# FES Control of a Finger MP Joint with a Proxy-Based Super-Twisting Algorithm
## 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, [PERSON] and [PERSON] [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:  
\[\dot{x}=a(t,x)+b(t,x)\mu^{*}, \tag{4a}\] \[\mu^{*}=\text{proj}(\mathcal{F},\mu),\sigma=\sigma(t,x) \tag{4b}\]  
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 \(\text{proj}()\) 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 [PERSON] [31], let us define the dynamics of the sliding variable \(\sigma\):  
\[\dot{\sigma}=f(t,x)+g(t,x)\mu^{*} \tag{5}\]  
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], \(\forall t>0,x\in\mathbb{R}\):  
\[|f(t,x)|\leq F_{0},\,|f(t,x)|\leq F_{1},\,0<g_{m}\leq g(t,x)\leq g_{M}, \tag{6}\]  
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 \(\mu^{*}=\text{proj}(\mathcal{F},\mu)\). 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].