## 3 Nonlinear Model Predictive Control Formulation
### Two-Link Robot Arm
The results show that all schemes can track the reference signals but the response of the MPC-TSF is much slower than that of the other schemes. To distinguish the three schemes, Figure 11 shows their transients, and the results show that the overshoot of the LQT-FL is greater than those of the other two schemes. Table 6 shows the control performances by applying the four control schemes. The results show that the LQT-FL has the smallest rising and settling time. The SDRE and MPC-TSF have near-zero overshoots, and the overshoot of the proposed MPC-FL is smaller than that of the MPC-TSF. The proposed MPC-FL has the smallest input range, which implies that its control effort is smallest. The RMSEs of the proposed MPC-FL are smallest. Speaking overall, the proposed MPC-FL has the first or second places in the six performance indices, and its performances are better than others in this case. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Controller** & **Joint** & \begin{tabular}{c} **Rising Time** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Settling Time** \\ **(s)** \\ \end{tabular} & \begin{tabular}{c} **Max. Overshoot** \\ **(Rad)** \\ \end{tabular} & \begin{tabular}{c} **Input Ranges** \\ **(N-m)** \\ \end{tabular} & \
\begin{tabular}{c} **Computational** \\ **Time (s)** \\ \end{tabular} &
\begin{tabular}{c} **RMSE (Rad)** \\ \end{tabular} \\ \hline SDRE & \(1\) & \(4.4800\) & \(6.6400\) & \(0.0000\) & \((-0.9893,6.4839)\) & \(167.4830\) & \(0.3419\) \\  & \(2\) & \(4.3040\) & \(5.8200\) & \(0.0000\) & \((-0.3345,6.1237)\) & \(167.4830\) & \(0.2807\) \\ \hline MPC-TSF & \(1\) & \(110.7220\) & \(147.5750\) & \(0.0000\) & \((0.0000,5.8805)\) & \(39.4840\) & \(0.1776\) \\  & \(2\) & \(107.8150\) & \(137.2700\) & \(0.0000\) & \((0.0000,6.1647)\) & \(39.4840\) & \(0.1045\) \\ \hline LQT-FL & \(1\) & \(1.1840\) & \(2.4400\) & \(0.0250\) & \((-10.4705,18.5682)\) & \(6.4400\) & \(0.2572\) \\  & \(2\) & \(1.1360\) & \(2.4100\) & \(0.0260\) & \((-3.5592,6.3192)\) & \(6.4400\) & \(0.2543\) \\ \hline MPC-FL & \(1\) & \(3.9240\) & \(5.4250\) & \(0.0028\) & \((-0.5592,1.9739)\) & \(0.0409\) \\ (proposed) & \(2\) & \(3.7480\) & \(5.3320\) & \(0.0028\) & \((-0.1901,0.6711)\) & \(6.5200\) & \(0.0407\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparisons of control performances by applying the four control schemes without input constraints to the two-link robot arm. Figure 9: Two-link robot arm. Figure 11: Two-link robot arm by applying the four control schemes with input constraints \(\pm 6.1\) N-m: (**a**) output responses of joint 1; (**b**) output responses of joint 2; (**c**) input signals of joint 1; (**d**) input signals of joint 2. Figure 10: Two-link robot arm by applying the four control schemes without input constraints: (**a**) output responses of joint 1; (**b**) output responses of joint 2; (**c**) input signals of joint 1; (**d**) input signals of joint 2. Case D: Control with the input constraints for the two-link robot arm. Similar to case C, one reduces the input constraint range to test the capabilities of enduring the minimum range, and the ranges are selected as \(\pm\)6.1, \(\pm\)6.0, and \(\pm\)1.5/\(\pm\)0.3 N\(\cdot\)m for both joints. Regarding the range \(\pm\)6.1 N\(\cdot\)m for both joints, the results are shown in Figure 11. The results show that all control schemes can track the reference signals, where the input constraints for the MPC-TSF, LQT-FL, and SDRE are active. Besides this, the proposed MPC-FL converges faster than the others, and it has smaller overshoot. Regarding the range \(\pm\)6.0 N\(\cdot\)m for both joints, the results are shown in Figure 12. Although the range is a little smaller than the previous range, the results are much different. The results show that the input constraints for all control schemes are active, but the SDRE cannot track the reference signals in contrast to other control schemes. Furthermore, the proposed MPC-FL has less rising and settling time.