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. Meyer 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]. Schumacher et al. proposed a line segment insertion approach to generate an ideal path for the task assignment problem with certain timing constraints [
31]. Yao et al. proposed a homotopy-based approach to design an expected path by deforming specific Dubins paths for curvature-bounded vehicles [
32]. Sun et al. presented a predictive-control-based algorithm to generate and track the path based on the requirements for the security and the path length for an unmanned surface vehicle [
33]. In more general cases, Stodola proposed a smooth algorithm to connect a series of waypoints and generate a feasible path for UAVs in reconnaissance missions [
34]. Shao et al. proposed a path planning method based on the particle swarm optimization algorithm on three dimensional UAV formation tasks [
35]. Moreover, Thanh et al. solved the collision avoidance problem based on the combination of geometrical constraints and kinematics equations during the path planning process [
36,
37]. Cong et al. proposed a formation control algorithm based on the backstepping design method and a finite-time attitude tracking law to deal with the collision avoidance problem in path planning [
38].
Although the existing works are effective in different situations, the aerial recovery problem of a UAV swarm has, to the authors’ knowledge, not yet been researched. In this problem, the coupling mechanism between the scheduling and the path planning is bidirectional and it has not been elaborated. For recovery scheduling, an appropriate method should be proposed. Moreover, an effective path planning approach needs to be proposed and adopted to design the flight paths for the UAV swarm.
To address the above issues, we studied the aerial recovery problem of a UAV swarm by one mother aircraft. To describe the coupling mechanism between the scheduling process and the path planning process, a recovery planning framework is proposed. In the recovery planning framework, the outputs of the scheduling are delivered to the path planning to design the paths. Conversely, the intermediate results in path planning are treated as the feedback input for scheduling. A GA is employed to resolve the scheduling problem and generate the recovery position series and the recovery path lengths owing to its convenience. Moreover, a homotopic approach is proposed for the path planning problem based on two different path homotopies. To validate the performance of the homotopic path planning approach and the recovery planning framework, a sufficient number of simulations for stochastic scenarios were conducted.
This paper is structured as follows. The preliminaries and the models of the aerial recovery problem are presented in
Section 2. The recovery planning framework is proposed and introduced in
Section 3.
Section 4 presents the details of the genetic algorithm and the homotopic path planning approach in the recovery planning framework.
Section 5 examines the proposed approaches and presents the overall performance of the recovery planning framework. Finally,
Section 6 delivers the conclusions from this investigation and it describes our future research.
4. Methodology
Based on the recovery planning framework, the GA is adopted to find the optimal recovery sequence, and a homotopic approach is proposed for path planning with the expected length. The details of the methods are presented and discussed in this section.
4.1. Genetic Algorithm
In this section, the GA is presented based on the specialty of the recovery scheduling problem. The GA begins by creating an original population based on the encoding method. Then the elite mechanism, the crossover operation, and the mutation operation are carried out to generate a new population. The execution process is terminated when reaching a maximum generation number
or achieving the optimal solution. Herein, the optimal solution in the last generation is the expected recovery sequence. The flow chart of the GA is demonstrated in
Figure 3.
The core components of the GA that is employed in the recovery scheduling problem are explained in detail as follows:
Encoding. To accomplish the genetic operation, an effective representation of the chromosomes must be proposed. As discussed in the second section, each chromosome can be directly encoded by a recovery sequence . The encoding method is, to a certain extent, similar to the path representation in the traveling salesman problem.
Initialization. In the GA, the searching space, which is called the population, is the collection of recovery sequences. Utilizing the encoding method, the recovery sequences in the original population are randomly created to guarantee the diversity.
Objective evaluation. During the aerial recovery process, the UAVs need to keep a safe distance between each other to avoid collisions. The fitness value of each recovery sequence can be obtained from Equation (10) if the corresponding trajectories will not cause collisions. Conversely, the recovery sequences that will cause collisions are penalized with the “death penalty”, i.e., they are given a fitness value of −1. Thus, these invalid recovery sequences are eliminated during the evolutionary process.
Elitism. In each generation, the best recovery sequences form the elite group, and the elite group is copied to the next generation without any other genetic operations. By applying the elite mechanism, the good recovery sequences are reserved during the evolutionary process so that the optimal recovery sequence can be obtained faster.
Crossover. To perform the crossover operation, two recovery sequences are first selected based on the roulette wheel operator. The order crossover operator (OX1) is used to carry out the crossover operation due to its convenience [
45] and the two recovery sequences are applied with the crossover operation, with a relatively high probability
.
Figure 4 shows an example of the crossover operation. During the crossover operation, a sub-sequence is selected from one of the recovery sequences and it is copied to the same location of the offspring. Then, the other recovery serial numbers are filled with the same order as they appear in another recovery sequence starting from the second crossover point. The other offspring is created through the same process.
Mutation. The exchange mutation operator (EM) is selected to perform the mutation operation and each recovery serial number is affected by the mutation operator with a low probability
. As demonstrated in
Figure 5, the mutated recovery serial number swaps its position with another randomly selected recovery serial number in the same recovery sequence.
4.2. Homotopic Path Planning Approach
In this section, a homotopic approach is proposed for planning paths with an expected length. Two path homotopy construction patterns are proposed and they are named as path homotopy 1 and path homotopy 2. In this investigation, the aerial recovery of the UAV swarm is considered to be a long-range mission. As a result, the Dubins path, which connects the initial position and recovery position of a UAV, is generated in stage 1 of the path planning layer and it contains both line and arc segments. Thus, we can only re-plan the line segment to generate the curvature-bounded paths with an expected length.
Suppose that are two curvature-bounded paths that connect and . The continuous function can be defined as a single-layer path homotopy if the following conditions hold:
.
and .
is the space of all curvature-bounded paths between and .
Thus, the output of the function
indicates a curvature-bounded path function within the path homotopy, and it can be given by
where
is the mapping from the homotopic description to the path
. The role of the homotopic description is to reserve the boundary points and to use only one independent parameter
to describe the path. Moreover, for the convenience of computation, we can define another mapping as
where
indicates the controlling parameters of the path if the path satisfies
. We can determine that the curvature vector
and the length vector
can be used to express the path, since the path consists of straight line segments and arc segments. Thus, the mapping can be rewritten as
where
. Compared with the homotopic description, the ending boundary point in this description is not reserved; hence, it is obtained by iteration. However, this description is much more intuitive to express the path.
Based on these two path descriptions, two path homotopy construction patterns are presented for the path planning. As shown in
Figure 6, the axis defined by the line segment of the Dubins path divides the plane into two half planes. For simplicity, path homotopy 1 is in the upper plane and it is the first to be considered. It can be observed that path homotopy 1 consists of a curve and a straight line, and the curve consists of four equal arcs. Based on the geometrical relationship, the boundary conditions of path homotopy 1 can be denoted as
where
is the curvature bound of the arcs in the curve;
and
are the curvatures of arc segments along the Dubins path;
is the length of the line segment along the Dubins path;
and
are the lengths of arc segments along the Dubins path.
On this basis,
indicates a path with a curvature vector
for the arbitrary
. To describe this path, a linear function between
and
can be defined as
Based on the definition of path homotopy 1 and the geometrical relationship, there exists . Since , and the curvature vector are known variables, the length vector can be obtained by iteration. Moreover, path homotopy 1 in the lower half plane can also be constructed, and the paths within this path homotopy are described in a similar way.
To cover all of the lengths of the curvature-bounded path, another path homotopy is proposed. As shown in
Figure 7, path homotopy 2 consists of circles. Based on the geometrical relationship, the boundary conditions of path homotopy 2 in the upper half plane can be written as
where
and
are the lengths of two line segments, and
represents the lap number of the circles.
Similarly,
indicates a path with a curvature vector
for the arbitrary
. To describe this path, a linear function between
and
can be defined as
Since the UAVs are modeled as curvature-bounded vehicles, there exists . Then, the length vector can be obtained by iteration as well. As before, path homotopy 2 in the lower half plane can be constructed and the paths within this path homotopy are described in a similar way.
Based on the two path homotopy construction patterns, the curvature-bounded paths within the homotopy are described. In particular, the path varies monotonously based on , so that the path with the expected length can be planned by solving Equation (16) using the Newton–Raphson iterative method.
5. Simulation Results
To verify the performance of the recovery planning framework, representative simulations were conducted using MATLAB. The simulations in this section are divided into three major parts. First, the path coverage evaluation of the homotopic path planning approach in long-range missions was carried out. Then, the verification of the recovery scheduling process was conducted. In the end, the overall performance of the multi-UAV recovery planning framework was validated for stochastic scenarios. A Monte Carlo study that considered the quantity of the UAVs is also presented.
5.1. 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 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°. Moreover, the heading angle of the trajectory of the mother aircraft is separately set as 45°, 135°, −135°, and −45°. 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.
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 if 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.
5.2. Verification of the Recovery Scheduling
To verify the recovery scheduling process, representative simulations are conducted. In this section, simulations are carried out in a scenario of 100 km × 100 km and the trajectory of the mother aircraft is fixed. The sample distance of the mother aircraft trajectory is 0.1 km. The velocity of each UAV is 100 m/s and the curvature bound is 2 km
−1. The velocity of the mother aircraft is 150 m/s and the recovery time cost
of each UAV is 60 s. For more intuitive simulation results, three UAVs are set to be recovered by the mother aircraft. The settings of the UAVs, including the initial position and the initial direction, are given in
Table 3.
Figure 9 shows the simulation results of the recovery scheduling. As seen in
Figure 9a–c, the UAVs are simply recovered at the minimum value for their recovery time windows. When the mother aircraft recovers all of the UAVs, the recovery scheduling is executed and the optimal recovery sequence is (2,3,1). In
Figure 9d, the recovery trajectory of UAV 1 is different to that in
Figure 9a. The recovery process of UAV 1 is influenced by UAV 3 since the recovery process cannot be performed instantly.
Figure 10 intuitively illustrates the timeline of the recovery process. In particular, the recovery time window of UAV 3 consists of two regions due to the geometric relationship between its initial position and the recovery position. Moreover, the results show that the minimum value of the recovery time window of UAV 1 is occupied by the recovery time cost of UAV 3; hence, UAV 1 is immediately recovered after completing the recovery process of UAV 3. In this case, the recovery scheduling process is simple, but it can be difficult when the size of the UAV swarm increases. The recovery processes of the UAVs can influence each other. In addition, UAVs with multi-region recovery time windows do not necessarily need to be recovered urgently and other UAVs that are in need should be preferentially taken into account. As discussed in the previous sections, the GA is adopted to deal with the large-scale scheduling problem.
5.3. Overall Performance of the Recovery Planning Framework
In this section, simulations are carried out in a scenario that is 200 km × 200 km to validate the overall performance of the recovery planning framework. The size of the UAV swarm is set to 10, 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
of each UAV is 60 s. The coefficients
and
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 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. The configuration of the UAV swarm and the trajectory of the mother aircraft is the same as the previous simulation. In each case, the Monte Carlo simulation performs 100 runs. The results of the mean value, the standard deviation, and the best value of the objective function is shown in
Table 6. Clearly, the mean values with a non-maneuvering mother aircraft are lower than those with a maneuvering mother aircraft, respectively. This is caused by the inappropriate distances and the heading angles of the UAVs, so that they do not have recovery time windows. Besides, the mean values are close to the best values, which indicates that the UAVs can be recovered efficiently by using the recovery planning framework. In addition, we can see that there is a slight increase in the standard deviation as the size of the UAV swarm increases, but they are in an ideal range. Overall, the results of the Monte Carlo study further validate that the recovery planning framework has a good performance in the aerial recovery problem for the UAV swarm. This framework ensures that the mother aircraft recovers the most UAVs in the most efficient manner despite the quantity change for the UAVs, and the flyable paths of the UAV swarm can be designed by using the homotopic path planning approach.
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. 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.