FES Control of a Finger MP Joint with a Proxy-Based Super-Twisting Algorithm

Abstract

To improve motion accuracy through functional electrical stimulation (FES) of forearm muscles, feedback control laws are applied to the index finger’s metacarpophalangeal (MP) joint. This paper introduces a proxy-based super-twisting algorithm (PSTA) for precise servo control of MP joints via FES. The PSTA combines first-order sliding mode control with a second-order super-twisting algorithm, effectively preventing windup during FES saturation and ensuring robust, accurate control. An implicit Euler method minimizes numerical chattering in the digital implementation. Experiments with Arduino and volunteers confirm the algorithm’s effectiveness.

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 and the 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, Sakaino 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]. 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. Zhang et al. proposed a robust finite-time command-filtered backstepping control for flexible-joint robots using only position measurements [21]. Zhang et al. also developed a rapid swing control strategy for 5-DOF tower cranes [22]. Additionally, Qian 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, Kikuuwe 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 $\mathrm{proj}\left(\right)$:

$$\begin{array}{c}\hfill \mathrm{proj}(\mathcal{A},x)=max(min(B,x),A),\forall x\in \mathbb{R}\end{array}$$

(1)

where $x\in \mathbb{R}$. This paper also uses the following definition of signum functions, where the function $\mathrm{sgn}\left(·\right)$ is the set-valued version of the signum function that differs from the single-valued version $\mathrm{sign}\left(·\right)$. 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, $\mathrm{sign}\left(·\right)$ is defined as the single-value function: $\mathrm{sign}\left(x\right):=\mathrm{sgn}\left(x\right)$ for $x\ne 0$ and $\mathrm{sign}\left(x\right):=0$ for $x=0$.

With a nonnegative scalar $F\ge 0$, the following relation is satisfied:

$$\begin{array}{cc}\hfill x& \in F\mathrm{sgn}(y-x)\iff x=\mathrm{proj}(\mathcal{F},y),\hfill \end{array}$$

(2)

with $\mathcal{F}=[-F,F]$, of which the proof can be found in previous papers [24,26]. The following mathematical equivalence is strictly satisfied [28]:

$$\begin{array}{c}\hfill z\in -{\alpha }_1\mathrm{sgn}(z-x)-{\alpha }_2\mathrm{sgn}(z-y)\iff z=\mathrm{proj}(\mathcal{B},x)\end{array}$$

(3)

where ${\alpha }_1,{\alpha }_2$ are parameters, $0<{\alpha }_2<{\alpha }_1$ and $x,y,z\in \mathbb{R}$, $\mathcal{B}:=\left[\mathrm{proj}\right(-\mathcal{C},y),\mathrm{proj}(\mathcal{C},y\left)\right]$, $\mathcal{C}:=[{\alpha }_1-{\alpha }_2,{\alpha }_1+{\alpha }_2]$ is a compact set, and $\mathrm{proj}\left(\right)$ is defined as in (1).

The structure of this paper is arranged as follows. The problem statement is introduced in Section 2. The proposed proxy-based super-twisting algorithm (PSTA) and its stability analysis are given in Section 3. Subsequently, Section 4 presents the proposed implicit Euler implementation of the proposed PSTA. The proposed approach is evaluated through experimental results of MP joint control in Section 5. Finally, conclusions and some future work are given in Section 6.

2. Problem Statement

In muscle modeling literature, the Hill-type model is a well-known approach [29]. This model represents the muscle as a second-order system with two poles and one zero. However, M. Ferrarin and A. Pedotti [30] studied the relationship between pulse and torque in knee joint muscle models. They found that in many cases, a first-order model with only one pole is more optimal than a second-order system model. Therefore, assuming that this simplified first-order system can also be applied to the finger MP joint, we can define the angle-torque model using an actuation saturation first-order system:

$$& \dot{x}=a(t,x)+b(t,x){\mu }^{*},\hfill$$

(4a)

$${\mu }^{*}=\mathrm{proj}(\mathcal{F},\mu ),\sigma =\sigma (t,x)\hfill$$

(4b)

where $x\in \mathbb{R}$ is the measurable angle of the finger MP joint, as shown in Figure 1, $\mu$ is the FES signal, i.e., the time duration of a fixed magnitude of voltage, determined by a control law $\mu$ to be designed, ${\mu }^{*}$ is the maximum FES signal, i.e., the projections of the control input $\mu$ to the set $\mathcal{F}=[-F,F]$ with the maximum duration of time $F>0$, $a(t,x)$ and $b(t,x)>0$ are smooth functions, $\sigma$ is the sliding variable to be defined [31], and $\mathrm{proj}\left(\right)$ is the projection function defined as in (1). The object is to develop a control law $\mu$ that can be realized within the single board computer (SBC) of Arduino such that the sliding variable converges to zero, i.e., $\sigma (t,x)=0$.

The challenge lies in the fact that the values of terms $a(t,x)$ and $b(t,x)$ vary over time and across individuals due to the model representing the muscle of the MP joint in humans. This variability makes it difficult to obtain an accurate model. Despite this uncertainty, it is established that the finger MP joint model (4) is a first-order system with a relative degree of 1. Then, as noted by Levant [31], let us define the dynamics of the sliding variable $\sigma$:

$$\begin{array}{c}\hfill \dot{\sigma }=f(t,x)+g(t,x){\mu }^{*}\end{array}$$

(5)

where $f(t,x)$ and $g(t,x)$ are the Lie derivatives derived from (4), i.e., $f(t,x)=\frac{\partial \sigma }{\partial x}a(t,x),$$g(t,x)=\frac{\partial \sigma }{\partial x}b(t,x)$. The two functions are unknown smooth but bounded functions and let us assume [31], $\forall t>0,x\in \mathbb{R}$:

$$\begin{array}{c}\hfill \left|f(t,x)\right|\le F_0,\left|\dot{f}(t,x)\right|\le F_1,0<g_m\le g(t,x)\le g_M,\end{array}$$

(6)

that is, $f(t,x)$ and its derivative are bounded by constants $F_0$ and $F_1$, respectively, while the input gain $g(t,x)$ is a constant or slowly varying unknown value that is bounded by two constants: $g_m$ and $g_M$. The objective is to design a control law $\mu$ that drives the state variable, denoted as x, to a desired trajectory $x_d$, i.e., $\sigma =0$, even when under actuation saturation ${\mu }^{*}=\mathrm{proj}(\mathcal{F},\mu )$. This must be achieved without prior knowledge of the functions $f(t,x)$, $g(t,x)$ mentioned above or existing but unknown constants $F_0$, $F_1$, $g_m$, and $g_M$. Previous works in SMC [14,31] have made similar assumptions to (6). Instead of focusing on the finger MP joint model (4), it would be reasonable to design an SMC controller with respect to (5) using (6), which varies for different individuals, but remains within a certain range.

With these bounded assumptions, it is straightforward to design some strategies of SMC to stabilize the system (4), because the problem of stabilizing the systems like (4) has been solved by different SMC methods under the appropriate assumptions, such as terminal SMC, adaptive SMC, HOSMC, and implicit-Euler SMC [32]. However, there are three challenges to design the strategies of SMC here:

• Without the knowledge of boundness parameters $F_0$, $F_1$, $g_m$, and $g_M$ and only the actuation saturation level F, adaptive SMC methods can be designed here, while their implementations require high-frequency sampling and switching rate of actuation, which is not suitable for the embedded platform of a low computation resource and sampling rate.
• One strategy is to set the gains of the gains of terminal SMC and HOSMC strategies as large as possible such that the stability of the uncertain closed-loop systems (5) and (6) is guaranteed. The problem is that the magnitude of chattering is proportional to the size of gains and the sampling period, which deteriorates the control accuracy, especially for the embedded system with a low sampling frequency.
• Even with large gains, the implicit-Euler discretizatoins of SMC can remove the numerical chattering in the absence of noises, but the method requires the knowledge of unavailable term $g(t,x)$, making it difficult to be implemented in such cases.

3. Proposed Proxy-Based Super-Twisting Algorithm (PSTA)

Continuous-Time Expression of PSTA

To improve the accuracy of PID control while retaining the anti-windup property of PSMC [24], we propose incorporating first-order sliding mode control (SMC) with the super-twisting algorithm (STA) [33,34], one of second-order SMC strategies. This combination exploits the set-valueness of extended SMC in the Filippov sense and is expected to enhance second-order control accuracy when saturation is inactive. It can be described as follows:

$$\sigma =e+{\kappa }_3{\int }_0^t{\lfloor e\left(\tau \right)\rceil }^{\xi }\mathrm{d}\tau ,{\mu }^{*}=\mathrm{proj}(\mathcal{F},\mu )\hfill$$

(7a)

$$\mu \in -Fsgn\left(s_2\right),s_2=\sigma -s_1,\hfill$$

(7b)

$$\mu =-{\kappa }_1{\lfloor s_1\rceil }^{1/2}+v,\dot{v}\in -{\kappa }_{2}sgn\left(s_1\right),\hfill$$

(7c)

where “∈” means that the signum function $\mathrm{sgn}\left(·\right)$ is set-valued instead of the conventional single-valued version, $e=x-x_d$, $\sigma$ is the defined sliding variable, and when $\xi =1$, it changes to the linear one $\sigma =e+{\kappa }_3{\int }_0^{t}e\left(\tau \right)\mathrm{d}\tau$, $\mu$ is defined in (4) and (5), while ${\mu }^{*}$ is the saturated input, ${\kappa }_3>0$ is a parameter of determining the convergence rate of e, $s_2$ is an intermediate variable and can be viewed as the sliding variable of the first-order SMC, $s_1$ is an another intermediate variable and can be viewed as the sliding variable of the STA, and their summation always satisfies $s_1+s_2=\sigma$, $F>0$ is the gain of the sliding mode control (SMC) and is also the saturation level of actuator, v is also an intermediate variable of the STA, ${\kappa }_1,{\kappa }_2>0$ are the gains of the conventional STA and its selection can refer to the work in [35], and the control input $\mu$ is determined by the set-valued first-order SMC (7b) and the set-valued STA (7c) simultaneously. The notation ${\lfloor ·\rceil }^{\xi }$ is defined as $\forall x\in \mathbb{R}$, ${\lfloor x\rceil }^{\xi }={\left|x\right|}^{\xi }\mathrm{sign}\left(x\right)$; $\forall \xi \in (0,1]$, ${\lfloor 0\rceil }^{\xi }=0$; ${\lfloor x\rceil }^0=\mathrm{sign}\left(x\right)$.

It should be noted that $\mu$ appearing in both (7b) and (7c) means that it is subjected to the constrains of (7b) and (7c) simultaneously, which is possible due to the set-valued mappings of (7b) and (7c). Due to the special structure of the above controller (7), it is referred as the proxy super-twisting algorithm (PSTA) thereafter. Additionally, from (7), one has $\mu ={\mu }^{*}$, that is, $\mu \in \mathcal{F}=[-F,F]$ is always satisfied, which is not the case of the conventional STA.

One desired property of the PSTA (7) is that, due to the saturation constraint by (7b), it has a similar property of the PSMC [24] and the conditioned STA [36], i.e., anti-windup effect with the saturation level $F>0$, which is inherited from the PSMC. However, the differences is that, when $\left|\mu \right|<F$, the control input $\mu$ is determined by the STA (7c), which enables it to achieve a second-order asymptotical accuracy and higher than that of the PMSC with the same three number of parameters. Moreover, comparing to the conventional STA [33], it has the properties of anti-windup property and can deal with nondifferentiable and differentiable disturbances due to the combination of (7b) and (7c).

Let us see the dynamics (5) interconnected by the PSTA (7), by first checking the derivative of $s_2$ in (7) with $\mu ={\mu }^{*}$, that is, ${\dot{s}}_2=\dot{\sigma }-{\dot{s}}_1$ and:

$$\begin{array}{c}{\dot{s}}_2=f(t,x)+g(x,u){\mu }^{*}-{\dot{s}}_1=\dot{e}+{\kappa }_3{\lfloor e\rceil }^{\xi }-{\dot{s}}_1\hfill \\ \in [-F_0,F_0]+[g_m,g_M]\mu -{\dot{s}}_1,\hfill \end{array}$$

(8)

from which one can see the closed-loop system with the control input $\mu$ subjected to the constraints $\mu \in -Fsgn\left(s_2\right)$, $s_2=\sigma -s_1$, and $\mu =-{\kappa }_1{\lfloor s_1\rceil }^{1/2}+v,\dot{v}\in -{\kappa }_{2}sgn\left(s_1\right)$. Since $\mu$ is always and simulatively subjected to (7b) and (7c), (8) can be equivalently rewritten as the following system:

$${\dot{s}}_2\in [-F_0,F_0]+[g_m,g_M]\mu -{\dot{s}}_1,\mu \in -F\mathrm{sgn}\left(s_2\right),\hfill$$

(9a)

$$& \mu =-{\kappa }_1{\lfloor s_1\rceil }^{1/2}+v,\dot{v}\in -{\kappa }_{2}sgn\left(s_1\right),\hfill$$

(9b)

from which one can see that the control input $\mu$ is simultaneously subjected to constraints of the first-order SMC and the second-order STA and is independent on the unknown model (4). Alternatively, one can say that the input $\mu$ computed by the STA always satisfies $\left|\mu \right|\le F$, which is constraint by the first-order SMC. The closed-loop system (9) has the stability property shown in Theorem 1.

Theorem 1.

Consider the dynamical system (5) with the uncertain smooth functions $f(t,x)$ and $g(t,x)$ satisfying $sup\left|f(t,x)\right|\le F_0$ and $0<g_m\le g(t,x)\le g_M$, $\forall t\ge t_0\ge 0,x\in \mathbb{R}$. If $F>F_0/g_m$, ${\kappa }_2\ge F_1/g_m$, ${\kappa }_1>\sqrt{{\kappa }_2+F_1}/g_m$, and ${\kappa }_3>0$ are satisfied for all $t\ge 0$, then, with bounded initial condition $s_2\left(t_0\right)+s_1\left(t_0\right)=\sigma \left(t_0\right)$, the equilibrium of the sliding variable $\sigma =s_1+s_2=0$ of the closed-loop system (5) interconnected with (7) is asymptotically attained.

Proof. 

Here, only the sketch of the proof is provided, because both the stabilities of first-order SMC and the STA are well studied in the literature. Because the closed-loop system (5) interconnected with (7) can be equivalently expressed as (9), one can only check the stability of (9). Let us first check the stability of (9a):

$$\begin{array}{c}\hfill {\dot{s}}_2\in [-F_0,F_0]+[g_m,g_M]\mu -{\dot{s}}_1,\mu \in -F\mathrm{sgn}\left(s_2\right),\end{array}$$

(10)

of which the stability can be easily recognizable when $F>F_0/g_m$ and $|{\dot{s}}_1|$ remains bounded, because it is the standard first-order SMC. The asymptotical satiability of (10) is straightforward to show by defining the Lyapunov function $V_1\left(s_2\right)=s_2^2/2$ according to ([26], Prop. 1), when only (10) is considered and ${\dot{s}}_1$ is considered as a bounded disturbance. If the assumptions in (6) and $F>F_0/g_m$ hold, there exists a finite time $t_1\ge 0$ (depending on initial conditions $s_2\left(t_0\right)$), $\forall t\ge t_1$, ${\dot{s}}_2=s_2=0$ ([26], Prop. 1). This convergence period $t\in [t_0,t_1)$ of (10) can be considered as the first stage of (9). Therefore, in the fist stage, both $|s_2|$ and $|{\dot{s}}_2|$ are shrinking if $|{\dot{s}}_1|$ is bounded. Then, for $t\ge t_1$, $s_2={\dot{s}}_2=0$, i.e., in the second stage, from (9), due to $\sigma =s_1+s_2$ and $\dot{\sigma }={\dot{s}}_1+{\dot{s}}_2$, one has the following expressions:

$${\dot{s}}_1\in [-F_0,F_0]+[g_m,g_M]\mu =f(t,x)+g(t,x)\mu \hfill$$

(11a)

$$\mu =-{\kappa }_1{\lfloor s_1\rceil }^{1/2}+v,\dot{v}\in -{\kappa }_{2}sgn\left(s_1\right),\hfill$$

(11b)

which can be further rewritten as follows:

$${\dot{s}}_1\in -[g_m,g_M]{\kappa }_1{\lfloor s_1\rceil }^{1/2}+z_1\hfill$$

(12a)

$$& {\dot{z}}_1\in -[g_m,g_M]{\kappa }_{2}sgn\left(s_1\right)+[-F_1,F_1],\hfill$$

(12b)

with definition $z_1:=f(t,x)+g(t,x)v$. Note from (11) to (12), $\dot{g}(t,x)=0$ has been used, because the input gain $0<g_m\le g(t,x)\le g_M$ in (6) assumes to be a constant or slowly varying but bounded parameter. The system (12) is finite-time stable if the above gain conditions satisfied, if ${\kappa }_2\ge F_1/g_m$, ${\kappa }_1>\sqrt{{\kappa }_2+F_1}/g_m$, and ${\kappa }_3>0$ hold [37].

The above proof is based on the assumption that $|{\dot{s}}_1|$ is bounded. To demonstrate that $|{\dot{s}}_1|$ and $|s_1|$ are shrinking and bounded, one can equivalently rewrite (10) as the dynamics about $s_1$ with the control law $\mu$ replaced by the STA (7c) and consider ${\dot{s}}_2$ as the disturbance. Then, by following previous works of STA [33,37], one can see the assumption holds that $|{\dot{s}}_1|$ and $|s_1|$ are shrinking and bounded. From the above analysis of (10), $|{\dot{s}}_2|$ is decreasing, and thus, the size of bound of $|{\dot{s}}_1|$ is decreasing. Therefore, $\sigma =s_1+s_2=0$ is attained for a finite time $t\ge t_s=max(t_1,t_2)$. □

Remark 1.

From Theorem 1, it is evident that even if the models $f(t,x)$ and $g(t,x)$ are unknown, and the bounds $g_m$, $g_M$, $F_0$, and $F_1$ are not available in practice, setting large enough gains such as F, ${\kappa }_1$, and ${\kappa }_2$ satisfies the conditions of Theorem 1. This guarantees stability of the closed-loop system (5) interconnected with (7). However, due to these large gains, numerical chattering may occur, which needs to be resolved using numerical methods. As mentioned in Section 2, conventional SMC strategies face three challenges in this situation. However, these challenges can be overcome by using PSTA (7) and its implicit-Euler discretization method without depending on $g(t,x)$. Further details are provided in subsequent sections.

4. Discretization Scheme of PSTA

In the first stage, i.e., $s_2\ne 0$ or $s_1\ne \sigma$, the dynamics (5) interconnected by the control law (7) is actually governed by the system (9), which is a perturbed first-order dynamical system controlled by the first-order SMC.

One can see that the PSTA (7) is an algebraic inclusion and cannot be directly implemented due to the set-valued signum functions. To obtain the discretization scheme (7), let us first assume that x is available and $x_d$ is known. Then, let us eliminate $\mu$ by rewriting the system (7) as follows:

$$-{\kappa }_1{\lfloor s_1\rceil }^{1/2}+v\in -Fsgn(\sigma -s_1),\hfill$$

(13a)

$$\dot{v}\in -{\kappa }_{2}sgn\left(s_1\right),\hfill$$

(13b)

which is an algebraic inclusion about the sliding variable $s_1$. Let us discretize (13) with the implicit Euler method:

$$v_{k+1}\in F\mathrm{sgn}(s_{1,k+1}-{\sigma }_{k+1}) + {\kappa }_1{\left|s_{1,k+1}\right|}^{\frac{1}{2}}\mathrm{sgn}\left(s_{1,k+1}\right)$$

(14a)

$$\hfill v_{k+1}\in v_k-h{\kappa }_2\mathrm{sgn}\left(s_{1,k+1}\right),$$

(14b)

where $h>0$ represents the sampling period of time and the subscript index $k\in \{\mathbb{N}\cup 0\}$ is the index of time-stepping, i.e., $t_k=hk$ and $t_k-t_{k-1}=h$. Substituting $v_k$ in (14a) with the expression in (14b) leads to the following expression:

$$\begin{array}{c}\hfill v_k-{\kappa }_1{\left|s_{1,k+1}\right|}^{\frac{1}{2}}\mathrm{sgn}\left(s_{1,k+1}\right)\in \\ \hfill F\mathrm{sgn}(s_{1,k+1}-{\sigma }_{k+1})+h{\kappa }_2\mathrm{sgn}\left(s_{1,k+1}\right),\end{array}$$

(15)

which is equivalent to the following one:

$$\begin{array}{c}\hfill v_k - {\kappa }_1|s_{1,k+1}{|}^{\frac{1}{2}}\mathrm{sgn}\left(s_{1,k+1}\right)\in h{\kappa }_2\mathrm{sgn}\left(\right|s_{1,k+1}{|}^{\frac{1}{2}}\mathrm{sgn}\left(s_{1,k+1}\right))\\ \hfill F\mathrm{sgn}\left[\right|s_{1,k+1}{|}^{\frac{1}{2}}\mathrm{sgn}\left(s_{1,k+1}\right)-\left|{\sigma }_{k+1}{|}^{\frac{1}{2}}\mathrm{sgn}\left({\sigma }_{k+1}\right)\right]\end{array}$$

(16)

due to the property $\mathrm{sgn}\left(x\right)=\mathrm{sgn}\left(\alpha x\right),\forall \alpha >0,x\in \mathbb{R}$. If ${\kappa }_2$ and h are properly selected such that $F>h{\kappa }_2$ is satisfied, inspired by the work in [38], the mathematical equivalence (3) can be applicable to (16), leading to:

$$\begin{array}{c}{v}_k-{\kappa }_1{\left|s_{1,k+1}\right|}^{\frac{1}{2}}\mathrm{sgn}\left(s_{1,k+1}\right)=\mathrm{proj}({\mathcal{D}}_k,y_k)\hfill \end{array}$$

(17)

where $y_k:=v_k+{\kappa }_1{\left|{\sigma }_{k+1}\right|}^{\frac{1}{2}}\mathrm{sgn}\left({\sigma }_{k+1}\right)$, ${\mathcal{D}}_k:=[{\mathcal{C}}_{1,k},{\mathcal{C}}_{2,k}]$ with ${\mathcal{C}}_{1,k}:=\mathrm{proj}(-\mathcal{F},v_{k-1})$, ${\mathcal{C}}_{2,k}:=\mathrm{proj}(\mathcal{F},v_{k-1})$, and $\mathcal{F}:=[F-h{\kappa }_2,F+h{\kappa }_2]$. From (17), one has the updated law of $a_{2,k}$:

$$\begin{array}{c}\hfill s_{1,k+1}=\frac{1}{{\kappa }_1^2}{(\mathrm{proj}({\mathcal{D}}_k,y_k)-v_k)}^2\mathrm{sign}(v_k-\mathrm{proj}({\mathcal{D}}_k,y_k))\end{array}$$

where $\mathrm{sign}\left(\right)$ is the single-valued signum function. Then, from (17), because of $\mu =-{\kappa }_1{\lfloor s_1\rceil }^{1/2}+v$, the first-stage of the proposed implicit-Euler digital implementation scheme for (7) is as follows:

$${\mu }_k=\mathrm{proj}({\mathcal{D}}_k,y_k)-v_k+v_{k+1}\hfill$$

(18a)

$$v_{k+1}\in v_k-h{\kappa }_2\mathrm{sgn}\left(s_{1,k+1}\right).\hfill$$

(18b)

One can see that “∈” still exists in (18b) and one straightforward way is to approximate it with the single-valued signum function $\mathrm{sign}\left(\right)$. Here, to avoid the numerical chattering caused by the single-valued signum function, an alternative way is to reuse (17): if $\mathrm{proj}({\mathcal{D}}_k,y_k)-v_k\ne 0$:

$$\begin{array}{c}\hfill \mathrm{sgn}\left(s_{1,k+1}\right)=\frac{v_k-\mathrm{proj}({\mathcal{D}}_k,y_k)}{|v_k-\mathrm{proj}({\mathcal{D}}_k,y_k)|},\end{array}$$

(19)

due to ${\kappa }_1|s_{1,k+1}{|}^{\frac{1}{2}}=|\mathrm{proj}({\mathcal{D}}_k,y_k)-v_k|$ from (17). Finally, one has the proposed digital implementation scheme of (7) without any signum function:

$$v_{k+1}=v_k-h{\kappa }_2\frac{v_k-\mathrm{proj}({\mathcal{D}}_k,y_k)}{|v_k-\mathrm{proj}({\mathcal{D}}_k,y_k)|}\hfill$$

(20a)

$$& {\mu }_k=\mathrm{proj}({\mathcal{D}}_k,y_k)-h{\kappa }_2\frac{v_k-\mathrm{proj}({\mathcal{D}}_k,y_k)}{|v_k-\mathrm{proj}({\mathcal{D}}_k,y_k)|}.\hfill$$

(20b)

Note that the variable $y_k$ contains the prediction value of ${\sigma }_{k+1}$ as in [12], which can be approximated with ${\sigma }_{k+1}\simeq {\sigma }_k-hF\mathrm{proj}(\mathcal{I},{\sigma }_k/\left(Fh\right))$ with $\mathcal{I}=[-1,1]$ [14].

Remark 2.

From the digital implementation algorithm (20), one can see that it is different from the discrete-time realizations of the conventional STA in the work [12]. The discrete-time updating laws of both $u_k$ and $v_k$ have some mathematical similarities to the quasi-continuous SMC [39] and the discrete-time equivalents of STA in [40]. In all these cases, the numerical chattering have been significantly reduced.

In the second stage, i.e., when $s_2=0$ and $\sigma =s_1$, the dynamics (5) interconnected by the control law (7) is actually governed by the system (12), which is a perturbed first-order dynamical system controlled by the STA with uncertain input gain $0<g_m\le g(t,x)\le g_M$. That means during the second stage, the properties of the proposed control law (7), such as the control accuracy and convergence, are finally determined by the STA.

There are some works related to the digital realizations of the STA in the literature [12,40]. However, as illustrated and compared in [32], these works have different properties, such as some have lost the asymptotical second-order control accuracy, and some are sensitive the overestimation of gains. To obtain the discrete-time implementation algorithm during the second stage, for simplicity, let us assume that $g_t=g(t,x)$$k_1=g_t{\kappa }_1$, $k_2=g_t{\kappa }_2$, $\varphi \left(t\right)=f(t,x)$, $△\left(t\right)=\dot{f}(t,x)$ and consider the discretization and realization of (12) with the implicit Euler method:

$$s_{1,k+1}=s_{1,k}+h{\mu }_k+h{\varphi }_{k+1},{\varphi }_{k+1}={\varphi }_k+h{△}_{k+1},\hfill$$

(21a)

$$v_{k+1}\in v_k-h{k}_2\mathrm{sgn}\left(s_{1,k+1}\right),\hfill$$

(21b)

$${\mu }_k=-k_1{\lfloor s_{1,k+1}\rceil }^{\frac{1}{2}}+v_{k+1}\hfill$$

(21c)

where one should be noted that the discretizations ${\varphi }_k=\varphi \left(t_k\right)$ and ${△}_k=△\left(t_k\right)$ do not mean that ${\varphi }_k$ is a piece-wise constant, but refers to the fact that it has no effect on $e_k$ within the period $(t_k,t_{k+1})$ after sampled at $t_k$ by analog-to-digital (ADC) devices in practice and (21) is an implicit Euler discretization of (12), while other schemes such as the simple explicit Euler and more accurate zero-order-hold (ZOH) are also possible. Substituting (21c) to (21a) leads to:

$$s_{1,k+1}=s_{1,k}-h{\mu }_k+h{z}_{2,k}+h^2{△}_{k+1}\hfill$$

(22a)

$${\mu }_k\in \left(k_1\sqrt{|s_{1,k+1}|}+h{k}_2\right)\mathrm{sgn}\left(s_{1,k+1}\right)\hfill$$

(22b)

where $z_{2,k}:={\varphi }_k+v_k$, ${△}_{k+1}$ and $s_{2,k}$ are unknown to us because $s_{2,k}:={\varphi }_k+v_k$ contains the external unknown disturbance ${\varphi }_k$. By following the work [41], one has the following results:

$$s_{1,k+1}=s_{1,k}-h{\tilde{\mu }}_k-h{\varphi }_k-h^2{△}_{k+1}\hfill$$

(23a)

$${\tilde{\mu }}_k={\tilde{u}}_{1,k}+v_{k+1};{\tilde{u}}_{1,k}=\frac{1}{h}\mathrm{proj}\left(C_k,s_{1,k}\right)\hfill$$

(23b)

$$v_{k+1}=v_k+h{\tilde{u}}_{2,k};{\tilde{u}}_{2,k}=\frac{1}{h^2}\mathrm{proj}\left(C_{2,k},s_{1,k}\right)\hfill$$

(23c)

where $C_{2,k}:=[-D_{2,k},D_{2,k}]$ while $C_k$ is defined as follows:

$$\begin{array}{c}{C}_k:=\left\{\begin{array}{cc}[-D_{1,k},D_{1,k}]\hfill & \mathrm{if}|s_{1,k}|>D_{1,k},\hfill \\ [-D_{2,k},D_{2,k}]\hfill & \mathrm{else}\hfill \end{array}\right.\hfill \end{array}$$

(24)

with $D_{1,k}:=h{k}_1\sqrt{|s_{1,k}|}+h^2{k}_2$ and $D_{2,k}:=h^2{k}_2$.

To make the proposed discrete-time realization algorithm (23) easy to be implemented, the condition $s_2=0$ or $\sigma =s_1$ is relaxed as $|\sigma -s_1|\le \epsilon$, where the parameter $\epsilon >0$ is a small real number as the threshold of the practical sliding motion. If $|{\sigma }_k-s_{1,k}|>\epsilon$, it means that the discrete-time system (23) is still within the first stage, while if $|{\sigma }_k-s_{1,k}|<\epsilon$, then it shows that (23) has attained the sliding motion of the first stage, i.e., ${\sigma }_k\approx s_{1,k}$. The input gain $g_t$ is unknown, and thus, $k_j,j=1,2$ are replaced with ${\kappa }_j$ in practice.

5. Experiments

5.1. Experiment Setup

The Ethics Committee of the School of Engineering at Kyushu University approved the experiments (Approval no. 2021-02), which were meticulously conducted with strict adherence to protocols. The experiment setup is illustrated in Figure 1, which includes an Arduino microprocessor (Arduino Mega, the manufacturer is Laplace, located in Miyagi, Japan), a driver (L298n), electrodes (IVES ELR-6211), and a joint angle acquisition unit (CP45 Single-turn Potentiometer). Six volunteers, $A,B,C,D,E,F$, four females and two males ranging in age from 18 to 60 years, participated in the study. Preliminary experiments revealed that electrical stimulation elicited a more sensitive response in the MP joint of the index finger than other finger joints; therefore, this joint was selected for further investigation to enable clearer comparison between different control methods. The location of each participant’s Flexor digitorum superficialis muscle, responsible for flexion movement, and Extensor digitorum muscle, responsible for extension movement, were identified on their arm’s surface, as these target muscles are located relatively superficially. Surface electrode patches provided by OGwellness (model ELR-615) were used after simple adjustments were made to their positions upon application; if no motion response was observed upon current application, electrodes were slightly relocated within close proximity until a motion response was achieved in the finger.

As studied in our previous research [42], the biphasic wave can effectively stimulate the nerve-muscles, and thus, leads to joint movements. Therefore, a stimulation signal is employed in experiments with alternating positive and negative square waves. The magnitude of the stimulation voltage remains constant, i.e., $V_{max}$ or $V_{min}$, followed a zero voltage of with constant duration of 50 $\mu$$\mathrm{s}$. By changing the duration of the square wave with the input $\mu$, the actual average voltage is changed and its magnitude of the actual average voltage determines the amount of joint motion and the sign of u determines the motion direction. In the experiments, the output $\mu$ of the PSTA is as the actuation input, i.e., $\mu$ $\mu$$\mathrm{s}$ with the saturation level $F=150$ $\mu$$\mathrm{s}$, i.e., $\left|\mu \right|\le F=150$ $\mu$$\mathrm{s}$, and the feedback of the joint angle $x=\theta$ as the output is sampled by an analog angle sensor, one analog potentiometer, with a period of 5 $\mathrm{m}$$\mathrm{s}$ from the Analog Pin of the Arduino equipped with ADC device. As shown in (7), we have adjusted the gain $F=150\mathsf{\mu }$s according to practical actuation saturation levels. For ${\kappa }_1$ and ${\kappa }_2$, we set ${\kappa }_1=1.5\sqrt{L}$ and ${\kappa }_2=1.1L$ based on literature about STA. Due to its insensitivity to gain overestimation, we increased L until tracking error convergence was achieved. Regarding ${\kappa }_3$ in (7), we tuned it through trial and error to ensure there was no overshooting in experiments.

The proposed PSTA for the MP joint control is shown in Figure 1, where the PSTA implemented in the Arduino board and $\mu$ is calculated in a real-time manner with $h=25$ ms. Then, the stimulation signal with a duration of $\mu$ $\mu$$\mathrm{s}$ is generated by the Arduino board to the motor driver connected with an external power supply. The outputs of the two motor drivers are transmitted to electrode patches that are applied to the index finger extensor muscle group, i.e., Channel 1, and the index finger flexor muscle group, i.e., Channel 2, respectively. One circular potentiometer, CP45 Single-turn Potentiometer, as the encoder with a resolution of $0.{35}^{\circ }$ is placed at the MP joint of the index finger. To prevent the wrist from rotating with the index finger, a fixation device is also placed on the wrist, as shown in Figure 1.

5.2. Experimental Results

The experimental results are shown in Figure 2 and Figure 3 for $x_d={\theta }_r=$ 30°. During the experiments, six different volunteers, $A,B,C,D,E,F$, were required to keep their index finger relaxed without tension throughout the experiments and the index MP joint was driven by the proposed algorithm “PSTA”, the STA of second-order SMC realized with the explicit-Euler discretzation, denoted as “Explicit”, and the PID, respectively, to the target angle ${\theta }_r$. With the usage of PID, we set its gains through trial and error to achieve the optimal performance. To show the performances of closed-loop control, the MP joint is also controlled by the self-conscious movement of the healthy volunteers, denoted as “Voluntary”, that is, each volunteer tries to control their MP joint to move from zero degree to 30° as quick as possible by observing a marker of 30° at the angle sensor with their own eyes. During the experiments of the six volunteers, the parameters of the proposed “PSTA” are kept as the same, i.e., ${\kappa }_1=2.2\sqrt{{\kappa }_2},{\kappa }_2=0.08,{\kappa }_3=50,\xi =0.6$. One can see that, without the information of $f(t,x)$ and $g(t,x)$ in (5), with the proposed algorithm PSTA for different experimenters, the controlled joint of MP finger quickly and exactly converges to ${\theta }_r$ with little overshooting, while other methods have lower accuracy, as shown in Figure 2 and

Table 1. RMSE and SSE of six volunteers under different control methods in Figure 3.

Volunteer	Metric	PSTA	Explicit	PID	Voluntary
A	RMSE	0.39	2.48	0.47	0.67
	SSE	0.33	0.33	0.33	0.67
B	RMSE	0.35	1.84	1.54	0.38
	SSE	0.02	4.16	0.02	0.02
C	RMSE	0.33	1.04	0.59	0.95
	SSE	0.08	0.02	0.08	1.34
D	RMSE	0.33	2.76	1.06	0.70
	SSE	0.02	1.04	0.08	1.04
E	RMSE	0.35	1.74	1.35	0.83
	SSE	0.02	8.46	1.36	0.44
F	RMSE	0.38	1.74	1.76	0.42
	SSE	0.08	0.08	0.08	0.08

.

The results presented in Figure 2 indicate that the conventional explicit-Euler discretization of the STA, referred to as “Explicit”, causes numerical chattering–oscillation around the target degree. On the other hand, the proposed PSTA effectively reduces chattering and leads to accurate tracking results. The tracking errors of four strategies are displayed in Figure 3, which show fluctuations and small magnitude of chattering around zero during steady-state due to sensor noise and varying muscle strength.

Increasing PID or SMC gains (or force in “Voluntary”) to improve control accuracy can result in overshooting and oscillations. However, thanks to its set-valued extension and implicit-Euler discretization, even with high gains, the proposed PSTA demonstrates high robustness when controlling MP joints across different volunteers without any overshooting or oscillations. The corresponding results for six volunteers are shown in Table 1, with statistical data presented in Figure 4. It shows that the proposed PSTA has the smallest average tracking error and variance for the six volunteers, even better than the deliberate “Voluntary” movement.

In

Table 2. Computational load of PSTA, Explicit, and PID controllers.

	PSTA	Explicit	PID
Average value [$\mu \mathrm{s}$]	420	199	265

, we compare the computational load of the proposed PSTA method, the Explicit method, and the PID method. The implicit Euler discretization of PSTA increases the computational load to average 420 $\mu \mathrm{s}$, compared to the explicit Euler discretization of STA and PID. However, given a sampling period of 5 ms and an update period of 25 ms for control input ${\mu }_k$, this increase is relatively minor. Despite higher computational complexity, the proposed PSTA offers benefits such as low chattering, anti-windup capabilities, and high control accuracy.

To further validate the robustness and effectiveness of the proposed method, we conducted additional experiments using a more complex reference trajectory, changing the target angle from 30° to 45°. The results of these experiments are shown in Figure 5. Similar to the previous experiments, the volunteers were instructed to keep their index finger relaxed throughout the experiments.

Even with the changing target angle, the PSTA was able to maintain a low level of chattering and accurate tracking performance. These findings further confirm the superiority of the PSTA in achieving precise control of MP joints under different conditions.

6. Conclusions

This manuscript presents the proxy-based super-twisting algorithm (PSTA) and an implicit Euler realization of the PSTA for highly accurate control of MP joint through FES. The proposed method maintains the accuracy and robustness of SMC without requiring an accurate model of a MP joint, while eliminating numerical chattering. Experimental results from different volunteers demonstrate that the PSTA is more effective than other control schemes, including conventional STA, PID, and voluntary movement. Qualitative analysis shows that this method is one of the most effective and accurate in the experiments. With limited computation resources and sampling rates, the PSTA still performs well in terms of response speed and control accuracy, which are sufficient for controlling MP joints. Future investigations should include more experiments on time-varying angles of MP joints through FES as well as other muscles, such as triceps brachii, using PSTA.

Future research includes a strict Lyapunov stability check and extending the work to control multiple finger joints simultaneously. This will involve developing advanced algorithms to manage increased complexity and ensure robust, accurate control across all joints. Addressing these challenges will enhance the applicability and effectiveness of the proposed method in practical scenarios.