Distributed observer-based cooperative guidance with appointed impact time and collision avoidance

doi:10.1016/j.jfranklin.2021.06.030

Journal of the Franklin Institute

Volume 358, Issue 14, September 2021, Pages 6976-6993

https://doi.org/10.1016/j.jfranklin.2021.06.030 Get rights and content

Abstract

The problem of cooperative guidance is considered with appointed impact time and collision avoidance for the leader-following flight vehicles, which consist of one leader with the target seeker and the other seeker-less followers. A fixed-time convergent guidance law is presented for the leader to achieve appointed impact time. To guarantee the simultaneous arrival of all the flight vehicles, a cooperative guidance law is proposed to make the follower-leader ranges keep proportional consensus with the range-to-go of leader. A distributed observer is put forward for the followers to estimate the range-to-go of leader. Moreover, the collision avoidance can be reliably fulfilled by the collaborative action of the direction-based and distance-based means.

Introduction

The simultaneous arrival of multiple flight vehicles is viewed as a powerful tool for striking the target guarded by the defensive systems. The main approaches to realize the simultaneous arrival are impact time control guidance (ITCG) and cooperative guidance [1], [2].

The principle of ITCG law is to allocate a common impact time command for every flight vehicle to reach the target simultaneously [3], [4]. In consideration of the nonlinear kinematics and singularity issue, an ITCG law was presented in Kim et al. [2] to hit the target at an appointed impact time. With the aid of the proportional navigation guidance (PNG) and an impact time error term, an ITCG law in Jeon et al. [5] derived the closed form solution in basis of the linear formulation. Taken into account the aerodynamic effect, a data-driven impact time control guidance law was given in Guo et al. [3] with the help of a database. The angle requirements, such as the field-of-view (FOV) angle and the impact angle, are often taken into consideration in ITCG laws. By adjusting the control gain, an ITCG law in Kim et al. [6] can hit the target at a specified look angle with FOV constraint. Making use of the backstepping control, the ITCG law in Kim and Kim [7] restricts the leading angle to a prescribed range. To meet the demands of both impact time and impact angle, a three-dimensional ITCG law in Yan et al. [8] is given by tracking the pre-designed trajectory.

Nevertheless, the flight vehicles in above ITCG laws have no interconnected communications. In contrast, the cooperative guidance approach employs the communication network to improve the collaborative capacity [9], [10]. Under the structure of PNG, a cooperative guidance strategy was put forward in Jeon et al. [11] for multiple missiles to achieve simultaneous arrival by adjusting a time-varying gain. Exploiting a local filtering algorithm, the robust cooperative guidance law in Wang et al. [12] is given to hit the maneuvering target even when some missiles are break down. In view that the continuous communication leads to high energy cost, the synchronous of impact time in Zhou et al. [13] was achieved under the discrete-time communication among the flight vehicles. By leveraging the fixed-time sliding mode theory, a robust three-dimensional cooperative guidance law is presented in Chen et al. [14] to attack a maneuvering target at desired impact angles. Beyond that, a two-stage cooperative guidance law in He et al. [15] discussed the salvo attack problem without time-to-go estimation.

Though these cooperative guidance laws can realize the simultaneous arrival with the communication network, a few key issues are of great concerns in current research. Firstly, the target seeker must be equipped for all the flight vehicles, i.e., taking up a substantial cost of the cooperative operation [16]. It is significant to accomplish the mission with fewer target seekers. Furthermore, the collision among the members ought to be avoided for mission security [17], [18]. A direction-based collision avoidance strategy for distributed guidance is given in Zhu et al. [19] for multi-UAV systems to intercept the multiple targets, in which the collision avoidance is handled by allocating different line-of-sight (LOS) angles for the UAVs. Whereas, the collision may still occur in Zhu et al. [19] during the stage before the LOS angles converged to their reference values.

Inspired by these issues, a cooperative guidance strategy is proposed in this paper for leader-following flight vehicles with appointed impact time. In comparison with the previous works, the main features of our research are formulated as follows.

1)
Dominated by the proposed cooperative guidance strategy, the follower-to-leader ranges will be forced proportional consensus with the range-to-go of leader. A distributed observer is presented for the followers to address the issue that the range-to-go of leader is unavailable to part of the followers.
2)
Though the followers are unable to obtain the target information for the absence of target seekers, the simultaneous arrival with appointed impact time can still be achieved under the proposed cooperative guidance approach.
3)
The collision avoidance between the flight vehicles can be fulfilled reliably by the collaborative actions of the direction-based and distance-based means.

Section 2 states the preliminaries and model description. In Section 3, the guidance approaches are proposed in consideration of the appointed impact time and collision avoidance. The simulation is given in Section 4 to verify the proposed strategy. Conclusion is provided in Section 5.

Section snippets

Preliminaries

In accordance with the graph theory, the communication links among the flight vehicles can be described by the adjacent matrix $A = [a_{i j}]$ for $i, j = 1, 2, \dots, n$ , in which $a_{i i} = 0$ , $a_{i j} = 1$ if the $i$ th flight vehicle can communicate with the $j$ th flight vehicle, $a_{i j} = 0$ otherwise. The graph is said to be undirected on condition that the communication links are all bi-directional, that is, $a_{i j} = 1$ and $a_{j i} = 1$ [20]. An undirected graph is said to be connected if there exists a route between any of two nodes.

Lemma 1

[9]

The

The guidance law for the leader

The guidance command of leader is proposed in this subsection so that the leader can hit the target at the appointed impact time.

We estimate the time-to-go of leader by: ${\hat{t}}_{g, 0} = \frac{r_{0}}{V_{0}} (1 + \frac{ϕ_{0}^{2}}{2 (2 N_{s} - 1)})$ where $N_{s} > 2$ denotes the navigation constant.

For the fact that $ϕ_{0}$ is small in regular situation, $1 - ϕ_{0}^{2} / 2 = \cos ϕ_{0}$ and $ϕ_{0} = \sin ϕ_{0}$ can be approximately established. As given in Li et al. [36], it yields ${\dot{\hat{t}}}_{g, 0} = - 1 + \frac{ϕ_{0}^{2}}{2} - \frac{ϕ_{0}^{2}}{2 (2 N_{s} - 1)} + \frac{r_{0} ϕ_{0} a_{0}}{V_{0}^{2} (2 N_{s} - 1)} + \frac{ϕ_{0}^{2}}{2 N_{s} - 1}$ where $ϕ_{0}^{l} = 0$ for $l \geq 3$ is utilized in Eq. (6).

We define an

Numerical simulation

The simulations are implemented to verify the effectiveness of the proposed strategy. One leader and three followers are employed to hit a stationary target at the appointed impact time $T_{d} = 65$ s. The communication links among the flight vehicles are shown in Fig. 3.

The parameters are selected as $k_{1} = 1.5$ , $k_{i, 1} = 0.5$ , $k_{i, 2} = 0.01$ , $k_{i, 3} = 10$ , $k_{i, 4} = 0.01$ , $N_{s} = 4$ , $μ_{1} = α = b_{1} = b_{2} = 0.5$ , $μ_{2} = β = 1.5$ , $δ = 0.3$ , $c_{1} = 10$ , $c_{2} = 1$ , $l_{i} = 0.1$ , $k_{i, 5} = 2$ , $p_{1} = 0.3$ , $p_{2} = 0.6$ , $p_{3} = 0.8$ , $T_{θ} = 50$ s, $θ_{f, 1} = π / 3$ rad, $θ_{f, 2} = π / 4$ rad, $θ_{f, 3} = π / 6$ rad, ${\hat{r}}_{0}^{1} (0) = 7000$

Conclusion

A cooperative guidance approach is proposed for the leader-following flight vehicles with the constraints of impact time and collision avoidance. The flight vehicles are composed of one leader with the target seeker and the other seeker-less followers. Only a portion of followers can communicate with the leader. Dominated by the proposed guidance laws, all the flight vehicles can simultaneously hit the target at the appointed impact time. The collision between the flight vehicles can be avoided

CRediT authorship contribution statement

Guofei Li: Conceptualization, Data curation, Formal analysis, Writing – review & editing. Jinhu Lü: Methodology, Writing – original draft, Formal analysis, Writing – review & editing. Guoliang Zhu: Conceptualization, Data curation. Kexin Liu: Formal analysis, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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^☆: This work was supported by China Postdoctoral Science Foundation under Grant 2020M670094, and also supported by the National Natural Science Foundation of China under Grants 62003021 and 61803357.

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Distributed observer-based cooperative guidance with appointed impact time and collision avoidance☆

Abstract

Introduction

Section snippets

Preliminaries

[9]

The guidance law for the leader

Numerical simulation

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

J. Frankl. Inst. Eng. Appl. Math.

J. Frankl. Inst. Eng. Appl. Math.

Aerosp. Sci. Technol.

J. Frankl. Inst. Eng. Appl. Math.

Aerosp. Sci. Technol.

J. Frankl. Inst.

Aerosp. Sci. Technol.

Aerosp. Sci. Technol.

J. Frankl. Inst. Eng. Appl. Math.

J. Frankl. Inst. Eng. Appl. Math.

Mech. Syst. Signal Process.

J. Frankl. Inst. Eng. Appl. Math.

Lyapunov-based impact time control guidance laws against stationary targets

IEEE Trans. Aerosp. Electron. Syst.

Data-driven method for impact time control based on proportional navigation guidance

J. Guid. Control Dyn.

Impact-time-control guidance law for anti-ship missiles

IEEE Trans. Control Syst. Technol.

Look-angle-shaping guidance law for impact angle and time control with field-of-view constraint

IEEE Trans. Aerosp. Electron. Syst.

Backstepping-based impact time control guidance law for missiles with reduced seeker field-of-view

IEEE Trans. Aerosp. Electron. Syst.

Fixed-time cooperative guidance law with input delay for simultaneous arrival

Int. J. Control