## 1 Introduction  
High autonomy and large-scale swarms are two development trends of unmanned aerial vehicles (UAVs) [1,2]. However, UAV swarms can fly into enemy territory and sometimes they cannot safely return by themselves. In addition, their range is not sufficient to execute long-range missions. To overcome these shortcomings and to maximize the advantages of autonomous UAV swarms, especially to reduce the service cost and improve the reuse rate and efficiency, aerial recovery technology is important for UAV swarms. For insurance, the UAV swarm in the Gremlins program of the Defense Advanced Research Projects Agency has an operation mode called \"launch-recovery-relaunch\" [3]. In the future, this operation mode will become the mainstream for UAV swarms. In this operation mode, a mother aircraft is employed to deliver the UAVs to the designated locations for launch. Then, the mother aircraft recovers them after completing the missions and prepares to deliver them to the next locations. Compared with other UAV recovery methods on land or sea, aerial recovery has the greatest value, which can greatly expand the endurance and service time and comprehensively improve the flexibility and effectiveness of the UAV swarms. However, there exist technical challenges because aerial recovery technology involves a variety of complex problems, including scheduling, path planning, rendezvous, and acquisition. In recent years, some researchers have studied aerial recovery technologies for UAVs [4; 5; 6; 7]. These studies focus on modeling the UAV recovery system and rendezvous guidance laws for the mother aircraft and the UAVs. The commonality between these works is that they focus on recovering a single UAV by the mother aircraft. With respect to the multi-UAV aerial recovery problem, it consists of two sub-problems: scheduling and path planning. Considering the dynamic constraints of UAVs and the recovery capability of the mother aircraft, scheduling and path planning of the multi-UAV aerial recovery problem will have high complexity and involve strong state information coupling. The recovery schedule and the recovery position series in the scheduling results will directly influence the performance of the path planning. Similarly, the rough path estimation can influence the scheduling process. For the scheduling problem in the aerial recovery process, it has similarities with the traveling salesman problem with time windows [8; 9], the vehicles routing problem with time windows [10], and the multi-robot recharging problem [11], which is considered from the perspective of combinatorial optimization. Many exact algorithms, such as the branch-and-bound algorithm [12; 13], the cutting plane algorithm [14; 15], integer linear programming [16; 17], and dynamic programming [18; 19], have been studied and applied to solve these problems at a small scale. However, the solution quantity will increase exponentially in large-scale combinatorial optimization problems. To solve such problems efficiently, many heuristic algorithms have been studied, including the genetic algorithm (GA) [20; 21; 22], the ant colony algorithm [23; 24], the simulated annealing algorithm [25; 26; 27], and the tabu search algorithm [28; 29]. Another major concern in the aerial recovery process is the usage of path planning to generate a path with an expected length for each UAV based on the scheduling results. Some researchers have considered path planning with an expected length. [PERSON] et al. studied the problem of intercepting a moving target with a Dubins vehicle at a given time, and they proposed three approaches, including a turning radius variation approach, a path deformation approach, and a final orientation rotation approach, to generate paths with an expected length [30]. [PERSON] et al. proposed a line segment insertion approach to generate an ideal path for the task assignment problem with certain timing constraints [31]. [PERSON] et al.