Identification of a Modified Wiener-Hammerstein System and Its Application in Electrically Stimulated Paralyzed Skeletal Muscle Modeling

Modeling Ankle Torque and Stiffness Induced by Functional Electrical Stimulation

Functional electrical stimulation (FES) is commonly used for individuals with neuromuscular impairments to generate muscle contractions. Both joint torque and stiffness play important roles in maintaining stable posture and resisting external disturbance. However, most previous studies only focused on the modulation of joint torque using FES while ignoring the joint stiffness. A model that can simultaneously modulate both ankle torque and stiffness induced by FES was investigated in this study. This model was composed of four subparts including an FES-to-activation model, a musculoskeletal geometry model, a Hill-based muscle-tendon model, and a joint stiffness model. The model was calibrated by the maximum voluntary contraction test of the tibialis anterior (TA) and gastrocnemius medial (GAS) muscles. To validate the model, the estimated torque and stiffness by the model were compared with the measured torque and stiffness induced by FES, respectively. The results showed that the proposed model can estimate torque and stiffness with electrically stimulated TA or/and GAS, which was significantly correlated to the measured torque and stiffness. The proposed model can modulate both joint torque and stiffness induced by FES in the isometric condition, which can be potentially extended to modulate the joint torque and stiffness during FES-assisted walking.

Two-Stage Shape Memory Alloy Identification Based on the Hammerstein-Wiener Model

Thanks to characteristics, such as high force and light weight, a good biocompatibility, noiseless operation and simplicity, and relatively low-cost compared with other conventional actuators, actuators based on shape memory alloy are currently one of the most interesting research topics. They have been introduced in applications such robotics, medicine, automation, and so on. For a good actuator integration of these types of applications, proper control is needed, which seems to be a difficult task due to the hysteresis, dilatory response, and non-linear behavior. This work presents a new form of modeling of this type of actuator based on Hammerstein-Wiener model. This has been identified in two stages of the operation. When the activation temperature for the actuator is obtained by the Joule effect, electrically energy is transformed into thermal energy. In the second stage, the thermal energy is transformed into mechanical work. To fulfill this objective, experimental data [e.g., the input signal (pulse-width modulation), temperature signal, and position signal] from the two stages was obtained for a specific shape memory alloy wire and for specific environmental conditions. This data was used in the modeling process. The final model consists of a combination of the models from the two stages, which represent the behavior of the shape memory alloy actuator where the input signal is the pulse-width modulation signal and the output signal are the position of the actuator. Our results indicate that our model has a very similar response to the behavior of the real actuator. This model can be used to tune different control algorithms, simulate the entry system before manufacture and test on real devices.

Identification of nonlinear Wiener-Hammerstein systems by a novel adaptive algorithm based on cost function framework

This paper investigates parameter identification of nonlinear Wiener-Hammerstein systems by using filter gain and novel cost function. Taking into account the system information is corrupted by noise, the filter gain is exploited to extract the system data. By using several auxiliary filtered variables, an extended estimation error vector is developed. Then, based on the discount term of the extended estimation error and the penalty term on the initial estimate, a novel cost function is developed to obtain the optimal parameter adaptive law. Compared with the conventional cost function which is composed of the square sum of output error, the proposed algorithm based on the cost function of this paper can provide faster convergence rate and higher estimation accuracy. Furthermore, the convergence analysis of the proposed scheme indicates that the parameter estimation error can converge to zero. The effectiveness and practicality of the proposed scheme are validated through the simulation example and experiment on the turntable servo system.

Modeling the Nonlinear Cortical Response in EEG Evoked by Wrist Joint Manipulation

Joint manipulation elicits a response from the sensors in the periphery which, via the spinal cord, arrives in the cortex. The average evoked cortical response recorded using electroencephalography was shown to be highly nonlinear; a linear model can only explain 10% of the variance of the evoked response, and over 80% of the response is generated by nonlinear behavior. The goal of this paper is to obtain a nonparametric nonlinear dynamic model, which can consistently explain the recorded cortical response requiring little a priori assumptions about model structure. Wrist joint manipulation was applied in ten healthy participants during which their cortical activity was recorded and modeled using a truncated Volterra series. The obtained models could explain 46% of the variance of the evoked cortical response, thereby demonstrating the relevance of nonlinear modeling. The high similarity of the obtained models across participants indicates that the models reveal common characteristics of the underlying system. The models show predominantly high-pass behavior, which suggests that velocity-related information originating from the muscle spindles governs the cortical response. In conclusion, the nonlinear modeling approach using a truncated Volterra series with regularization, provides a quantitative way of investigating the sensorimotor system, offering insight into the underlying physiology.

Inverting Monotonic Nonlinearities by Entropy Maximization

This paper proposes a new method for blind inversion of a monotonic nonlinear map applied to a sum of random variables. Such kinds of mixtures of random variables are found in source separation and Wiener system inversion problems, for example. The importance of our proposed method is based on the fact that it permits to decouple the estimation of the nonlinear part (nonlinear compensation) from the estimation of the linear one (source separation matrix or deconvolution filter), which can be solved by applying any convenient linear algorithm. Our new nonlinear compensation algorithm, the MaxEnt algorithm, generalizes the idea of Gaussianization of the observation by maximizing its entropy instead. We developed two versions of our algorithm based either in a polynomial or a neural network parameterization of the nonlinear function. We provide a sufficient condition on the nonlinear function and the probability distribution that gives a guarantee for the MaxEnt method to succeed compensating the distortion. Through an extensive set of simulations, MaxEnt is compared with existing algorithms for blind approximation of nonlinear maps. Experiments show that MaxEnt is able to successfully compensate monotonic distortions outperforming other methods in terms of the obtained Signal to Noise Ratio in many important cases, for example when the number of variables in a mixture is small. Besides its ability for compensating nonlinearities, MaxEnt is very robust, i.e. showing small variability in the results.

Identification of ankle plantar-flexors dynamics in response to electrical stimulation

Modeling the muscle response to functional electrical stimulation (FES) is an essential step in the design of closed-loop controlled neuroprostheses. This study was aimed at identifying the dynamic response of ankle plantar-flexors to FES during quiet standing. Thirteen healthy subjects stood in a standing frame that locked the knee and hip joints. The ankle plantar-flexors were stimulated bilaterally through surface electrodes and the generated ankle torque was measured. The pulse amplitude was sinusoidally modulated at five different frequencies. The pulse amplitude and the measured ankle torque fitted by a sine function were considered as input and output, respectively. First-order and critically-damped second-order linear models were fitted to the experimental data. Both models fitted similarly well to the experimental data. The coefficient of variation of the time constant among subjects was smaller in the case of the second-order model compared to the first-order model (18.1%vs. 79.9%, p<0.001). We concluded that the critically-damped second-order model more consistently described the relationship between isometric ankle torque and surface FES pulse amplitude, which was applied to the ankle plantar-flexors during quiet standing.

Fatigue and non-fatigue mathematical muscle models during functional electrical stimulation of paralyzed muscle

Electrical muscle stimulation demonstrates potential for preventing muscle atrophy and for restoring functional movement after spinal cord injury (SCI). Control systems used to optimize delivery of electrical stimulation protocols depend upon the algorithms generated using computational models of paralyzed muscle force output. The Hill-Huxley-type model, while being highly accurate, is also very complex, making it difficult for real-time implementation. In this paper, we propose a Wiener-Hammerstein system to model the paralyzed skeletal muscle under electrical stimulus conditions. The proposed model has substantial advantages in identification algorithm analysis and implementation including computational complexity and convergence, which enable it to be used in real-time model implementation. Experimental data sets from the soleus muscles of fourteen subjects with SCI were collected and tested. The simulation results show that the proposed model outperforms the Hill-Huxley-type model not only in peak force prediction, but also in fitting performance for force output of each individual stimulation train.