# FES Control of a Finger MP Joint with a Proxy-Based Super-Twisting Algorithm
## 1 Introduction  
The functional electrical stimulation (FES) is a type of neuromuscular electrical stimulation that induces muscle contraction by passing an electric current through motor nerves. FES is clinically applied to assist in the rehabilitation training of hemiplegic patients due to strokes. After a stroke, patients often face upper limb impairments like muscle weakness and loss of coordination. Functional Electrical Stimulation (FES) can assist in restoring these functions by stimulating the neuromuscular system electrically [1]. In recent years, it has become popular as an interface for Augmented Reality (AR) [2] and Brain Computer Interface (BCI) systems [3]. Most FES rehabilitation systems use open-loop control schemes, limiting them to simple movements only. However, closed-loop feedback control with joint angle measurement is necessary to achieve complex and accurate movements such as the extension-flexion symmetrical exercise, which effectively reconstructs motor functions [4; 5]. There are numerous closed-loop feedback control strategies available for FES control in the literature, but when applied to the case of low-computation such as Arduino boards, each one presents unique challenges. The backstepping control method [6], the model predictive control (MPC) [7] and the control law based on the recurrent neural network (RNN) [8] were also proposed for the FES control. However, they require an accurate model of muscle or large amount of data, which are difficult to obtain or difficult to implement in embedded systems. In recent years, due to the robustness to disturbances and model uncertainties, the sliding mode control (SMC) has been widely employed for the control of joint angles through the FES [8; 9; 10; 11]. These SMC methods have the robustness to disturbances and finite-time convergence rate, but also fall into the trouble of numerical chattering problem, which deteriorates the control accuracy [12]. The higher-order SMC (HOSMC) attenuate the magnitude of chattering, but it requires the model of muscle andthe knowledge of the upper bound of the uncertainties and disturbances. It also requires the unknown disturbance to be differentiable, which limits the scope of its applications. Furthermore, even in the noise-free cases, the conventional discrete-time implementation of HOSMC also requires a high sampling rate and powerful real-time computation platform to avoid numerical chattering [13; 14]. In these designed control schemes for FES, an accurate muscle model is crucial for controlling joint angles. For instance, [PERSON] et al. developed a second-order linear FES model and proposed a full-state feedback control scheme based on it [15]. However, the challenge lies in the fact that the muscle model varies with time and differs among individuals, making accuracy difficult to achieve. Developing a high-fidelity muscle model of finger metacarpophalangeal (MP) joint, which is only a single joint with one degree of freedom (DoF), for different individuals is still cumbersome due to the nonlinearity, time-varying, and individual-depending characteristics of the musculoskeletal system. Furthermore, some of the control strategies, e.g., the MPC [7], the RNN [8], and the HOSMC [11], all require a high-frequency sampling rate, large amount of data, or powerful computation resource to achieve a high performance of the FES control, which is not suitable for embedded devices of MP joint control with limited computation power and sampling rate. Among the various control strategies, the super-twisting algorithm (STA) is a popular method within higher-order sliding mode control (HOSMC). It achieves second-order sliding mode accuracy while only requiring first-order information of the sliding variable. STA has been widely used in mechatronic systems such as fish robots [16] and exoskeleton systems [17]. To reduce chattering effects while maintaining robustness, an adaptive version of STA was developed and is commonly employed for real-time control in practical systems [18; 19; 20].