## 5 Simulation Results
### Evaluation of the Path Length Coverage  
The path length coverage of the homotopic approach is discussed through simulations in the representative scenarios. To evaluate the homotopic approach, the path length coverage interval is  
Figure 7: Path homotopy 2.  
used. It is calculated by subtracting the shortest curvature-bounded path length; that is, the Dubins path length, from the planned path length using the homotopic approach. In the simulations, the initial position and the recovery position are taken into consideration since the planned path is affected by the boundary points. The initial coordinate of the UAV is fixed at (\(-\)25,0) and the heading angle is 135\({}^{\circ}\). Moreover, the heading angle of the trajectory of the mother aircraft is separately set as 45\({}^{\circ}\), 135\({}^{\circ}\), \(-\)135\({}^{\circ}\), and \(-\)45\({}^{\circ}\). The recovery positions on the trajectories of the mother aircraft are 32.5 km and 50 km away from the initial position of the UAV, respectively. Moreover, the minimum turning radius of the UAV is set to 2 km\({}^{-1}\).  
The settings of the recovery positions are listed in Table 1 and the simulation results are shown in Figure 8. As demonstrated, the Dubins path is used as the initial curvature-bounded path. The paths are generated by using the homotopic approach, and then the path length coverage of the homotopic approach can be obtained.  
The path length coverage intervals under the different conditions are given in Table 2. For each case in the simulations, we can find that all of the expected path lengths can be covered by using a combination of the two path homotopies.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Coordinate of the Recovery Position (km)** & **Heading Angle of the Recovery Position (Degree)** \\ \hline (a) & (12.5,0) & 45 \\ (b) & (12.5,0) & 135 \\ (c) & (12.5,0) & \(-\)135 \\ (d) & (12.5,0) & \(-\)45 \\ (e) & (25,0) & 45 \\ (f) & (25,0) & 135 \\ (g) & (25,0) & \(-\)135 \\ (h) & (25,0) & \(-\)45 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Settings of the recovery positions.  
Figure 8: _Cont._  
Figure 8: Paths generated using the homotopic approach. (**a–d**) The coordinates of the recovery position are (12.5,0) and the heading angles are 45\({}^{\circ}\), 135\({}^{\circ}\), \(-\)135\({}^{\circ}\), and \(-\)45\({}^{\circ}\); (**e–h**) The coordinates of the recovery position are (25,0) and the heading angles are 45\({}^{\circ}\), 135\({}^{\circ}\), \(-\)135\({}^{\circ}\), and \(-\)45\({}^{\circ}\).  
As we can see, the path length coverage interval of path homotopy 1 ranges from zero to its maximum value, and path homotopy 2 ranges from its minimum value to positive infinity. To ensure that all of the path lengths are covered based on the two path homotopies, the maximum path length coverage value of path homotopy 1 needs to be greater than the minimum path length coverage value of path homotopy 2. Based on the geometrical relationship, we can get \(l>4\pi/[(\pi-2)\mu_{\text{max}}]\) if \(l\) is the length of the straight line segment of the Dubins path. Since the aerial recovery of the UAV swarm is considered to be a long-range mission, the above condition is satisfied so that all of the path lengths are covered based on the two path homotopies.