Robust switched control design for electrically stimulated lower limbs: A linear model analysis in healthy and spinal cord injured subjects

Abstract

Functional electrical stimulation (FES) has been used to restore and aid motor functions in paraplegics, promoting better therapeutic results for its users. From experimental results, one can observe that there exists an uncertain term added to the control signal for a given operating point, because of the plant uncertainties. An experimental setup is presented to identify a linear model containing uncertainties. Then, robust single-gain controllers and suitable switched controllers are designed for compensating the uncertain term added to the control signal. Open-loop technique, robust single-gain and robust switched controllers are numerically compared. The experimental results show the regulation for five healthy and four paraplegic individuals. The successful run time when the robust switched control is used along with a smooth switching signal is higher than those of other studies presented in the literature. In addition, the results indicate that compared with the robust single-gain controller, the robust switched controller minimizes the influence of the parametric uncertainties, returns the smallest time-derivative value of the Lyapunov function, and presents higher feasibility and lower gain norm to control the system.

Introduction

Functional electrical stimulation (FES) is a very important technique for paralyzed limbs rehabilitation and to minimize muscular spasticity in paraplegic individuals.

Different researchers have attempted to overcome the challenges of applying electrical stimuli to paralyzed limbs. Among the difficulties, the use of FES has a great limitation due to muscular fatigue. Artificially activated muscles fatigue at a faster rate than those naturally activated by the physiological system. Several closed-loop control attempts have been developed to minimize these problems.

Functional electrical stimulation approaches using closed-loop control have been employed in the last decades. The state-of-the-art controllers include proportional–integral–derivative (PID) (Abbas and Chizeck, 1991, Ferrarin et al., 2001), fuzzy logic (Ibrahim et al., 2011), sliding mode (Ajoudani & Erfanian, 2009), neural network (Abbas and Chizeck, 1995, Gwo-Ching Chang et al., 1997, Wang et al., 2012), adaptive (Ajoudani and Erfanian, 2009, Ferrarin et al., 2001), and robust controllers (Ajoudani and Erfanian, 2009, Santos et al., 2015, Sharma et al., 2009). In addition, other strategies have also been used to minimize fatigue, such as closed-loop control coupled with electrode switching strategy (Downey et al., 2017).

The understanding of electrical stimulation experimental results for a longer time along with the development of modern control techniques will allow FES to benefit a greater number of spinal cord-injured individuals and different applications such as walking, FES cycling (Azevedo Coste & Wolf, 2018), sitting pivot transfer (Lopes et al., 2016), and sit to stand (Jovic et al., 2016).

In the literature, some controllers results have been obtained from simulations (Gaino et al., 2017, Santos et al., 2015, Schauer et al., 2005, Yang and de Queiroz, 2018). Some other studies have presented experimental validation results, but the number of individuals is small and the volunteers are generally healthy (Kirsch et al., 2017, Sharma et al., 2017). In addition, when a step-type input signal is applied, the controller performance is adequate for a time interval not more than $25$ s (Kirsch et al., 2017). The challenge is to maintain a satisfactory step-type signal regulation for a longer interval using only electrical stimulus applied to the muscle.

In this work new techniques of robust control for electrically stimulated lower limbs are proposed. An experimental procedure is presented to identify the polytopic uncertainties of a dynamic model of the lower limb movement provided by electrical stimuli applied to the quadriceps. Robust single-gain controller and robust switched controller are validated experimentally. The designs of the controllers are based on linear matrix inequalities (LMIs). For the first time, a robust switched controller is applied in lower limb electrical stimulation. Experimental results are obtained from healthy individuals ($N_h=5$) and paraplegic individuals ($N_p=4$). Due to plant parametric uncertainties (Ajoudani and Erfanian, 2009, Kirsch et al., 2017, Nunes, 2019, Sharma et al., 2017), such as system parameter variations, and unmodeled dynamics (e.g., activation dynamics, nonlinear muscle fiber recruitment, muscle spasms, muscle fatigue, and multiplicative torque–angle and torque–velocity scaling factors), there exists an uncertain term added to the control signal, for a given operating point. Thus, this study contributes to improve the solution of this problem, by adding a ${\gamma }_{\xi }$ switching law to compensate this uncertain term added to the control signal. This strategy is adopted in the robust single-gain and robust switched controllers, and the performance indexes root-mean-square error (RMSE) and successful run time (SRT) are compared to those of other studies. The successful run time when the proposed robust switched control is used along with a smooth switching signal is higher than those of other studies presented in the literature. In addition, the results indicate that compared with the robust single-gain controller, the robust switched controller minimizes the influence of the parametric uncertainties, returns the smallest time-derivative value of the Lyapunov function, and presents higher feasibility and lower gain norm to control the system.

This article is organized as follows: Section 2 presents the dynamic model in the state space considering polytopic uncertainties. Section 3 describes the identification procedure for polytopic uncertainties. Section 4 shows the robust single-gain and robust switched control design for electrically stimulated lower limb. Then, the experimental results are presented and discussed in Section 5. Finally, in Section 6 some concluding remarks are reported.