## 5 Experimental Results
### Experimental Results  
The real-time control experiments are conducted to verify the performance of the proposed method. Figures 10 and 11, respectively, show the star position error trajectories using the GWO and IWO algorithms. The controller parameters are obtained from the optimization process in the simulations. Figures 10a and 11a illustrate the tracking errors on the X-axis, Figures 10b and 11b show the tracking errors of the master and slave Y-axes together, and Figures 10c and 11c show the synchronization errors, which are within 20 \(\upmu\) \begin{table}
\begin{tabular}{c c c c c c c} \hline value & \(K_{e}\) & \(\mu\) & \(K_{de}\) & \(\lambda\) & \(\beta\) & \(\alpha\) \\ \hline PSO & 1.8931 & 0.1097 & 0.4520 & 1.5940 & 1.9000 & 0.9726 \\ IWO & 1.8237 & 0.2757 & 1.8330 & 1.6256 & 1.4351 & 1.1889 \\ BBO & 1.8980 & 0.1006 & 1.8914 & 1.6927 & 1.9000 & 0.6967 \\ GWO & 1.3132 & 0.1285 & 1.8970 & 1.6103 & 1.8326 & 1.0344 \\ \hline value & \(K_{\mathrm{cl}}\) & \(\mu_{1}\) & \(K_{dc1}\) & \(\lambda_{1}\) & \(\beta_{1}\) & \(\alpha_{1}\) \\ \hline PSO & 0.4488 & 0.1000 & 1.9000 & 1.4877 & 1.9000 & 0.8245 \\ IWO & 1.8866 & 0.1012 & 1.6841 & 1.5311 & 1.8837 & 0.7184 \\ BBO & 1.7297 & 0.4819 & 0.7888 & 1.4647 & 1.6594 & 0.8007 \\ GWO & 1.8825 & 0.1085 & 1.4989 & 1.3023 & 1.8645 & 1.4217 \\ \hline value & \(K_{\mathrm{cl}}\) & \(\mu_{2}\) & \(K_{dc2}\) & \(\lambda_{2}\) & \(\beta_{2}\) & \(\alpha_{2}\) \\ \hline PSO & 1.2828 & 0.142 & 1.9000 & 1.4889 & 1.4156 & 0.6118 \\ IWO & 1.8302 & 0.1000 & 1.7325 & 1.5320 & 1.9000 & 0.7267 \\ BBO & 1.8914 & 0.4146 & 1.1917 & 1.4872 & 1.3578 & 0.6606 \\ GWO & 1.8992 & 0.2610 & 1.7539 & 1.3435 & 1.8493 & 1.3241 \\ \hline \end{tabular}
\end{table} 
Table 4: Optimal scaling factor parameters for all trajectory types. \begin{table}
\begin{tabular}{c c c c} \hline
**Method** & **ATE (\(\upmu\))** & **TESD (\(\upmu\))** & **ITAE** \\ \hline \multicolumn{4}{c}{Circle} \\ PSO & 1.4719 & 3.2174 & 728.4400 \\ IWO & 0.9100 & 1.6635 & 689.4438 \\ BBO & 1.5367 & 3.0456 & 732.9247 \\ GWO & 1.1919 & 1.4536 & 732.1176 \\ \hline \multicolumn{4}{c}{Heart} \\ PSO & 1.3767 & 7.3135 & 687.1533 \\ IWO & 1.2314 & 5.5600 & 662.0899 \\ BBO & 1.4962 & 7.7701 & 706.2485 \\ GWO & 1.2854 & 4.7161 & 716.1556 \\ \hline \multicolumn{4}{c}{Star} \\ PSO & 2.9381 & 9.7067 & 758.2886 \\ IWO & 2.9195 & 6.0844 & 732.7888 \\ BBO & 2.9834 & 9.8334 & 822.2761 \\ GWO & 2.8224 & 6.8502 & 800.8768 \\ \hline \multicolumn{4}{c}{Bow} \\ PSO & 6.2185 & 30.6246 & 726.3240 \\ IWO & 2.0803 & 12.5113 & 686.4565 \\ BBO & 5.9139 & 27.0573 & 778.0170 \\ GWO & 5.0548 & 18.6534 & 795.2960 \\ \hline \multicolumn{4}{c}{Average} \\ PSO & 3.0013 & 12.7156 & 725.0515 \\ IWO & 1.7852 & 6.4548 & 692.6948 \\ BBO & 2.9826 & 11.9266 & 759.8666 \\ GWO & 2.5886 & 7.9183 & 761.1115 \\ \hline \end{tabular}
\end{table} 
Table 5: The simulation results of all trajectories. and decay gradually. Figures 10d and 11d demonstrate the contour trajectories. The smooth trajectories and small steady-state error can be seen in the results. The circle contour tracking results using the GWO and IWO algorithms are shown in Figures 12 and 13, respectively. Figures 12c and 13c show the synchronization errors, which are always within 40 \(\upmu\)m. Figures 14 and 15 are the position error trajectories of the bow tie with GWO and IWO, respectively. Figures 14c and 15c show that the synchronization errors are within 12 \(\upmu\)m and 15 \(\upmu\)m, respectively. The heart contour tracking results are shown in Figures 16 and 17. The synchronization errors are shown in Figures 16c and 17c, and are within 40 \(\upmu\)m and 15 \(\upmu\)m, respectively. The ATE and TESD tracking indices for the star, circle, bow, and heart contours are displayed in Table 6. From the average ATE and TESD indices in the experimental results, it can be seen that GWO exhibited better tracking performance than IWO. Although IWO has better tracking performance than the other algorithms in the simulations, the experimental results are worse than the others.