## 3 Nonlinear Model Predictive Control Formulation
### Flexible-Joint Mechanism
Regarding the input constraints \(\pm\)1.9, the constraints are active for the LQT-FL and SDRE. Although both can reach the desired angle, the LQT-FL has huge overshoots. Regarding the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Controller** & **Rising Time** & **Settling Time** & **Max. Overshoot** & **Input Ranges (N-m)** & **Computational** & **RMSE** \\  & **(s)** & **(s)** & **(Rad)** & **Input Ranges (N-m)** & **Time (s)** & **(Rad)** \\ \hline SDRE & 1.1120 & 15.8700 & 0.3589 & (\(-\)1.9091, 0.9438) & 43.0110 & 0.0688 \\ MPC-TSF & 13.3184 & 20.1470 & 0.0002 & (\(-\)0.0004, 0.5991) & 45.0750 & 0.1333 \\ LQT-FL & 6.1280 & 48.8000 & 0.4638 & (\(-\)4.9458, 5.5785) & 1.5310 & 0.1918 \\ MPC-FL & 9.2668 & 12.6885 & 0.0000 & (\(-\)0.0004, 1.0480) & 15.5830 & 0.1564 \\ (proposed) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparisons of applying the four control schemes to the flexible-joint mechanism without input constraints. Figure 3: Flexible-joint mechanism by applying the four control schemes without input constraints: (**a**) output responses; **(b**) input signals. input constraints \(\pm\)1.8, the constraints are active for the LQT-FL and SDRE, where the LQT-FL has not only huge overshoots but also fluctuations at the steady state, while the SDRE cannot track the reference signal. Regarding the input constraints \(\pm\)0.59, the constraints are active for all controllers, where only the LQT-FL, MPC-TSF and proposed MPC-FL can track the reference signal, but the LQT-FL has huge overshoots and the MPC-TSF has a longer settling time, compared with the proposed MPC-FL. Regarding the input constraints \(\pm\)0.56, the constraints are active for all controllers but only the proposed MPC-FL and LQT-FL could track the reference signal effectively, although the LQT-FL still has huge overshoots. Table 4 shows the performance indices, which are the rising time, the settling time, the maximum overshoots, the ranges of input signals, and the RMSEs. The results show that a larger input constraint range can be satisfied by all control schemes. However, as the range increases, some control schemes may not track the reference signal. Even though some of them track the signal effectively, the performance indices are worse than those obtained by the proposed MPC-FL. Figure 4: Flexible-joint mechanism by applying the four control schemes with input constraints \(\pm\)4.5 N\(\cdot\)m: (**a**) output responses; (**b**) input signals. Figure 5: Flexible-joint mechanism by applying the four control schemes with input constraints \(\pm\)1.9 N\(\cdot\)m: (**a**) output responses; (**b**) input signals. Figure 8: Flexible-joint mechanism by applying the four control schemes with input constraints \(\pm\)0.56 N-m: (**a**) output responses; (**b**) input signals. Figure 6: Flexible-joint mechanism by applying the four control schemes with input constraints \(\pm\)1.8 N-m: (**a**) output responses; (**b**) input signals. Figure 7: Flexible-joint mechanism by applying the four control schemes with input constraints \(\pm\)0.59 N-m: (**a**) output responses; (**b**) input signals.