## 4 Simulation Study
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Controller Parameters.** & **Symbol** & **APPSMC** & **ASMC1** & **ASMC2** & **SMC1** & **SMC2** \\ \hline Initial error bound & \(p_{\mathrm{ori}},i=1,2\) & 1 & N/A & N/A & N/A & N/A \\ Steady state error bound & \(p_{\mathrm{ori}},i=1,2\) & 0.01 & N/A & N/A & N/A & N/A \\ Convergence rate & \(l_{i},i=1,2\) & 0.5 & N/A & N/A & N/A & N/A \\ coefficient & \(\delta_{i},i=1,2\) & 25 & N/A & N/A & N/A & N/A \\ Parameter of the sliding & \(\mathbf{\lambda}\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}0.5&0\\ 0&0.5\\ \end{array}\right]\) \\ Robust gain & \(\mathbf{\gamma}\) & \(\left[\begin{array}{cc}5&0\\ 0&5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}5&0\\ 0&5\\ \end{array}\right]\) & \(\left[\begin{array}{cc}5&0\\ 0&0\\ 0&0.5\\ \end{array}\right]\) \\ Gain of adaptive law (32) & \(\eta_{1}\) & 0.01 & 0.01 & 10 & N/A & N/A \\ Gain of adaptive law (33) & \(\eta_{2}\) & 0.01 & 0.01 & 10 & N/A & N/A \\ Gain of adaptive law (34) & \(\eta_{3}\) & 0.01 & 0.01 & 10 & N/A & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the proposed APPSMC controller. Choosing SMC2 and ASMC2 for comparison, system tracking performance using APPSMC is provided in Figures 8 and 9. It can be seen that system dynamic can track the command trajectory after several milliseconds. Furthermore, it can be found that blue, red and purple line almost overlap with each other after time = 15 ms. Figures 10 and 11 depict trajectory tracking performances using APPSMC, SMC2, and ASMC2. Parameters of the controllers are listed in Table 3. In Figures 10 and 11, blue and green solid line are upper and lower bound described by performance bound function. Red dotted line, purple dotted line and black dotted line depict tracking error performance of SMC2, ASMC2, and APPSMC, respectively. It can be seen from Figures 10 and 11 that all three control methods can achieve trajectory tracking target while only APPSMC and SMC1 can maintain tracking error to be within the performance bound. It shall be pointed out that the great tracking error performance of SMC1 is derived on the basis that all system parameters are well known. Besides, the robust gain in SMC2 is chosen to be very big and the big robust gain will result in severe chattering phenomenon. At the same time, APPSMC can still  
Figure 8: X axis trajectory tracking comparisons of SMC2, ASMC2, and APPSMC. Figure 7: Y axis tracking error comparisons of SMC2, SMC1, ASMC2, and ASMC1. Figure 9: Y axis trajectory tracking comparisons of SMC2, ASMC2, APPSMC. achieve desired error performance using adaptive laws to tune controller parameters even if system parameters are not exactly known and there are no severe chattering in control force for the fact that the robust gain in APPSMC is very small. Table 4 compares system mean absolute errors and root mean square errors of APPSMC, ASMC1, ASMC2, SMC1, and SMC2. It can be concluded from Table 4 that the proposed control method can achieve excellent tracking error performance compared with conventional ASMC. SMC2 have a better tracking error performance on the basis that all system parameters are well known. Besides, the robust gain in SMC2 is larger than the robust gain used in APPSMC and the big robust gain will result in severe chattering in control forces. Figure 16 depicts control forces in the X and Y direction. It can be seen from above figure that there are some slight chattering in the control forces and the chattering can be further reduced by substituting sign function in the control force with saturated function.