Abstract
This paper investigates two guidance laws of standoff cooperative tracking static and moving of multiple autonomous unmanned aerial vehicles for targets from the perspective of the control system design. In the scheme of the proposed guidance laws, one vehicle is chosen as leader and others as followers. The leader only needs the measurement of the target, and the followers only measure the leader and its neighbors in the communication topology network. By using the proposed guidance laws, it is guaranteed that all vehicles can track a static or moving target with an evenly spaced formation of circle. Considering the coupling of tracking and cooperation, the stability analysis is performed by constructing two relatively independent subsystems based on Lyapunov theory, and the corresponding rigorous proofs of stability are given. By comparing with the Lyapunov vector field guidance law, the simulation results verify the effectiveness and superiority of the proposed guidance laws.
1. Introduction
In the past few decades, due to the rapid development of science and technology, the use of unmanned aerial vehicles (UAVs) is from surveillance and reconnaissance to rescue and communication relay [1–3], and its application fields have been greatly expanded. At present, one of the most important applications of UAVs is the use of tracking ground targets; through the UAV guidance system, the target can be always in the field of view, so as to achieve real-time tracking purpose of ground targets [4–6]. Compared with single UAV, multi-UAV cooperative tracking can increase the coverage of the sensor on the ground target and reduce the target state estimation error, and it has a wider range of application prospects [7, 8].
Unlike rotary-wing UAV, fixed-wing UAV must maintain a minimum velocity to produce enough lift. A typical way to accomplish a tracking mission is standoff tracking which means to monitor the ground target by orbiting around it at the desired distance; this tracking mode is called standoff tracking [9, 10]. The guidance problem in cooperative standoff tracking can be divided into two aspects: relative distance control and inter-UAV angle control, which are realized by the lateral and longitudinal guidance law, respectively.
Lawrence proposed a guidance law based on the Lyapunov vector field guidance (LVFG) method [11]. In the study by Frew et al. [12], it is assumed that the target is moving at a constant velocity, and the LVFG has been proposed to improve the guidance accuracy of the relative distance; then, another Lyapunov function is used to solve the flight velocity command as the longitudinal guidance law of the UAV to control the inter-UAV angle. A weighted relative distance and inter-UAV angle error are used as the objective function, and the horizontal and vertical guidance commands for the next optimization step are searched by the method of model predictive control (MPC) [13]. This method not only needs a lot of calculation time but also still has some difficulties to achieve online applications. A cyclic pursuit for coordinated target tracking applications based on sliding mode control and a virtual leader is presented, but the guidance law is too complicated to achieve in practical projects [14]. A variant cycle of standoff tracking applications has also been investigated, while only fixed and uniform moving targets were considered for this method [15]. Cheng proposed a composite impact time control guidance law considering the problem of simultaneous attack against a ground target [16]. It is also necessary to consider the effect of disturbances, so the robust leader-follower formation guidance laws are derived [17–19].
From the above previous work, we can conclude that the existing research results only study the static target instead of the moving target which limits the practical application of the guidance laws. To the best of the author’s knowledge, few research results can consider both scenarios in a unified control framework. The developed results in this paper not only realize the objective of the tracking static target but also can track the moving target in a unified analytical control framework. In this paper, a new leader-follower formation tracking mode is proposed, which is based on the principle that a leader UAV is used to track a ground target in the standoff mode. Besides, when multiple follower UAVs track the leader UAV and are evenly distributed in a circle, the leader UAV and follower UAVs adopt the same tracking distance. As a result, the leader UAV maintains a certain distance from the ground target to avoid being exposed, and the follower UAVs can be closer to monitoring the ground target. The novelty of this paper lays in extending the practical application of standoff cooperative tracking of multi-UAVs not only for static targets but also for moving targets.
This paper is organized as follows. Section 1 describes the mathematical model of a ground target tracking problem. Section 2 provides the new guidance law of the leader UAV and mathematical stability proof. Section 3 provides the new guidance law of the follower UAVs and mathematical stability proof. The results of simulation are discussed in Section 4. The paper ends with some conclusions.
2. Problem Formulation
In order to develop guidance laws for the fixed-wing UAV that tracks a ground target, some assumptions have been stated. Firstly, the UAV is considered at a constant velocity and altitude over the ground. Furthermore, a separate inner loop (stabilization loop) and an outer loop (guidance loop) control approach are designed, similarly as in most applications [20–23].
In Figure 1, let denote the range between the UAV and the target, and is bounded. The bearing angle is defined as the angle between the UAV forward direction and the direction from the UAV to the target, which is measured counterclockwise. denotes the desired radius, and is the velocity of the UAV.
The system dynamics can be described by the following kinematical model:where is the 2D location of the UAV, denotes the heading angle, and are the angular rate and acceleration as the control inputs, respectively, and denotes the 2D location of the target; then, the range . The objective is to design control input and such that as .
The dynamics (1) can be rewritten as
In dynamics (2), the state variables are changed from to . Furthermore, it can be seen that if the velocity of the UAV is constant, and can be determined by eachother. When , or , which represents the clockwise motion and counterclockwise motion of the UAV, respectively.
3. Leader UAV Guidance Law Design
Firstly, a guidance law of the leader UAV to track a static target is designed, and then, the guidance law is expanded to track a moving target.
3.1. Static Ground Target Tracking
In this paper, the guidance law for the leader UAV tracking a static ground target in the standoff mode is designed as
Then, we give the following conclusion.
Theorem 1. Consider UAV dynamics in (2) subject to the guidance law in (3). If k > 0, is the asymptotically stable equilibrium point of the closed-loop system.
Proof. consider the following candidate Lyapunov function:It can be verified that , and only at .
Taking the derivative of yields thatWith (2), we haveSubstituting into , we haveObviously, when , for any , there is , and it can be concluded that if and only if . Let ; when , we can obtain that , and no other solution can stay identically in other than . According to LaSalle’s invariance principle (Khalil, 2002), is the asymptotically stable equilibrium point of the closed-loop system.
Remark. this Lyapunov function is well designed as its equilibrium point is , . By designing appropriate guidance laws, we can make till , and the control objective can be realized.
3.2. Constant Velocity Ground Target Tracking
When the ground target moves at velocity , the UAV kinematics’ model can be written aswhere the velocity vector of the UAV can be decomposed into relative velocity, and the target velocity vector (depicted in Figure 2)where is the relative velocity vector. Then, the dynamics in (8) can be rewritten as the relative motion:
Velocity vector is constant when the target is in uniformly moving motion. Let be the unit tangent vector of and be the angular velocity vector corresponding to ; taking the derivative of (9) yields that
The symbol “” denotes the cross product of vectors. Take the norm of both sides because ; thus, we can obtain that
Then, we get
According to the parallelogram law of the vector, the area of the two triangles which are constituted by and and and , respectively, are equal, that is,
Then, we can obtain that
We have
Finally, we can obtain the guidance law for tracking the constant velocity ground target:and the corresponding closed-loop system can be written as
Then, we give the following conclusion.
Theorem 2. Consider UAV dynamics in (8) subject to the guidance law in (17). If , is the asymptotically stable equilibrium point of the closed-loop system in (18).
Proof. consider the following candidate Lyapunov function:Taking the derivative of yields thatSubstitute (10) and (17) into , thenIt can be seen that if , always exit and only if . and are both bounded and is also bounded according to Barbalat’s lemma [20], when , . Furthermore, because is bounded, still according to Barbalat’s lemma, we can conclude that when , , and is the asymptotically stable equilibrium point of the closed-loop system.
3.3. Variable Velocity Ground Target
When the ground target moves at variable velocity , then both are also variable, and equation (9) can be rewritten in the following scalar form:
Taking the derivative of both sides’ yields, one can have
Solving the quadratic equation and eliminating , we can obtain the guidance law for tracking the variable velocity ground target:where
The corresponding closed-loop system can be written as
We can see that equations (26) and (18) are the same, thus still satisfy Theorem 2 for the variable velocity ground target.
4. Follower UAV Guidance Law Design
Consider N follower UAVs in the formation problem, the ith (i = 1, 2, ..., N) UAV’s Dubins model can be obtained as follows:
The geometry between the follower UAVs and the leader UAV is depicted in Figure 3.
In this paper, the follower UAVs are required to maintain a circular formation with the centre at the leader UAV and hold equal angular separation. Meanwhile, the follower UAV’s velocity and heading angle should gradually converge to the leader UAV’s. The relative motion can be written aswhere the subscript ‘0’ denotes the leader UAV.
The follower UAV’s angular rate control is used to carry out distance and heading angle tracking for the leader UAV, while formation’s inter-UAV angle control and velocity tracking are performed by velocity control. We use such a communication topology, that is, the follower UAVs can perceive the state of the leader UAV, and a circular communication structure is implemented among the follower UAVs.
4.1. Angular Rate Guidance Law Design
In this paper, the following angular rate guidance law is designed for the ith follower UAV:
Then, we give the following conclusion.
Theorem 3. Consider the UAV formation dynamics in (28) subject to the guidance law in (29). If , the distance between the follower UAV and the leader UAV will gradually converge to , and the heading angle of the follower UAV will also gradually converge to the heading angle of the leader UAV.
Proof. consider the following candidate Lyapunov function:Taking the derivative yields thatwhere , andand we getSo is bounded, as is uniformly continuous, and according to Barbalat’s lemma. Furthermore, as is uniformly continuous, we have according to Barbalat’s lemma; then, we can conclude that .
4.2. Velocity Guidance Law Design
In this paper, the following velocity guidance law is designed for the ith follower UAV:
Then, we give the following conclusion.
Remark. it is easy to find that the Lyapunov function is bounded, so we can conclude that and simultaneously according to Barbalat’s lemma. In practice, even if , it is not an attractor which does not affect the convergence of the whole guidance system. It is the same for the case of . Hence, we can conclude that there exists no problem of singular values.
Theorem 4. Consider the UAV formation dynamics in (28) subject to the guidance law in (34); if , the inter-UAV angle between follower UAVs will gradually converge to , and the velocity of the ith follower UAV will gradually converge to the velocity of the leader UAV.
Proof. consider the following candidate Lyapunov function:Taking the derivative yields thatwhere , andand we getSimilar with Theorem 3, we can also conclude that and .
5. Simulation Results
In this section, some simulation results are presented in order to demonstrate the effectiveness of the proposed leader-follower formation guidance laws. In these simulation cases, we consider a 4-UAVs’ formation as an example, UAV #0 is the leader UAV and UAV #1–UAV #3 are the follower UAVs. The ground target motion is considered as stationary, linear with constant velocity, and linear with variable velocity.
Firstly, the simulation applies the leader UAV to track a ground target and then the UAV formation to track the ground target in the same way.
5.1. Track a Ground Target Using One UAV
The initial states of the UAV are set as follows:(i)Position coordinate: (800, 0)(ii)Heading angle: −60°(iii)Cruising velocity: 45 m/s(iv)Maximum heading angular rate: 0.1 rad/s
The initial states of the ground target are set as follows:(i)Position coordinate: (1000, 1500)(ii)Heading angle: 30°(iii)Velocity: 15 m/s
The gain is chosen as .
5.1.1. Static Ground Target
Figure 4 shows that the trajectory of standoff tracking of one UAV around the static target, and it can be seen that the trajectory of the UAV converges to the circle surrounding the target under the proposed guidance law. Figure 5 shows the changes of the bearing angle and relative range.
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5.1.2. Constant Velocity Ground Target
Figure 6 shows the standoff tracking trajectory of the UAV when the target moves with constant velocity. It can be seen that the trajectory of the UAV is the combination of circular motion and linear motion. The changes of the bearing angle and relative range during the process of the convergence are shown in Figure 7.
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5.1.3. Variable Velocity Ground Target
The velocity of the ground target is chosen as
The angular velocity of the ground target is set as
In this scenario, the target moves with the sine trajectory. It can be seen that the proposed guidance law can also realize the standoff tracking of the target. The trajectory of the UAV and the ground target are shown in Figure 8, and the convergence curves of the bearing angle and relative range are shown in Figure 9.
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From Figures 4 to 9, it can be seen that whether the target is still or moving, and the UAV always can track the ground target well in the standoff mode.
5.2. Track a Ground Target Using Formation UAVs
The initial states of the UAV #1 are set as follows:(i)Position coordinate: (150, −1000)(ii)Heading angle: −30°(iii)Cruising velocity: 40.5 m/s(iv)Maximum heading angular rate: 0.1 rad/s
The initial states of the UAV #2 are set as follows:(i)Position coordinate: (−75, −870)(ii)Heading angle: 80°(iii)Cruising velocity: 41 m/s(iv)Maximum heading angular rate: 0.1 rad/s
The initial states of the UAV #3 are set as follows:(i)Position coordinate: (−75, −1130)(ii)Heading angle: 100°(iii)Cruising velocity: 40.5 m/s(iv)Maximum heading angular rate: 0.1 rad/s
The gains are chosen as and .
5.2.1. Static Ground Target
Under the proposed guidance laws, the multi-UAVs can converge to the desired standoff trajectories when the target is static. The trajectories of the UAVs’ formation are shown in Figure 10, the velocities and heading angles of the formation are shown in Figure 11, and the inter-UAV angles and relative range between leader and follower UAVs are shown in Figure 12.
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5.2.2. Constant Velocity Ground Target
As shown in Figure 13, the target moves with constant velocity and UAVs can accomplish the objective of standoff tracking, and the changes of the velocities and heading angles are shown in Figure 14, and the inter-UAV angle and relative range between leader and follower UAVs are shown in Figure 15.
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5.2.3. Variable Velocity Ground Target
In this scenario, the target moves with the sine trajectory. The proposed guidance laws can also guarantee the convergence of the formation. The trajectories of the UAVs are shown in Figure 16, the changes of the velocities and heading angles are shown in Figure 17, and the inter-UAV angles and relative ranges between the leader and follower UAVs are shown in Figure 18.
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From Figures 10 to 18, it can be seen that regardless of the ground target is static or moving, the desired formation always can be achieved, the follower UAVs’ velocities and heading angles gradually converge to the leader, and the followers always can track the ground target well.
5.3. Simulation Analysis and Comparison
In order to verify the tracking performance, we adopt the well-known LVFG guidance algorithm proposed in Ref. [12] to run the same simulation case of variable velocity ground target tracking. We choose the tracking performance of UAV #1 as an example.
The velocity of UAV #1 and relative range between UAV #1 and the ground target using proposed method in this paper and Ref. [12] are shown in Figure 19.
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When other performances are relative equivalent, from Figure 19, the proposed guidance law in this paper makes the velocity and heading angle of the follower UAVs always be consistent with the leader UAV, thus the convergence characteristics of velocity and relative range of the UAV #1 are better than the results in Ref. [12].
6. Conclusion
In this paper, a new leader-follower formation tracking scheme is proposed, which is based on the principle that a leader UAV is applied to track a ground target in the standoff mode, while multiple follower UAVs to track the leader UAV and are evenly distributed in a circle. As a result, the leader UAV maintains a certain distance to the ground target to avoid being exposed, while the follower UAVs can be closer to monitoring the ground target. The stabilities of the new guidance laws are proved using Lyapunov functions. Numerical simulations of a 4-UAVs’ formation show that the new leader-follower formation can track the static and moving targets well, and its performance is better than the well-known classic LVFG algorithm [24].
Data Availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The work was supported by National Natural Science Foundation of China 61903192 and Natural Science Foundation of Jiangsu Province BK20190402.