## 6 Conclusions  
In this paper, the multi-UAV aerial recovery problem with a single mother aircraft is examined. A recovery planning framework is proposed to establish the coupling mechanism between the scheduling layer and the path planning layer. In the recovery planning framework, the outputs of the scheduling layer are delivered to the path planning layer to design paths. Conversely, the path estimation stage in the path planning layer is treated as the feedback input to the scheduling layer to obtain the maximum recovery time window for each UAV. A GA is introduced to optimize the recovery sequence. Based on the GA, the recovery planning framework can realize efficient and precise scheduling for the UAV swarm. Moreover, a homotopic approach, which searches paths of the expected length within a certain homotopy, is proposed for the path planning of the UAV swarm.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Objective Value 1**} & \multicolumn{3}{c}{**Non-Maneuvering Mother Aircraft**} & \multicolumn{3}{c}{**Maneuvering Mother Aircraft**} \\ \cline{2-7}  & \(\mathbf{N=10}\) & \(\mathbf{N=20}\) & \(\mathbf{N=30}\) & \(\mathbf{N=10}\) & \(\mathbf{N=20}\) & \(\mathbf{N=30}\) \\ \hline Mean & 52.56 & 72.82 & 71.50 & 72.66 & 88.65 & 90.77 \\ St Dev & 0 & 0 & 1.73 & 0 & 0.35 & 2.29 \\ Best & 52.56 & 72.82 & 73.66 & 72.66 & 89.22 & 92.97 \\ \hline \hline \end{tabular}  
* All of the objective values should be multiplied by \(10^{-2}\).  
\end{table}
Table 6: Results of the Monte Carlo study.  
Figure 12: Distances from the UAVs to the mother aircraft. (**a**) 10 UAVs with a non-maneuvering mother aircraft; (**b**) 10 UAVs with a maneuvering mother aircraft; (**c**) 20 UAVs with a non-maneuvering mother aircraft; (**d**) 20 UAVs with a maneuvering mother aircraft; (**e**) 30 UAVs with a non-maneuvering mother aircraft; (**f**) 30 UAVs with a maneuvering mother aircraft.  
Two path homotopy construction patterns based on deforming the line segment of the Dubins path have been presented in detail. The simulation results show that all of the paths with expected lengths are covered in a long-range aerial recovery by using the homotopic path planning approach. Then, the recovery scheduling process is verified in a representative scenario. Furthermore, the simulation results validate the overall performance of the recovery planning framework. In conclusion, this framework can ensure that the mother aircraft recovers the most UAVs in the most efficient manner despite the quantity change of the UAVs.  
Future work based on the findings from this research will include an investigation on extending the path homotopy construction patterns to a generalized form. By doing this, the proposed homotopic path planning approach can be applied to more complicated situations. Then, aerial recovery with multiple mother aircrafts will be considered to improve the effectiveness of the UAV swarms even further. Moreover, it is necessary to focus on the aerial recovery problem with the unknown mother aircraft trajectories in advance, since the motions of the mother aircrafts are difficult to obtain in the real world. To make the summary and conclusion more sufficient and convincing, experiments on a real system which employs the aerial recovery planning framework can also be considered in the future.  
Methodology, Y.L. and W.Y.; software, Y.L. and W.Y.; validation, Y.L., S.X., and J.Z.; formal analysis, Y.L. and S.X.; investigation, Y.L. and N.Q.; writing--Original draft preparation, Y.L. and J.Z.; writing--Review and editing, W.Y. and N.Q. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors would like to thank the editors and reviewers for their hard work and valuable suggestions that improved this paper substantially.  
The authors declare no conflict of interest.