## 6 Simulation Test and Results Analysis
### Verification of the IBOA Trajectory Optimization  
Taking the evaluation indices of two optimization objectives as the test standard, the obtained results from IBOA, WOA, GA, HPSOBOS, and BOA were compared. In all \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Joint**} & **Velocity** & **Acceleration** & **Jerk** \\  & **(deg/s)** & **(deg/s\({}^{2}\))** & **(deg/s\({}^{3}\))** \\ \hline
1 & 600 & 2000 & 15,000 \\
2 & 600 & 2000 & 15,000 \\
3 & 600 & 2000 & 15,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Kinematic limits of a Delta robot. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Key Points** & **X (m)** & **Y (m)** & **Z (m)** \\ \hline \(p1\) & \(-0.02\) & \(0.30\) & \(-1.25\) \\ \(p2\) & \(-0.02\) & \(0.30\) & \(-1.20\) \\ \(p3\) & \(-0.02\) & \(0.12\) & \(-1.15\) \\ \(p4\) & \(-0.02\) & \(-0.12\) & \(-1.15\) \\ \(p5\) & \(-0.02\) & \(-0.30\) & \(-1.20\) \\ \(p6\) & \(-0.02\) & \(-0.30\) & \(-1.25\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Trajectory point sequence. Figure 7: NURBS curve motion parameters. (**a**) The drive arm angular displacement; (**b**) The drive arm angular velocity; (**c**) The drive arm angular acceleration; (**d**) The drive arm angular jerk. simulations, the population size and the maximum number of iterations were set to 50. The weight factors of the selected time and impact were set to 0 and 1, respectively. The simulation results are shown in Figure 8.  
Figure 8 reveals that WOA converges to 0.06 after 26 iterations, but the convergence is premature. Moreover, GA and HPSOBOA converge at the 6 th generation, while the original BOA converges at the 12 th generation, but they simply fall into the local convergence, so that their optimization results do not meet the requirements. The lowest optimization results were achieved from the IBOA, indicating the high optimization accuracy of this algorithm. More specifically, the IBOA reaches an optimized value of 0.03 after 32 steps, indicating that IBOA can achieve reasonable optimization results. Table 7 shows the optimization results of different algorithms for the motion time. It was found that the IBOA reduces the motion time by 16.2%, while WOA, GA, HPSOBOA, and BOA reduce the robot motion time by 14.1%, 15.5%, 12.7%, and 12.7%, respectively. The optimization results for the second optimization objective are shown in Table 8. It can be observed that the best optimization effect on the jerk was 87.6%, which was achieved using the IBOA. The obtained results demonstrate that the IBOA algorithm outperformed the other algorithms in terms of the time-jerk trajectory planning of the robot. The motion parameters of the drive arm optimized by IBOA are shown in Figure 9. It can be observed that the optimized motion time for IBOA decreases from 0.28 s to 0.2346 s. Moreover, the maximum jerk values of the driving arm before and after optimization were \(2.5\times 10^{3}\) deg/s\({}^{3}\) and \(0.3089\times 10^{3}\) deg/s\({}^{3}\), respectively. It is found that the displacement curve changes after optimization. Due to the change in the position of the control point, the trajectories before and after optimization are different, and therefore the displacement curve changes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **No Optimization** & **Optimization of the Algorithm** \\ \hline WOA & 0.28 & 0.2404 \\ GA & 0.28 & 0.2365 \\ BOA & 0.28 & 0.2442 \\ HPSOBOA & 0.28 & 0.2327 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results for time objectives before and after optimization. Figure 8: Distribution of convergence rates for different algorithms. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **No Optimization** & **Optimization of the Algorithm** \\ \hline WOA & 2.5 & 1.4472 \\ GA & 2.5 & 2.4435 \\ BOA & 2.5 & 1.8683 \\ HPSOBOA & 2.5 & 1.8245 \\ IBOA & 2.5 & 0.3089 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results for jerk objectives before and after optimization.