# Time-Optimal Trajectory Planning for Woodworking Manipulators Using an Improved PSO Algorithm
## 3 Experiments and Results
### Time Optimization Experiment of Woodworking Manipulator
Figure 10: Time-optimal 3-5-3 piecewise polynomial trajectory planning process based on GoldS-PSO. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **Joint** & \(T_{\text{total}}\) & \(t_{\text{1}}\) & \(t_{\text{2}}\) & \(t_{\text{3}}\) \\ \hline \multirow{4}{*}{PSO} & 1 & 7.2210 & 1.7301 & 3.5905 & 1.9008 \\  & 2 & 7.4472 & 2.0576 & 2.2782 & 3.0889 \\  & 3 & 7.0339 & 2.6648 & 2.3479 & 2.0272 \\  & 4 & 6.9658 & 2.0908 & 2.5381 & 2.3368 \\ \hline \multirow{4}{*}{Gold-SA} & 1 & 7.1210 & 1.9488 & 3.1077 & 2.0645 \\  & 2 & 7.5203 & 2.2059 & 2.2647 & 3.0497 \\ \cline{1-1}  & 3 & 6.9526 & 2.4483 & 2.4637 & 2.0406 \\ \cline{1-1}  & 4 & 7.1518 & 2.3862 & 2.4097 & 2.3560 \\ \hline \multirow{4}{*}{GoldS-PSO} & 1 & **7.0672** & 2.1107 & **2.9540** & 2.0025 \\  & 2 & **7.2177** & 2.0009 & 2.6121 & **2.6047** \\ \cline{1-1}  & 3 & **6.8970** & **2.3189** & 2.5846 & 1.9935 \\ \cline{1-1}  & 4 & **6.8713** & 2.0205 & 2.5096 & 2.3412 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Interpolation time after optimizing each joint. curve for each joint is depicted in Figure 11. It better reflects the performance of each algorithm in the optimization process. Figure 11 shows that GoldS-PSO had a faster rate of convergence than PSO. Out of all the optimization processes, the iterations of GoldS-PSO were within the 60 generations to complete global convergence, whereas PSO required at least the 120 generations. Compared to Gold-SA, GoldS-PSO's rate of convergence did not significantly improve; however, its ability to jump out of local extreme values was stronger and its convergence accuracy was the highest. Although Gold-SA achieved global convergence within the 50 generations, it was hard to adjust the evolutionary direction in the middle or later iteration and easily fell into the dilemma of local extremum. Even in the optimization processes of joints 2 and 4, Gold-SA's convergence accuracy was lower than that of PSO. In a word, GoldS-PSO has a good convergence rate and is not easily trapped in local convergence. It balances the rapidity and accuracy of convergence, which indicates better performance for such engineering optimization problems. Subsequently, we selected the maximum interpolation time of each joint segment as the final \(t_{1}\), \(t_{2}\), and \(t_{3}\). Then, the total interpolation time obtained after optimizing the PSO, Gold-SA, and  
Figure 11: Four joints’ iterative process by using PSO, Gold-SA, and GoldS-PSO. Golds-PSO was 9.3442 s, 8.6057 s, and 7.8776 s, respectively. The GoldS-PSO algorithm reduced time consumption by 47.483% compared to before optimization, 15.695% compared to PSO optimization, and 8.461% compared to Gold-SA optimization, demonstrating the effectiveness and superiority of the algorithm. The final planning times \(t_{1}=2.3189\) s, \(t_{2}=2.9540\) s, and \(t_{3}=2.6047\) s of each segment was substituted into the 3-5-3 piecewise polynomial interpolation function in Section 2.2 to obtain the displacement, velocity, and acceleration simulation curves of each joint.