Nonlinear identification of skeletal muscle dynamics with Sigma-Point Kalman Filter for model-based FES Mitsuhiro Hayashibe, Philippe Poignet, David Guiraud, Hassan El Makssoud Abstract— A model-based FES would be very helpful for the adaptive movement synthesis of spinal-cord-injured patients. For the fulfillment, we need a precise skeletal muscle model to predict the force of each muscle. Thus, we have to estimate many unknown parameters in the nonlinear muscle system. The identification process is essential for the realistic force predic- tion. We previously proposed a mathematical muscle model of skeletal muscle which describes the complex physiological sys- tem of skeletal muscle based on the macroscopic Hill-Maxwell and microscopic Huxley concepts. It has an original skeletal muscle model to enable consideration for the muscular masses and the viscous frictions caused by the muscle-tendon complex. In this paper, we present an experimental identification method of biomechanical parameters using Sigma-Point Kalman Filter applied to the nonlinear skeletal muscle model. Result of the identification shows its effective performance. The evaluation is provided by comparing the estimated isometric force with experimental data with the stimulation of the rabbit medial gastrocnemius muscle. This approach has the advantage of fast and robust computation, that can be implemented for online application of FES control. I. INTRODUCTION Functional Electrical Stimulation (FES) is well known as an effective technique to evoke artificial contractions of paralyzed skeletal muscles. It has been employed as a general method in modern rehabilitation medicine to partially restore motor function for the patients with upper neural lesions [1], [2]. Recently, the rapid progress in microprocessor technology provided the means for computer-controlled FES systems [3], [4], [5], which enable flexible programming of stimulation sequences. A fundamental problem concerning FES is to handle the high complexity and nonlinearity of the neuro-musculo-skeletal system [6], [7]. Moreover, effect such as muscle fatigue, spasticity, and limited force in the stimulated muscle complicate the control task further. The use of mathematical model would improve the development of neuroprosthetics by using optimized operation for individ- ual patients. A mathematical model may enable to describe the relevant characteristics of the patient’s skeletal muscle and predict the precise force against certain stimulation. Therefore it can enhance the design and functions of control strategies applied to FES. Until now, a great variety of muscle models has been proposed over the years, differing in the intended application, mathematical complexity, level of structure considered, and fidelity to the biological facts. Some of them have been attempted to exhibit the microscopic or macroscopic functional behavior like Huxley [8] and Hill The authors are with INRIA Sophia-Antipolis -DEMAR Project and LIRMM, UMR5506 CNRS UM2, 161 Rue Ada - 34392 Montpellier Cedex 5, France hayashibe,guiraud,poignet@lirmm.fr [9]. The distribution-moment model [10] constitutes a bridge between the microscopic and macroscopic levels. It is a model for sarcomeres or whole muscle which is extracted via a formal mathematical approximation from Huxley cross- bridge models. Models integrating geometry of the tendon and other macroscopic consideration can be found in [11]. A study, based on Huxley and Hill-Maxwell type model by Bestel-Sorine [12], proposed an explanation of how the beating of cardiac muscle may be performed through a chemical control input. It was connected to the calcium dy- namics in muscle cell that stimulates the contractile element of the model. Starting with this concept, we adapted it to the striated muscle [13]. We proposed a musculotendinous model considering the muscular masses and viscous frictions in muscle-tendon complex. This model is represented by differential equations where the outputs are the muscle active stiffness and force. The model input represents the actual electrical signal as provided by the stimulator in FES. Under general FES, you have to make detailed empirical tuning by actually stimulating the patient’s muscle for each task. If this adjustment can be calculated in the simulation, and if we can find best signal pattern using virtual skeletal muscle, it would be very helpful for the movement synthesis for paraplegic patient. However, to perform this simulation, a precise skeletal muscle model is required to produce the well-predicted force of each muscle. The skeletal muscle dynamics are highly nonlinear, and we have to identify many unknown physiological and biomechanical parameters. The principal objective of this study is then to develop an exper- imental identification method to identify unknown internal parameters from the limited information. This process is essential for realistic force prediction in the skeletal muscle modeling for FES. For the parameter estimation in our mus- cle model, the force information corresponding to isometric contractions was used along with the electrical input. Sigma- Point Kalman Filter (SPKF) was applied to the in-vivo rabbit experimental data to identify internal states in the nonlinear dynamics of skeletal muscle. SPKF has higher accuracy and consistency for nonlinear estimation than Extended Kalman Filter (EKF). The identification protocol and the detailed results are described to show the feasibility of our approach and the quality of the identification. II. SKELETAL MUSCLE MODEL Our approach is to provide a knowledge model based on the physiological reality to obtain meaningful internal parameters. Basically, our muscle model is composed of two elements in different nature: i) activation model which 2008 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-23, 2008 978-1-4244-1647-9/08/$25.00 ©2008 IEEE. 2049