## 5 Simulation Verification
### Comparative Experiment  
Through comparative experiments, it has been verified that the proposed method in this paper can achieve shorter robot-arm-running time compared to other methods. The comparative experiments all use the quintic NURBS curves to fit the path points of the manipulator, with the difference being that a single-objective genetic algorithm is used to find the optimal time based on both uniform and random paths. Specifically, uniform path refers to a path where the points between the start and final points increase or decrease uniformly. Random path, on the other hand, consists of arbitrary points that satisfy the constraints given in Table 7. The experimental results are shown in Table 9: \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Path Points**} & \multicolumn{6}{c}{**Joint**} & \multirow{2}{*}{**t\({}_{\text{i}}\) (s)**} \\ \cline{2-2} \cline{5-8}  & **1** & **2** & **3** & **4** & **5** & **6** & \\ \hline
1 & 10.000 & \(-\)10.000 & \(-\)30.000 & \(-\)25.000 & 20.000 & 0.000 & 0.000 \\
2 & 6.675 & \(-\)6.139 & \(-\)16.259 & \(-\)19.999 & 11.178 & 13.371 & 1.325 \\
3 & 3.389 & \(-\)2.235 & \(-\)3.317 & \(-\)15.010 & 2.339 & 26.820 & 1.802 \\
4 & 0.024 & 1.507 & 9.819 & \(-\)9.995 & \(-\)6.719 & 39.981 & 2.271 \\
5 & \(-\)3.298 & 5.546 & 23.202 & \(-\)5.010 & \(-\)15.416 & 53.244 & 2.720 \\
6 & \(-\)6.481 & 9.328 & 36.681 & 0.334 & \(-\)23.942 & 66.927 & 3.171 \\
7 & \(-\)9.894 & 13.365 & 49.564 & 5.021 & \(-\)33.380 & 80.028 & 3.645 \\
8 & \(-\)13.229 & 17.044 & 63.336 & 9.922 & \(-\)42.085 & 93.406 & 4.127 \\
9 & \(-\)16.659 & 21.081 & 76.673 & 15.034 & \(-\)51.122 & 106.819 & 4.598 \\
10 & \(-\)20.000 & 25.000 & 90.000 & 20.000 & \(-\)60.000 & 120.000 & 5.899 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Time-optimal path and time. Figure 6: Trajectory planning curves charts: (**a**) angular displacement chart; (**b**) angular velocity chart; (**c**) angular acceleration chart; (**d**) angular jerk chart. According to the data in Table 9, using the PUMA560 model generated from Figure 5, three trajectory planning images were plotted, as shown in Figure 7. The angular displacement, angular velocity, angular acceleration, and angular jerk curves for the random path are plotted in Figure 8. Similarly, the curves for the uniform path are plotted in Figure 9.  
According to the data in Table 9, it can be observed that the new time-optimal trajectory planning method proposed in this paper reduces the total running time by 46.23% compared to the traditional method of random path (or fixed path) and by 19.26% compared to the uniform path method. This effectively improves the efficiency of the manipulator. The four images in Figure 6 indicate that the obtained curves satisfy the kinematic constraints in Table 7, and the curves are uniform and continuous, demonstrating the effectiveness of the generated trajectory. By comparing the three trajectory planning methods in Figure 7, it can be observed that the proposed method passes through the start and final points, does not encounter singular points, and has a similar running trajectory to the uniform path method (which has a shorter running trajectory), demonstrating the effectiveness and rationality of the proposed method. In conclusion, the new time-optimal trajectory planning method proposed in this paper has a positive significance compared to conventional methods. To verify the search capability of the improved NSGA-II, trajectory planning for the manipulator was performed using NSGA-II, MOPSO, MOEA/D, SPEA2, KNSGA-II [25] and AONSGA-II [26]. The resulting running times for the manipulator optimization using different algorithms are shown in Table 10. Additionally, trajectory planning curve graphs were plotted for each algorithm, as shown in Figures 10-13.