## 3 Experiment and Application
### Engineering Application: Multi-UAVs Cooperative Path Planning
#### 3.2.4 Simulation Validation
The planned paths and voyages are close and can effectively avoid obstacles. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Range (km)** & **Flight Time (s)** \\ \hline
1 & 117.7185 & [1961.975, 2942.9626] \\
2 & 99.8391 & [1663.9855, 2495.9783] \\
3 & 106.0902 & [1768.1699, 2652.2548] \\
4 & 97.3108 & [1621.8469, 2432.7704] \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Range and flight time of each aircraft (case 2). Figure 9: The result of case 2. (**a**) 3D UAV cooperative path map; (**b**) Contour map of cooperative path; (**c**) Convergence curve of each UAV; (**d**) Path cost convergence curve. **Case 4:** Eight UAVs fly to four target points. The coordinates of the starting and the target points are listed in Table 15. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Range (km)** & **Flight Time(s)** \\ \hline
1 & 99.2486 & [1654.1433, 2481.215] \\
2 & 104.8832 & [1748.0538, 2622.0807] \\
3 & 115.4467 & [1924.111, 2886.1664] \\
4 & 110.7951 & [1846.5853, 2769.8779] \\
5 & 104.5509 & [1742.5156, 2613.7734] \\
6 & 100.002 & [1666.7006, 2500.0509] \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Range and flight time of each aircraft (case 3). Figure 10: The result of case 3. (**a**) 3D UAV cooperative path map; (**b**) contour map of cooperative path; (**c**) convergence curve of each UAV; (**d**) path cost convergence curve. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Start Point** & **Target Point** \\ \hline
1 & (1,1,0) & (99,10,70) \\
2 & (1,20,0) & (99,18,70) \\
3 & (1,40,0) & (99,30,70) \\
4 & (1,60,0) & (99,40,70) \\
5 & (1,75,0) & (99,70,70) \\
6 & (1,90,0) & (99,90,70) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Coordinates of the starting and target points of all UAVs (case 3). The result of case 4 was obtained as depicted in Figure 11. Through a simulation, the flight time intervals (unit: s) and range (unit: km) of each UAV are presented in Table 16. The time intersection is [1922.8382, 2493.2072]. The planned paths and voyages are close and can effectively avoid obstacles. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Range (km)** & **Flight Time (s)** \\ \hline
1 & 99.7531 & [1662.5517, 2493.8276] \\
2 & 100.9139 & [1681.8984, 252.8476] \\
3 & 115.3703 & [1922.8382, 2884.2573] \\
4 & 101.3747 & [1689.5775, 2534.3663] \\
5 & 101.0788 & [1684.6466, 2526.97] \\
6 & 102.4978 & [1708.2963, 2562.4445] \\
7 & 103.6654 & [1727.7567, 2591.635] \\
8 & 100.3576 & [1672.6264, 2508.9396] \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Range and flight time of each aircraft (case 4). Figure 11: The result of case 4. (**a**) 3D UAV cooperative path map; (**b**) contour map of cooperative path; (**c**) convergence curve of each UAV; (**d**) path cost convergence curve. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Start Point** & **Target Point** \\ \hline
1 & (1,1,0) & (99,10,70) \\
2 & (1,15,0) & (99,10,70) \\
3 & (1,30,0) & (99,32,70) \\
4 & (1,40,0) & (99,32,70) \\
5 & (1,55,0) & (99,68,70) \\
6 & (1,70,0) & (99,68,70) \\
7 & (1,80,0) & (99,85,70) \\
8 & (1,95,0) & (99,85,70) \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Coordinates of the starting and target points of all UAVs (case 4). The performance of the ans platryhynchos optimizer is further verified in this work. In particular, the PSO, DE, and GWO algorithms, the improved GWO based on chaos thought (HGWO) [71], and the improved GWO with dynamic weighted average combined with the static average method (IGWO) [50] are used to implement multi-cooperative path planning in different cases. The simulation result was subsequently compared with the algorithms in this study to further verify the effectiveness of the novel algorithm. Each algorithm was run 30 times to avoid the influence of randomness, with the number of populations set to 50 and the number of iterations set to 100.