## 5 Simulation Results
### Overall Performance of the Recovery Planning Framework  
In this section, simulations are carried out in a scenario that is 200 km \(\times\) 200 km to validate the overall performance of the recovery planning framework. The size of the UAV swarm is set to 10,  
Figure 10: Recovery time of UAV 1, UAV 2, and UAV 3 when recovering all of them. Figure 9: Simulation results of the recovery scheduling. **(a)** Recover UAV 1 individually; **(b)** Recover UAV 2 individually; **(c)** Recover UAV 3 individually; **(d)** Recover UAV 1, UAV 2, and UAV 3. 20, and 30, respectively. The settings of the UAVs are listed in Table 4. Similarly, the velocity of each UAV is 100 m/s and the curvature bound is 2 km\({}^{-1}\) in these simulations. The trajectory of the mother aircraft consists of straight-line segments and arc segments. The sample distance of the mother aircraft trajectory is 0.1 km. The flying velocity of the mother aircraft is 150 m/s and the recovery time cost \(\tau\) of each UAV is 60 s. The coefficients \(\alpha\) and \(\beta\) are set as 0.5 since they are equally important. When considering the GA that is used in the scheduling problem, the parameters are given in Table 5.  
Figure 11 presents the simulation results of the recovery planning framework. As demonstrated in Figure 11a, four UAVs do not have recovery time windows when the mother aircraft is a non-maneuvering target, because they are too close to the initial position of the mother aircraft or they have unsuitable heading angles. The rest of the UAVs can be recovered by the mother aircraft successfully. In Figure 11b, all 10 UAVs can be recovered when the mother aircraft is a maneuvering target. Based on the recovery planning framework, the optimal recovery sequence, along with the recovery position series and the recovery path lengths, are obtained. The flyable paths of the UAV swarm are designed and generated using the homotopic path planning approach, where the path homotopy is selected according to the geometrical relationship. Figure 11c,d shows the simulation results when 10 new UAVs join the UAV swarm. It can be observed that, as before, the recovery \begin{table}
\begin{tabular}{c c c} \hline
**UAV Index** & **Initial Position (km)** & **Initial Direction (degree)** \\ \hline
1 & (112.14,8.81) & \(-\)126.32 \\  
(-\)68.80 \\
19 & (178.55,82.65) & 130.13 \\
20 & (108.69,154.81) & \(-\)31.60 \\
21 & (145.63,34.32) & 90.13 \\
22 & (194.31,109.47) & \(-\)104.30 \\
23 & (34.63,180.37) & 23.16 \\
24 & (173.51,179.49) & 98.03 \\
25 & (38.05,27.44) & \(-\)62.11 \\
26 & (199.72,62.28) & 116.80 \\
27 & (95.72,187.33) & 41.86 \\
28 & (45.29,171.27) & \(-\)47.51 \\
29 & (78.91,66.34) & \(-\)42.99 \\
30 & (33.04,150.45) & 2.809 \\ \hline \end{tabular}
\end{table} 
Table 4: Initial settings of UAVs. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & \(N_{\mathbf{g}}\) & \(N_{\mathbf{p}}\) & \(N_{\mathbf{e}}\) & \(P_{\mathbf{c}}\) & \(P_{\mathbf{m}}\) \\ \hline Value & 3000 & 100 & 3 & 0.95 & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters of the GA.  
planning framework performs well. Figure 11e,f illustrates the performance of the recovery planning framework when 10 new UAVs once again join the UAV swarm. As shown in Figure 11e, 10 UAVs do not have recovery time windows. Moreover, 19 of the UAVs can be recovered and 1 UAV cannot be recovered. In Figure 11f, all of the UAVs in the scenario can be recovered by the maneuvering mother aircraft and the paths with the expected lengths are designed for the UAVs. As demonstrated, the recovery planning framework performs well even though the size of the UAV swarm increases. Figure 12 shows the distances from the UAVs to the mother aircraft in these simulations. As the figure shows, the results validate the effectiveness of the homotopic path planning approach since the flyable paths of the UAVs can be designed and generated for recovery. In addition, a Monte Carlo study is carried out for further validation.