# FES Control of a Finger MP Joint with a Proxy-Based Super-Twisting Algorithm
## 1 Introduction
However, conventional and adaptive STAs are not suitable for the input saturation cases and they have no anti-windup effects due to the integration action in the STA. To further strengthen disturbance rejection, recent studies have shown significant advancements. [PERSON] et al. proposed a robust finite-time command-filtered backstepping control for flexible-joint robots using only position measurements [21]. [PERSON] et al. also developed a rapid swing control strategy for 5-DOF tower cranes [22]. Additionally, [PERSON] et al. introduced a neural network-based sliding mode control for dual ship-mounted cranes, effectively suppressing sea wave disturbances [23]. To achieve safer control with anti-windup capabilities, [PERSON] et al. [24] proposed proxy-based sliding mode control (PSMC). This notable scheme combines first-order SMC with conventional PID control to provide robust control and damped motion without needing a plant model. PSMC has been widely applied in various scenarios, including exoskeleton suits [25], due to its desirable characteristics like the anti-windup effect and nonovershooting behavior when actuators are saturated. As noted in [24], when the input is not saturated or the closed-loop system remains on the sliding surface, PSMC functions equivalently to a conventional PID controller. One limitation of PSMC is that its equivalent PID controller's asymptotic accuracy depends on proportional, integral, and derivative gains that require careful tuning to achieve satisfactory performance. This paper intends to solve the control problem of FES for the MP joints with a new method of SMC by exploiting its robustness and its anti-windup property. It has a high control accuracy as well as a low computation complexity, such that it can be even implemented in embedded devices such as the Ardunio platform in a real-time manner. Meanwhile, the model uncertainties of the MP joints and actuation saturation are also taken into account to design the new control law. The main contributions are as follows:  
* **Robustness and High Control Accuracy**: Introducing the proxy-based super-twisting algorithm (PSTA) for FES control of MP joints, combining the robustness of Sliding Mode Control (SMC) and the high control accuracy of the Super-Twisting Algorithm (STA). * **Avoiding Numerical Chattering**: Proposing an implicit Euler discretization method to avoid numerical chattering while maintaining high control accuracy and robustness. * **Experimental Validation**: Validating the PSTA and its implicit discretization through experiments on MP joints using FES. Achieving high accuracy on a low-cost Arduino Mega board in real-time, demonstrating robustness and low computational complexity without detailed muscle model information. To the best of the authors' knowledge, this is the first time to achieve such a high control accuracy for the finger MP joint through the FES with such a low-cost hardware platform and without requiring the knowledge of the muscle model of the MP joint. **Notations**: Let \(\mathcal{A}\) be a closed interval of real numbers defined as \(\mathcal{A}:=[A,B]\) with \(A<B\in\mathbb{R}\). This paper uses the following projection function \(\text{proj}()\):  
\[\text{proj}(\mathcal{A},x)=\max(\min(B,x),A),\forall x\in\mathbb{R} \tag{1}\]  
where \(x\in\mathbb{R}\). This paper also uses the following definition of signum functions, where the function \(\text{sgn}(\cdot)\) is the set-valued version of the signum function that differs from the single-valued version \(\text{sign}(\cdot)\). This kind of set-valued definition, instead of the conventional single-value definition, has been employed by many previous papers [12; 26; 27]. To distinguish the difference, \(\text{sign}(\cdot)\) is defined as the single-value function: \(\text{sign}(x):=\text{sgn}(x)\) for \(x\
eq 0\) and \(\text{sign}(x):=0\) for \(x=0\).