# FES Control of a Finger MP Joint with a Proxy-Based Super-Twisting Algorithm
## 5 Experiments
### Experimental Results  
The experimental results are shown in Figures 2 and 3 for \(x_{d}=\theta_{r}=\)30\({}^{\circ}\). 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\({}^{\circ}\) as quick as possible by observing a marker of 30\({}^{\circ}\) 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. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Volunteer** & **Metric** & **PSTA** & **Explicit** & **PID** & **Voluntary** \\ \hline \multirow{2}{*}{A} & RMSE & 0.39 & 2.48 & 0.47 & 0.67 \\  & SSE & 0.33 & 0.33 & 0.33 & 0.67 \\ \hline \multirow{2}{*}{B} & RMSE & 0.35 & 1.84 & 1.54 & 0.38 \\  & SSE & 0.02 & 4.16 & 0.02 & 0.02 \\ \hline \multirow{2}{*}{C} & RMSE & 0.33 & 1.04 & 0.59 & 0.95 \\  & SSE & 0.08 & 0.02 & 0.08 & 1.34 \\ \hline \multirow{2}{*}{D} & RMSE & 0.33 & 2.76 & 1.06 & 0.70 \\  & SSE & 0.02 & 1.04 & 0.08 & 1.04 \\ \hline \multirow{2}{*}{E} & RMSE & 0.35 & 1.74 & 1.35 & 0.83 \\  & SSE & 0.02 & 8.46 & 1.36 & 0.44 \\ \hline \multirow{2}{*}{F} & RMSE & 0.38 & 1.74 & 1.76 & 0.42 \\  & SSE & 0.08 & 0.08 & 0.08 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE and SSE of six volunteers under different control methods in Figure 3.  
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, 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 \(\upmu\)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.