of the stimulation parameters. Indeed, synthesis of stimulation sequences or control strategies to achieve movement can be efficiently computed and optimized using numerical models [
In actual FES system, the appropriate tuning is achieved empirically by intensively stimulating the patient’s muscle for each task. If this adjustment could be calculated in the simulation, and if we could find the best signal pattern using virtual skeletal muscle, such method would be very helpful for movement synthesis for spinal cord injured (SCI) patients. However, in order to perform the simulation, an accurate skeletal muscle model is required to reproduce a wellpredicted force for each muscle corresponding to the patientspecific characteristics.
For any biological systems, identification is a difficult problem due to the fact that (i) measurements must be as noninvasive, particularly on humans, (ii) some entities cannot be directly measured, (iii) an experimental setup and protocol have to be designed and certified, (iv) intersubject variations can be large, and (v) the large nonlinearity and complexity of the models cause some optimization algorithms to fail. Thus, few papers address biomechanical parameters identification in FES context, and they used macroscopic model for global force production [
The skeletal muscle dynamics are in particular highly nonlinear, and we need to identify many unknown physiological parameters if a multiscale model is applied. The main objective of this paper is to develop an experimental computational method to identify unknown internal parameters from the limited information. For such identification of nonlinear system, extended Kalman filter (EKF) has often been used for skeletal muscle [
The outline of the paper is as follows. The next section presents the formulation of the skeletal muscle model controlled by FES. The following section is devoted to the experimental identification of the model for isometric contraction, including the identification protocol. The experimental measurement was performed
Muscle modeling is complex, in particular when the model is based on biomechanics and physiological realities. Most of the muscle models have been based on phenomenological models derived from Hill’s classic work [
Our muscle model is composed of two elements of different nature: (i) an activation model which describes how an electrical stimulus generates an action potential (AP) and initiates the contraction and (ii) a mechanical model which describes the dynamics in force (Figure
Outline of skeletal muscle model and its identification.
The activation model describes the electrical activity of muscle which represents the excitationcontraction phenomena. In muscle physiology, it is known that two input elements dominate muscle contraction: fiber recruitment rates and temporal activation [
The temporal activation can be considered as the underlying physiological processes which describes the chemical input signal,
Chemical control input
The model is based on the macroscopic HillMaxwell type model and the microscopic description of Huxley [
The model is composed of macroscopic passive elements and a contractile element
Macroscopic mechanical configuration of the muscle model.
In this study, we have developed a method to identify the parameters in the mechanical part of a skeletal muscle model. The input controls of the model are set as the static recruitment rate
In isometric contraction, the differential equations of skeletal muscle dynamics are straightly given in (
For this kind of nonlinear identification, extended Kalman filter (EKF) is the wellknown standard method. In EKF, the nonlinear equation should be linearized to first order with partial derivatives (Jacobian matrix) around a mean of the state. The optimal Kalman filtering is then applied to the linearized system. When the model is highly nonlinear, EKF may give particularly poor performance and diverge easily. In skeletal muscle dynamics, its statespace is dramatically changed between the contraction and relaxation phases. At this time, partial derivatives would be incorrect due to the discontinuity. Therefore, we introduced the sigmapoint Kalman filter (SPKF). The initial idea was proposed by Julier and Uhlmann [
An outline of the SPKF algorithm is described. For details, the reader should refer to [
In the process update, the
Stimulation experiments were performed on two New Zealand white rabbits at Aarhus University Hospital in Aalborg, Denmark, as depicted in Figure
Overview of the rabbit experiment.
The left leg of the rabbit was anchored at the knee and ankle joints to a fixed mechanical frame using bone pins placed through the distal epiphyses of the femur and tibia. A tendon of the medial gastrocnemius (MG) muscle was attached to the arm of a motorized lever system (Dual mode system 310B Aurora Scientific Inc.). The position and force of the lever arm were recorded. An initial muscletendon length was established by flexing the ankle to 90°. A bipolar cuff electrode was implanted around the sciatic nerve, allowing the MG muscle to be stimulated. Data acquisition was performed at a 4.8 kHz sampling rate. The muscle force against the electrical stimulation was measured under isometric conditions.
In order to facilitate the convergence of the identification, the estimation process has been split into two steps. In the first step, we only estimate geometrical parameter
Parameter estimation was performed using two rabbit experimental data. The stimulation signal input used for the estimation is composed of two successive pulses (doublet) at 16 Hz and 20 Hz with amplitude 105
Force measurement for muscle passive stiffness.
The experimental muscle force against the doublet stimulation (20 Hz) was used for parameter estimation during measurement updates for
The covariance matrix initialization.
Initial covariance matrix (geometric)  







 
Initial covariance matrix for (dynamic)  
 





Estimated state of the stiffness
Estimated parameter for the original length of the contractile element
As can be seen from the resultant computational behavior on the graphs, the internal state vectors of the muscle model converged well to stationary values even with different initial values. After the complete estimation process for geometric and dynamic parameters, the estimated values are summarized in Table
Parameter estimation.
Parameters  Rabbit 1  Rabbit 2  

16 Hz  20 Hz  16 Hz  20 Hz  




 




 




 




 
 
Measured length of contractile element  7 cm  7 cm  
Body weight  4.5 kg  4.2 kg 
The extended Kalman filter is generally chosen for nonlinear system identification. However, in EKF, firstorder partial derivatives are used for the computation, which means that a matrix of partial derivatives (Jacobian) is computed around the estimate for each step. The detail of EKF is summarized in the appendix. When the process and measurement functions
Using the same computational conditions, such as initial values for states, parameters, and covariance matrices, parameter estimation was also performed with EKF to compare the estimation quality for this nonlinear system. The estimation results with EKF are shown in Figures
Quantitative comparison on identification (SPKF versus EKF) in 7 different initial conditions.
Rabbit 1  SPKF  EKF1  EKF2 









Mean 
7.4  4.7  7.5 
Deviation 



Average errors 
5.7  33.6  6.9 
 
Rabbit 2  SPKF  EKF1  EKF2 
 








Mean 
6.8  3.9  10.4 
Deviation 



Average errors 
2.9  44.9  47.9 
EKF1 and EKF2 represent EKF estimation with the observation noise covariance
Estimated parameter for the original length of the contractile element
Estimated state of the strain of the contractile element
The linear approximation in the EKF transformation matrix could be the cause of the lower quality of the state estimation. Finally, it resulted in a bias for parameter estimation. Thus, the converged value for
A crossvalidation of the identified model was carried out to confirm the validity of this method. The resultant muscle force was calculated using the identified models with the control input generated by the stimulation frequency. Figure
Measured and simulated isometric muscle force by the identified models with three successive stimulation pulses (in
Measured (red) and simulated isometric muscle force by the identified model (blue) with six successive stimulation pulses (in
The skeletal muscle model used in this identification protocol is based on both macroscopic and microscopic physiology which is unlike the blackbox or other approaches using simple Hill muscle model. The structured model requires more parameters in highly nonlinear dynamics, which have to be estimated by experiment. However, the great advantage of our model is the insight which it can give to a muscle’s biomechanical and physiological connections, where the parameters have a physical significance such as length and mass. This paper describes an identification method which uses experimental response and a nonlinear, physiological muscle model to obtain subjectspecific parameters. The advantage of animal experiment is to have direct access to extracted muscle for confirmation of the obtained parameter after the experiment.
This work was performed for application of FES stimulated muscles; however, since many living systems have nonlinear dynamics and subject specificity, this kind of identification approach itself could be applied to other organ models and clinical situations. In this work, both SPKF and EKF algorithms were applied for muscle dynamics identification. EKF showed a high dependency on initial settings of the computation and it was difficult to find the effective range of initial states and covariances. In SPKF, the obtained result was more consistent and robust with respect to various initial conditions. Moreover, for SPKF, the computation of a Jacobian is not necessary, so it can easily be applied even to complex dynamics. SPKF results in approximations that are accurate to at least second order in Taylor series expansion. In contrast, EKF results in firstorder accuracy. Further, the identification accuracy is clealy improved, especially for nonlinear systems, but the computation cost still remains the same as for EKF. Advanced and robust system identification, including designing experimental protocols, has a very important role to improve the control issues in neuroprosthetics. In addition, the main difficulty in understanding human systems is caused by their timevarying properties. The systems are not static and change over time. The function of a human being is not always the same; for example, muscle fatigue can easily change the expected force response. In order to deal with the timevarying characteristics of a human system, robust biosignal processing and modelbased control which corresponded to nonlinearity and time variance would provide a breakthrough in the development of neuroprosthetics [
We have developed an experimental identification method for subjectspecific biomechanical parameters of a skeletal muscle model which can be employed to predict the nonlinear force properties of stimulated muscle. The mathematical muscle model accounts for the multiscale major effects occurring during electrical stimulation. Thus, the identification method was required to deal with the high nonlinearlity and discontinous states between muscle contraction and relaxation phase. The identified model was evaluated by comparison with experimental measurements in crossvalidation. There was a good agreement between measured and simulated muscle force outputs. The result showed the performance which can contribute to the prediction of the nonlinear force of stimulated muscle under FES. The feasibility of the identification could be demonstrated by comparison between the estimated parameter and the measured value. In this study, the identification was performed by sigmapoint Kalman filter and extended Kalman filter. The performance was compared and summarized under the same computational conditions. SPKF gives more stable performance than EKF. The internal state in EKF estimation was not well estimated as the contracted muscle strain. In SPKF estimation, keeping the realistic state transition and independence from initial conditions, it could realize converged solution for each identification trial.
SPKF is a Bayesian estimation algorithm which recursively updates the posterior density of the system state as new observations arrive online. This framework can allow us to calculate any optimal estimate of the state using newly arriving information. We believe that the proposed identification method has also the advantage for human muscle identification while it provides not only better accuracy in nonlinear dynamics but also adaptability to timevarying systems. The preliminary result is reported as in [
The extended Kalman filter [