Abstract
This paper considers a pursuit-evasion game with multiple pursuers and a superior evader. A novel cooperative pursuit strategy is proposed to capture a faster evader while maintaining a formation. First, the initial states including position distribution and the minimum required number of pursuers for ensuring capture are obtained based on the idea of Apollonius circle. Second, a cost function is designed, and the cooperative hunting strategy is developed using the distances between the centers of multiple Apollonius circles. Finally, numerical simulation and UAVs flying tests are provided to demonstrate the effectiveness of the proposed cooperative hunting strategy.
1. Introduction
In recent years, the cooperative control of multiagent systems has emerged as an attractive area of research [1–3]. As an important application of cooperative control, cooperative pursuit-evasion problem has attracted significant attention due to its tremendous potentials in searching hazardous areas, rescuing lives, capturing criminals, etc. The pursuit-evasion problem can be classified into three types based on the speed relationship between the pursuer and evader: (i) a case where the speed of pursuer is higher than that of evader [4–10]; (ii) a case where the speeds of both pursuer and evader are equal [11–14]; and (iii) a case where the speed of evader is higher than that of pursuer Wei et al. [15]; Chen et al. [16]; Zha et al. [17]; Ramana and Kothari [18, 19]; Yan et al. [20]; Liang et al. [21]; Liang and Deng [22]; Fang et al. [23].
For the first case, a planar pursuit-evasion game in the presence of obstacles is studied in Oyler et al. [4]. Dominance regions are shown to provide a complete solution to the game. In Wei and Yang [8], by employing the optimal control theory, the optimal trajectories of the pursuers and evader are obtained, and the effect of pursuers’ and evader’s detection ability on pursuit performance is studied. A scenario where two pursuers chase an evader is considered in [10]. The state space regions where just an optimal pursuer captures the evader and both pursuers cooperatively capture the evader are characterized, respectively. For the second case, the area minimization policy is studied extensively [11, 24], where the goal of pursuers is to reduce safe-reachable area of the evader to zero in a finite time and guarantee successful capture. This approach is later extended to the case where the evader’s position is uncertain and supposed to lie within an ellipsoid in Shah and Schwager [14].
For the last case, the Apollonius circle is first used to solve the pursuit-evasion problem with superior evader in Isaacs [25]. A perfectly encircled formation is proposed in Ramana and Kothari [18] based on the idea of Apollonius circle, but it is unfortunate that the perfect formation can only last for a moment. In [23], an encirclement algorithm considering the balances between maintaining a formation and pursuing the faster evader is proposed, and the pursuers are divided into two groups, one for surrounding and the other one for hunting. The evader is allowed to move freely without any constraint. In [20], the optimal decision of pursuers with uncertain speed is obtained based on simulated annealing. In [21], a particular pursuit-evasion game with three players is addressed, and the players’ winning regions are obtained.
The optimal pursuit strategy in Wei and Yang [8] is proposed to capture a slower evader. However, the application of this strategy is limited because the evader usually has high mobility. In this paper, a cooperative hunting strategy is designed to capture a faster evader with multiple pursuers, which greatly improves the practicability of the pursuit strategy. The contributions of this paper are threefold. First, we clarify some necessary conditions for the capture, and the minimum required number of pursuers is obtained based on the idea of Apollonius circle. Second, based on the differential game theory, a cooperative hunting strategy is obtained for the pursuers to prevent the evader from escaping, and the necessary conditions for successful capture can always be satisfied. Third, the cooperative hunting strategy proposed in this paper can achieve the balance between maintaining a formation and capturing the faster evader.
The remainder of this paper is organized as follows. Problem statement and preliminary results are given in Section 2. Sections 3 presents the main results. Simulation and experimental results are given in Sections 4 and 5. Section 6 concludes the paper.
2. Problem Statement and Preliminary Results
Consider a group of pursuers and one faster free-moving evader in two-dimensional space. All pursuers have the same maximum speed. The relative motion between the pursuers and the evader can be described as follows:where is the distance between the -th pursuer and evader, and and are the speeds of the evader and the pursuers, respectively (see Figure 1). is the bearing angle between the line-of-sight (LOS) of the -th pursuer and the direction of the -th pursuer’s speed, is the bearing angle between the -th LOS and the direction of the evader’s speed. The distance between two adjacent pursuers iswhere is the included angle between the -th LOS and the + 1-th LOS. The derivative of is
In this paper, we are interested in two problems:(1)What initial conditions including position distribution and the minimum required number of pursuers can ensure successful capture?(2)What strategies should be adopted by the pursuers to maintain the formation and pursue the faster evader?
The idea of Apollonius circle is widely used to solve the pursuit-evasion problem when the evader is faster than all pursuers. The following definition, assumption, and lemma are necessary for the following study.
Definition 1. Consider the -th pursuer and an evader . The evader is captured whenwhere is the capture radius.
Assumption 1. The speeds of the pursuers and evader are constant, and only the direction of speed can be changed.
Lemma 1. Ramana and Kothari [18]. The set of points that the -th pursuer and evader can reach simultaneously in the plane can be expressed asThe locus formed by the set of points is called Apollonius circle. The center and radius of Apollonius circle are represented aswhere .
Remark 1. A special case that the Apollonius circle of each pursuer is tangent with its neighboring pursuers is shown in Figure 2. Based on the idea of the Apollonius circle, the pursuers can reach any location within corresponding Apollonius circle earlier, and the evader has no chance to escape at this moment. Notably, if a gap is found, the evader will travel through it to avoid being captured. It is proved in Ramana and Kothari [18] that this formation is only maintained instantaneously so that it does not ensure successful capture.
3. Main Results
In this section, we first provide the necessary conditions for capture based on the idea of Apollonius circle. Then, by employing the differential game theory, we solve the bearing angles of pursuers to capture the evader.
3.1. Necessary Conditions for Capture
To achieve a successful capture, arbitrary adjacent Apollonius circles must maintain intersecting during the pursuit process. This is called the formation that pursuers need to maintain. Therefore, the capture will be achieved if and only ifwhere is the distance between the centers of two adjacent Apollonius circles and has the following relationship with
According to Assumption 1, is a constant. The relationship between and is linear. The distance between the centers of two adjacent Apollonius circles can be changed by adjusting the positions of the pursuers.
In order to meet the requirements of equation (7), it is important to determine the number of pursuers required. An extreme case is shown in Figure 2. As described earlier, the -th pursuer and evader both take time to reach tangential point . It can be seen thatwhere is an angle formed by the lines and , and it can be expressed as
By substituting equations (10) into (9), the relationship between the number of pursuers and the speed ratio can be obtained
According to Remark 1, the intersection of adjacent Apollonius circles is necessary to ensure successful capture. Therefore, the minimum number of pursuers is
Remark 2. It is obvious that the lower the speed of pursuers, the more pursuers are required to achieve the capture. In fact, the speed and the number of pursuers can be chosen depending on the actual environment. Some simulation results will be given in Section 4 to verify the observation.
Therefore, the necessary conditions for capture include two aspects. First, the position distribution of pursuers should satisfy the intersectant requirement of the corresponding Apollonius circles. Second, the minimum required number of pursuers should be greater than .
3.2. Cooperative Hunting Strategy
Based on the idea of Apollonius circle, a cost function is defined aswhere . The cost function represents the square of distance between the centers of the adjacent Apollonius circles. When the evader is surrounded by all pursuers, the reduction of makes it less likely that the evader escapes from the gap between the pursuers. When all pursuers follow the hunting strategy, the safe-reachable area of the evader can be gradually reduced until capture is achieved.
In this paper, the pursuers and evader have independent control variables. The optimal control variables of pursuers will be solved separately, and the evader can move arbitrarily. Define the Hamiltonian of the -th pursuer aswhere , , and . The Hamiltonian multiplier and are n-dimensional row vectors. The costate dynamics of and are given asand
The terminal condition for this pursuit-evasion game problem is described as Assumption 1. The optimal bearing angle of the -th pursuer must satisfywhere
With , the solution of equation (17) can be obtained as
Theorem 1. The bearing angle in equation (15) is the optimal bearing angle of the -th pursuer.
Proof. In order to show that can reduce the cost function , the second partial derivative of the Hamiltonian function with respect to isSubstituting equations (19) into (20), we can getThe optimal bearing angle of the -th pursuer is obtained. This completes the proof.
It is worth noting that two adjacent Apollonius circles cannot be guaranteed to intersect when the cost function is reduced. That is to say, if only the above optimal bearing angle is adopted and the formation is not maintained, the evader may escape from the gap. Therefore, we should properly adjust and maintain the formation in the process of reducing the cost functional . That is, the condition equation (7) always needs to be satisfied. Condition equation (7) can be transformed into the following equation:For the -th pursuer, its purpose is to guarantee that the Apollonius circle intersects with that of + 1-th pursuer while reducing the cost functional . The corresponding Apollonius circles of adjacent pursuers are intersectant at the initial moment. In order for during pursuit, should be satisfied when is close to zero. Therefore, the bearing angle for cooperative hunting strategy can be obtained by solving when , where and are small positive numbers. In this case, the condition that the adjacent Apollonius circles intersect can always be satisfied. In other words, the evader will have no chance to escape.
4. Simulation
In this section, some numerical examples are provided to verify the effectiveness of the proposed strategy. We assume that the evader’s speed is , and the evader moves towards the largest gap between neighboring pursuers. The capture radius is set to . Other initial parameters of the pursuers and evader will be set distinctively in different cases.
Case 1. Six pursuers pursuit a superior evader.
Suppose , , and the initial positions of the players are , , , , , and , respectively. The simulation results are shown in Figures 3 and 4.
Figure 3 presents the trajectories of six pursuers and an evader. The initial positions of the players are marked by stars, and the final positions are denoted by triangles. The black circle represents pursuer’s capture area. The evader is captured when the position of any pursuer is within the circle or on the boundary. The evader is eventually captured by pursuer 2. Figure 4 shows the distances between pursuers and evader. The black line represents the sum of corresponding distances. The pursuers reduce the distances between each other and the evader continuously. Finally, the evader is captured when .
In order to illustrate the necessity of constraint condition equation (22), the results without constraint are shown in Figure 5. The pursuers just keep getting closer to the evader. The condition that the Apollonius circle of each pursuer intersects with that of its neighbor pursuers cannot be satisfied. Finally, the evader successfully escapes from the gap between pursuer 3 and pursuer 4.
In comparison, under the same initial conditions, the trajectories obtained by using the pursuit strategy proposed in Wei and Yang [8] are shown in Figure 6. It shows that the pursuers cannot capture the evader. The evader will escape from the gap between pursuer 1 and pursuer 6.
Figure 7 shows the comparison of cost function between the proposed method in this paper and the method in Wei and Yang [8]. It is obvious that the cooperative hunting strategy proposed in this paper can reduce the cost function while ensuring the capture of the evader.
Moreover, Table 1 gives the comparative results on the required number of pursuers with given . Compared with the results in Chen et al. [16], the cooperative hunting strategy proposed in this paper can achieve capture with fewer pursuers when .
Case 2. Nine pursuers pursuit a superior evader.
Suppose , and the initial positions of the players are , , , , , , , , , and , respectively. The simulation results are shown in Figures 8 and 9. The evader is eventually captured by pursuer 8 when .
It is similar to Case 1 that the pursuers cannot capture the evader by using the pursuit strategy proposed in Wei and Yang [8] as shown in Figure 10. Figure 11 shows the comparison of cost function between the proposed method in this paper and the method in Wei and Yang [8] when , . It can be seen that the cost function in this paper is much smaller. On the contrary, the cost function in Wei and Yang [8] is smaller, but the evader cannot be successfully captured.
5. Experiment Validation
5.1. Experimental Platform
An experimental platform is built to demonstrate the performance of the proposed hunting strategy with real UAV systems. The platform is comprised of the Parrot Bebop quadrotors, the OptiTrack real time tracking systems, and a Linux-based computer, as shown in Figure 12. The real time tracking system is comprised of an array of eight OptiTrack Prime 13 cameras and a host PC. The function of the system is to identify the position and orientation of the quadrotor. The Linux-based computer receives these pose information and calculates the control input of the quadrotor according to the proposed strategy. The inputs are transmitted to Bebop quadrotors via 2.4 GHz Wi-Fi network. The robot operating systems (ROS) environment is used to transfer the messages.
In this experiment, four quadrotors are used to pursuit a quadrotor in the horizontal XY plane. The initial positions of the quadrotors are , , , , and , respectively. In order to avoid collisions between quadrotors, the heights of the quadrotors are controlled to be , , , , and , respectively. The capture radius is set to be , and the speed ratio is 0.9.
5.2. Experimental Results
The experimental results are shown in Figures 13–15. Figure 13 shows the position trajectories of the pursuers and the evader. Figure 14 shows the flight geometries of the five quadrotors at time s, respectively. It can be seen that pursuit quadrotors keep hunting the escaping quadrotor and achieve the capture at as shown in Figure 15. Overall, the proposed cooperative hunting strategy is effective in the quadrotor application. The video of capture experiment is presented in the following figures.
6. Conclusions
In this paper, we investigate a pursuit-evasion game with multiple pursuers and a superior evader. Multiple pursuers capture the fast evader while maintaining formation. To ensure successful capture, the initial states including position distribution and the minimum required number of pursuers are obtained based on the idea of Apollonius circle. By designing the cost function as the distances between the centers of multiple neighboring Apollonius circles, the cooperative hunting strategy is obtained to capture the evader. Simulation results and UAVs flying experiments are provided to show the effectiveness of the proposed cooperative hunting strategy. Compared to the existing results, the proposed method can achieve capture with less number and lower speed of pursuers.
In the future, we plan to investigate the pursuit-evasion game with intercommunication, jamming, limited observation, and measurement noise.
Data Availability
The data used to support the findings of this study are partly included at https://v.youku.com/v_show/id_XNTA5MzU1MjM2NA==.html.
Disclosure
The material in this paper was not presented at any conference.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Nos. 62103045 and 61803032) and China Postdoctoral Science Foundation (No. 2021M700014).