Three-dimensional piecewise guidance strategy for multi-UAVs guaranteeing simultaneous arrival and field-of-view constraint

Abstract

This paper presents a three-dimensional (3D) piecewise guidance scheme for multiple UAVs) guaranteeing simultaneous arrival even under the field-of-view (FOV) constraint. The guidance law does not require to consider the estimation of the time-to-go and can be divided into two stages: cooperative stage and proportional navigation guidance (PNG) stage. In the cooperative stage, a decentralized consensus control approach is proposed for multi-UAV systems and does not rely on global information about the communication topography. Moreover, the neighbors’ new auxiliary states are introduced to ensure the states achieving consensus under FOV constrain. The guidance strategy transfers to the PNG stage while the UAV formation is close enough to the target and the simultaneous arrival to hit the target is guaranteed for UAVs via the PNG guidance law. Numerical simulation results demonstrate the effectiveness of the proposed cooperative guidance law.

Keywords

UAV formation cooperative guidance law FOV constraints simultaneous arrival three-dimensional

Introduction

With the continuous improvement of science and technology in recent years, the application of unmanned aerial vehicles (UAVs) has been greatly expanded, from target recognition, communication relay to reconnaissance and surveillance, disaster assessment, and so on. The rapid development in the aera of cooperative homing guidance has been observed. Due to its limited detection range and kill radius, one single UAV can barely strike the target effectively. Once to be exposed, it is likely to be in passive situation. On the contrary, multi-UAV coordinated operations can make up for the shortcomings of single-UAV operations, strengthening the combat efficiency, which presents a threat to the enemy. Utilizing multiple missiles to carry out multi-directional three-dimensional (3D) cluster raids on the target, which is able to cause loopholes in the defensive airspace and achieve sudden strike.

With the blooming development of antimissile technology, it is even more difficult for one single UAV to achieve penetration in the face of UAV defense systems equipped with enemy targets. Moreover, multi-UAV cooperative combat with the help of communication system forms a multi-UAV combat network, realizes cooperation and coordination through information sharing, and completes strike jointly, which greatly improves the missiles’ penetration probability. Tian et al. (2015) proposed an adaptive multi-variable controller to deal with the finite-time reentry attitude tracking problem, the tracking controller with unmatched disturbance considering the convergence of finite time is designed. Liu et al. (2018) designed a high-performance adaptive controller for the uncertain model of hypersonic flight vehicles considering the limited angle-of-attack. Hu et al. (2021) proposed an analytical 3D guidance law considering non-linear coupled dynamics guaranteeing impact angle and field-of-view (FOV) constraints. He et al. (2019) proposed a new guidance approach based on integral sliding mode control, which taken into account interception maneuvering target with impact angle constraint. Wang et al. (2015b) designed a continuous second-order sliding mode guidance law for intercepting a highly maneuverable target based on the super-twisting algorithm.

Therefore, the research on multi-UAV coordinated homing guidance is of important practical value, especially in two aspects: FOV constraints and finite-time consensus. To be more specific, (1) with the help of FOV constraint, multi-UAV is able to proceed against the target from the expected direction, thereby greatly improving the missiles’ penetration capability and strike effectiveness; (2) the finite-time consensus homing guidance law designed based on the multi-UAV cooperative control theory is capable of achieving consensus within finite-time, which is beneficial to accelerate the system response and improve the overall penetration probability.

One of the characteristics of multi-UAV cooperative homing guidance is that it is not necessary to assign the unified against-time, but to consensus the against-time through communication, and to strike the target synchronously. Aiming at the above problem of cooperative homing guidance, Wang et al. (2015a) designed a distributed cooperative homing guidance law, adding a control term for the hit time error between the missile and the adjacent missile on the basis of the proportional guidance law and the FOV-constrained guidance law. Zhang et al. (2015) proposed a centralized cooperative homing guidance law with consensus. The cooperative homing guidance law proposed in the above literature, although with the FOV constraint, barely takes into account the finite-time coordination.

Kang et al. (2018) estimated a cooperative homing guidance law with a model-predictive control (MPC). The constraints of the normal acceleration and FOV were also considered. Zhao and Zhou (2008) proposed a distributed cooperative homing guidance law with ITCG law. Zhou et al. (2017) designed distributed cooperative homing guidance within communication failure taking into account communication failure. Although the above guidance law has FOV constraints, it does not take into account 3D.

Zhao and Zhou (2015) designed a bias-proportional cooperative homing guidance law by adding target maneuvering term and time cooperative term on the basis of the traditional proportional homing guidance law, which enables multiple missiles to attack maneuvering targets in coordination. Although the cooperative homing guidance law proposed above can achieve striking maneuvering targets, it does not have the FOV constraint and does not take into account the problem of consensus within finite time.

Hou et al. (2015) proposed a cooperative homing guidance law based on the time-varying proportional guidance method. Dong et al. (2019) proposed a controller based on integral barrier Lyapunov functionals to control the system and keep boundary the states in the meanwhile. Yang and Song (2021) proposed two nonsingular distributed cooperative guidance laws to solve the 3D cooperative guidance of multi-agents against a stationary target. Yu et al. (2021) proposed an adaptive distributed cooperative guidance law, in order to solve multiple aerial vehicles system attacking a maneuvering target. Although the guidance rate can achieve consensus with finite-time, it does not have the FOV constraints.

He et al. (2018) proposed a consensus of missiles’ states to salvo attack, but the control law may occur singular problem in many situations. Although the cooperative homing guidance law designed above considering with the FOV constraint, barely take into account the finite-time coordination.

Generally, the design of cooperative guidance law for UAVs can be divided into two categories according to different attack targets: guidance law for pre-launch preset finite-time and real-time attack guidance law during flight. Harl and Balakrishnan (2012) developed the type of guidance law through optimal control and sliding mode control theories, respectively. However, there is an obvious shortcoming in the first type of guidance law, that is, it is difficult to design an estimated finite time for maneuvering targets, especially for maneuvering targets of which movement laws are uncertain. Therefore, the second type of guidance law emerges. In order to synchronize the finite time dynamically, researchers proposed letter-follower cooperative homing guidance strategy and centralized cooperative homing guidance strategy (Jeon et al., 2010). In addition, in order to improve the adaptability and scalability of the guidance system, Zhao et al. (2010) proposed a distributed cooperative homing guidance algorithm by controlling the finite time.

Aiming at the problem of cooperative homing guidance, Zhai et al. (2016) proposed an interception algorithm based on area coverage by considering the utilization of multiple interceptor agents on targets with decoys, which maximized the rate of joint interception. Shaferman and Oshman (2016) proposed a cooperative strategy between leading and following agents, which improved the interception performance of followings through the information sharing between agents and targets. Hassan et al. (2019) provided a distributed platform of guidance and control model, considering mission planning on the basis of cooperative guidance, to achieve the coordinated maneuvers of multiple UAVs. Saeed and Hassan (2019) designed a new approach to solving non-linear real constraints for symbolic execution. In order to reduce the transmission error, Wang et al. (2017b) proposed a mid-course cooperative homing guidance law, in which the mid-course agent can use the target information collected by the terminal agents.

The advantages and necessities of the method proposed in this paper are further explained. The method in this paper does not only take into consideration the problem of time-to-go coordination but also considers an important issue attached to FOV constraint. The main contributions and the novelty of this paper can be described in three aspects:

A consensus law for multi-UAV system while taking into account the FOV constraint is designed. Meanwhile, the proposed cooperative controller can be designed depending on the local neighbor’s information, differently from the other consensus algorithm depend on the global states.

The second contribution of this paper is the method is able to achieve consensus within time-to-go and the FOV convergence to the desired value within a limited time via a piecewise strategy.

Compared with the traditional cooperative guidance control method, the proposed control law does not requires to consider the information of the time-to-go, which avoids the estimation error of the time-to-go values and thus guarantees the accuracy and effectiveness.

The rest of the paper is organized as follows. The problem formulation is presented, and some basic model and useful background on graph theory are introduced in section “Preliminaries and guidance model.” In section “Consensus strategy and cooperative homing guidance laws,” the consensus strategy and guidance laws are proposed. Numerical results analyzed in different scenario are shown in section “Simulation results.” Finally, the conclusions are drawn in section “Conclusion and future research.”

Preliminaries and guidance model

Problem statement

Consider the scenario that a group of n UAVs attack a target, the cooperative homing guidance geometric relationship is described in Figure 1. To hit the target, it is necessary to ensure that the speed direction of the n UAV must coincide with the direction of the line of sight at the proportional navigation guidance (PNG) stage.

Suppose that the speed of the ith UAV is $V_{i}$ , $R_{i}$ denotes the relative distance between the i-UAV and the target. In order to ensure the UAV formation hit the target simultaneously, that is, the time-to-go values of each UAVs must be equal, we have

t_{go, i} = t_{go, j}

(1)

According to the definition of time-to-go, we can easily get

lim_{t \to \infty} R_{i} / V_{i} = lim_{t \to \infty} R_{j} / V_{j}

(2)

In addition, the FOV constraint of UAVs must to be guaranteed, that is

0 \leq σ_{i} \leq σ_{max}

(3)

Therefore, the purpose of this paper is to achieve cooperative homing guidance in equation (1) without violating the FOV constraint in equation (3).

Graph theory

The communication relationship between the subsystems in the UAV system can be represented by topology diagram $G (V, ε)$ , which consists of a set of nodes $V = {v_{1}, v_{2}, . . ., v_{N}}$ and a set of edges $ε \subseteq {(ν_{1}, ν_{2}) : ν_{1}, ν_{2} \in V}$ . Two adjacent nodes which can transmit information to each other are called neighbors, and the set of neighbors of node $ν_{i}$ is defined as $N_{i} = {j : (ν_{i}, ν_{i}) \in ε, j \neq i}$ . In the topological graph $G (V, ε)$ , the interconnection of each node can be expressed as an adjacency matrix $A = [a_{ij}] \in R^{n \times n}$ , if the edge between nodes i and j is connected, that is, $(ν_{i}, ν_{j}) \in ε$ , then there is $a_{ij} > 0$ ; otherwise, $a_{ij} = 0$ . The Laplacian matrix of the topology diagram $G (V, ε)$ is defined as $L = [l_{ij}] \in R^{n \times n}$ :

l_{ij} {\begin{matrix} - a_{ij}, i \neq j \\ \sum_{j = 1, j \neq i}^{n} a_{ij}, i = j \end{matrix}

(4)

Lemma 1

Assuming that the matrix L is a Laplacian matrix of a directed graph G or an undirected graph G, then the Laplacian matrix has at least one zero eigenvalue, and other non-zero eigenvalues have positive real parts (Wang et al., 2017a). If the graph G has a spanning tree, 0 is the only one eigenvalue of G, the others N – 1 eigenvalues have positive real parts.

In addition, if 0 is the eigenvalue of L, $1_{N}$ is the associated eigenvector, then, $L 1_{N} = 0$ .

Lemma 2

Suppose there is a vector $z = [z_{1}^{T}, z_{2}^{T}, . . ., z_{n}^{T}]$ , where $z_{i} \in R^{n}$ , L is the Laplacian matrix of a graph G, then (Ren and Cao, 2011)

(L \otimes I_{n}) z = 0 \Leftrightarrow z_{1} = z_{2} = . . . = z_{n}

The UAV homing guidance model

Assuming that the target is stationary, the UAV speed V is assumed to be constant, the UAV/target 3D engagement geometry is shown in Figure 2.

$O - X_{I} Y_{I} Z_{I}$ denotes the inertial frame, $O - X_{m} Y_{m} Z_{m}$ denotes the velocity frame, the relative range between the UAV and the target is R, the LOS angle is defined by $θ_{L}$ , $φ_{L}$ and $θ_{m}$ , $φ_{m}$ in the inertial frame and velocity frame, respectively. The leading angle $σ$ denotes the angle between LOS and missile velocity is given by

σ = \arccos (\cos θ_{m} \cos φ_{m})

(5)

The governing equations of relative motion about 3D homing guidance are described as follows (Song and Ha, 1994)

\overset{\cdot}{R} = - Vcos θ_{m} \cos φ_{m}

(6)

R {\overset{\cdot}{θ}}_{L} = - Vsin θ_{m}

(7)

{\overset{\cdot}{φ}}_{L} R \cos θ_{L} = - Vcos θ_{m} \sin φ_{m}

(8)

{\overset{\cdot}{θ}}_{m} = \frac{a_{z}}{V} - {\overset{\cdot}{φ}}_{L} \sin φ_{L} \sin φ_{m} - {\overset{\cdot}{θ}}_{L} \cos φ_{m}

(9)

\begin{matrix} {\overset{\cdot}{φ}}_{m} = \frac{a_{y}}{V \cos θ_{m}} + {\overset{\cdot}{φ}}_{L} \sin φ_{L} \cos φ_{m} \tan θ_{m} \\ - {\overset{\cdot}{θ}}_{L} \sin φ_{m} \tan θ_{m} - {\overset{\cdot}{φ}}_{L} \cos θ_{L} \end{matrix}

(10)

where $a_{y}$ and $a_{z}$ denote the acceleration components in the velocity frame.

Consensus strategy and cooperative homing guidance laws

In this section, a control strategy of consensus cooperative homing guidance with FOV constrains is put forward. The consensus cooperative homing guidance is divided into two stages: cooperative stage and PNG stage. In the cooperative stage, two new states are introduced, while ensuring the two stage achieve consensus with FOV constrains. The control strategy transfer to PNG stage while the UAV formation is close enough to the target, the leading angle reduced to zero through PNG laws to ensure that the UAVs will hit the target.

Consensus controller with FOV constrains

A consensus controller with FOV constrains is designed in this chapter.

This chapter mainly designs the formation controller with azimuth constraint in the cooperative stage. In this work, we consider two new auxiliary states (He et al., 2018)

γ_{i} = \frac{R_{i}}{V_{i}}

(11)

μ_{i} = - \cos σ_{i}

(12)

From equations (3) and (12), the FOV constraint be converted to

μ_{min} \leq - \cos σ_{i} \leq μ_{max}

(13)

According to equations (5) and (6), a UAV formation system with dynamic behavior is described by the following second-order system

{\begin{matrix} {\overset{\cdot}{γ}}_{i} = μ_{i} \\ {\overset{\cdot}{μ}}_{i} = u_{i} \end{matrix}

(14)

where $i = 1, 2, . . ., N$ , $u_{i}$ denotes the virtual input, the UAV can interact with each other with the topology diagram $G (V, ε)$ .

Therefore, the objective of this stage is aimed to achieve consensus with FOV constrains, that is

lim_{t \to \infty} R_{i} / V_{i} = lim_{t \to \infty} R_{j} / V_{j}

(15)

- \cos σ_{i} = - \cos σ_{j}

(16)

From equations (11) and (12), the above objective can be transformed to

lim_{t \to \infty} γ_{i} = lim_{t \to \infty} γ_{j}

(17)

μ_{i} = μ_{j}

(18)

Definition 1: Consider the following n UAV second-order system with individual dynamical behavior in equation (14)

{\begin{matrix} {\overset{\cdot}{γ}}_{i} = μ_{i} \\ {\overset{\cdot}{μ}}_{i} = u_{i} \end{matrix}

where $i \in n$ . Suppose the topology diagram $G (V, ε)$ is connected, K and $K_{e}$ denote positive-definite diagonal matrix, for a vector $z = [z_{1}, z_{2, . . .}, z_{n}]$ , and $\arctan (z) = [\arctan (z_{1}), \arctan (z_{1}), . . ., \arctan (z_{n})]$ , the consensus controller is designed

u_{i} = - K [μ_{i} - A + B \frac{2}{π} \arctan (K_{e} \sum_{j = 1}^{n} a_{ij} γ_{ij} + C)]

(19)

where

A = \frac{1}{2} (μ_{min} + μ_{max})

(20)

B = \frac{1}{2} diag {μ_{max}^{1} - μ_{min}^{1}, . . ., μ_{max}^{n} - μ_{min}^{n}}

(21)

C = \frac{1}{2} \tan [π B^{- 1} (a - μ_{d})]

(22)

Under the controller, the UAV formation will achieve consensus with the reference value $μ^{d}$ , that is, $lim_{t \to \infty} γ_{i} = lim_{t \to \infty} γ_{j}$ and $μ_{i} = μ_{j} \to μ_{d}$ as $t \to \infty$ .

Before the proof, there are two lemmas that are supposed to be deduced to assist the proof of Definition 1.

Lemma 3

Consider the following equation

h (x) = \int_{0}^{x} σ (τ) d τ

(23)

Notice that the function $1 / 2 \tan (π x)$ in C is an inverse function of $2 / π \arctan (x)$ in equation (19), $σ (τ)$ is an invertible function, then, it is easy to see that

lim_{x \to \infty} \frac{2}{π} \arctan (x) = - 1

(24)

lim_{x \to - \infty} \frac{2}{π} \arctan (x) = 1

(25)

From equations (24) and (25), it can be verified that $h (x) \geq 0$ .

Lemma 4

Consider the following equation

p_{i} = a_{ij} \frac{2}{π} \arctan (K_{e} \sum_{k = 1}^{n} a_{ik} γ_{ik} + C)

(26)

Let $p_{ij} = p_{j} - p_{i}$ , where $(i, j) \in ε$ , $K_{e}$ denotes positive-definite diagonal matrix, $a_{ij}$ is an element in an adjacency matrix $A$ , which is described in the topology diagram $G (V, ε)$ , C is already defined in equation (22). Then $p_{ij} = 0_{n}$ and $η_{ij} = 0_{n}$ are mutually sufficient and necessary conditions, that is

p_{ij} = 0_{n} \Leftrightarrow γ_{ij} = 0_{n}

Proof: Let $l_{i} = 2 / π \arctan (K_{e} \sum_{k = 1}^{n} a_{ik} γ_{ik} + c)$ and $l_{j} = 2 / π \arctan (K_{e} \sum_{l = 1}^{n} a_{jl} γ_{jl} + c)$ .

The above function can be converted to

\sum_{i = 1}^{n} p_{ij} = \sum_{l = 1}^{n} a_{ij} (l_{j} - l_{i})

Note that $a_{ij} \neq 0$ when $(i, j) \notin ε$ , take account for the relationship between Laplacian matrix and the state w, one has

\sum_{i = 1}^{n} p_{ij} = \sum_{l = 1}^{n} a_{ij} (l_{j} - l_{i}) \Leftrightarrow (L \otimes I_{n}) l

If $\sum_{i = 1}^{n} p_{ij} = 0$ , it is easy to see that $(L \otimes I_{N}) l = 0$ . From Lemma 2, one has

l_{1} = l_{2} = \dots = l_{n}

(27)

that is, $l_{j} - l_{i} = 0$ .

From equation (27), it follows that

\sum_{k = 1}^{n} a_{ik} γ_{ik} = \sum_{l = 1}^{n} a_{jl} γ_{jl}

(28)

Consider the error variable of i UAV as

e_{i} = \sum_{j = 1}^{n} a_{ij} γ_{ij}

(29)

By equation (28), one has $e_{i} = e_{j}$ . Take account for the relationship between Laplacian matrix and the state $γ$ , the function $e_{i} = e_{j}$ is equivalent to the following one

e = (L \otimes I_{n}) γ

(30)

Form Lemma 1, if 0 is the eigenvalue of L, $1_{N}$ is the associated eigenvector, then, $L 1_{N} = 0$ , it follows that

\begin{matrix} \sum_{i = 1}^{n} e_{i} = (1_{n} \otimes I_{n}) e \\ = (1_{n} \otimes I_{n}) (L \otimes I_{n}) r \\ = (1_{n} L \otimes I_{n}) r \\ = 0 \end{matrix}

(31)

If equation (31) holds, where $(i, j) \in ε$ , one has $e_{i} = e_{j} = 0_{m}$ , that is

e = (L \otimes I_{n}) γ = 0_{nn}

(32)

From Lemma 2, it can be verified that $γ_{1} = γ_{2} = \dots = γ_{n}$ , that is, $γ_{ij} = 0$ .

On the other hand, if $γ_{ij} = 0_{n}$ , which means $w_{j} = w_{i}$ , it is easy to see that $p_{ij} = 0_{n}$ .

From equations (14), (19), and (29), the error function represented by ${\overset{\cdot\cdot}{γ}}_{ij}$ can be obtained, that is

{\overset{\cdot\cdot}{γ}}_{ij} = - K ({\overset{\cdot}{η}}_{ij} - B g_{ij})

(33)

where $(i, j) \in ε$ , KB is invertible. Suppose the error function is in equilibrium, that is, ${\overset{\cdot\cdot}{γ}}_{ij} = 0$ . If ${\overset{\cdot}{γ}}_{ij} = 0$ and $g_{ij} = 0$ , then the equation (33) holds. From Lemma 3, $g_{ij} = 0$ is equivalent to $γ_{ij} = 0$ . Therefore, the unique equilibrium point of the error function is obtained

[\begin{matrix} γ_{ij} \\ {\overset{\cdot}{γ}}_{ij} \end{matrix}] = 0_{2 n}

(34)

Consider the following Lyapunov Function candidate

V = \frac{1}{4} \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} {\overset{\cdot}{γ}}_{ij} + ρ (i, j) \in ε

(35)

where

ρ = 1_{1 \times n} {KBK}_{e}^{- 1} \sum_{i = 1}^{n} h (K_{e} \sum_{j \in Ω_{i}} γ_{ij} + C)

(36)

In the above function $Ω_{i}$ denotes the set of neighbors of ith UAV, $h (x) = \int_{0}^{x} σ (τ) d τ$ is already defined in equation (25). V(x) is the Lyapunov function composed of state ${\overset{\cdot}{γ}}_{ij}$ and $γ_{ij}$ . It is clear that V(x) is a positive semi-definite equation. Taking the derivative of V(x) with respect to time, one has

\begin{matrix} \overset{\cdot}{V} (x) = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} {\overset{\cdot\cdot}{γ}}_{ij} + \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {(\frac{\partial ρ}{\partial h} \frac{\partial h}{\partial η_{ij}})}^{T} {\overset{\cdot}{γ}}_{ij} \\ = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} [- K ({\overset{\cdot}{γ}}_{ij} - B g_{ij})] \\ + \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} KB σ (K_{e} \sum_{l \in Ω_{i}} γ_{il} + c) \\ = - \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} K {\overset{\cdot}{γ}}_{ij} + \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} KB g_{ij} \\ + \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} KB σ (K_{e} \sum_{l \in Ω_{i}} γ_{il} + c) \\ = - \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in Ω_{i}} {\overset{\cdot}{γ}}_{ij}^{T} K {\overset{\cdot}{γ}}_{ij} \\ \leq 0 \end{matrix}

(37)

Therefore, $\overset{\cdot}{V} (x)$ is a negative semi-definite equation. According to the LaSalle’s invariant set theorem, the invariant set is given by

Γ_{1} = {{[\begin{matrix} γ_{ij}^{T} & {\overset{\cdot}{γ}}_{ij}^{T} \end{matrix}]}^{T} \in R^{2 m} | {\overset{\cdot}{γ}}_{ij} = 0_{m}}, (i, j) \in ε

(38)

Substituting equation (38) to the error function equation (33), it can be rewritten as

{\overset{\cdot\cdot}{γ}}_{ij} = KB g_{ij}, (i, j) \in ε

(39)

Note that if the error equation trajectory moves within the invariant set $Γ_{1}$ , for $(i, j) \in ε$ , one has that $γ_{ij} \neq 0_{m}$ , then, ${\overset{\cdot\cdot}{γ}}_{ij} \neq 0_{m}$ , it is clear that the error equation will leave the invariant set $Γ_{1}$ without satisfying equation (39). Therefore, the largest invariant set of the system can be obtained, that is

Γ_{2} = {{[\begin{matrix} γ_{i j}^{T} & {\dot{γ}}_{i j}^{T} \end{matrix}]}^{T} \in R^{2 m} | {\dot{γ}}_{i j} = γ_{i j} = 0_{m}}, (i, j) \in ε

(40)

By equation (40), it can be shown that the largest invariant set is the same as the unique equilibrium point in equation (34). For the reason that the error function is asymptotically stable, then, $lim_{t \to \infty} γ_{i} = lim_{t \to \infty} γ_{j}$ is proved.

From equations (14) and (19), one has

{\overset{\cdot}{μ}}_{i} = - K (μ_{i} - μ_{d}) + s

(41)

where $s = - K [μ_{d} - A + B (2 / π) \arctan (K_{e} \sum_{j = 1}^{n} a_{ij} γ_{ij} + c)]$ , $μ^{d}$ denotes the reference value of the multi-agent system. Based on the above result, one knows that $lim_{t \to \infty} γ_{i} = lim_{t \to \infty} γ_{j}$ , then $γ_{ij} \to 0_{n}$ as $t \to \infty$ . By equations (20)–(22), it is clear that $lim_{t \to \infty} s = 0_{n}$ . According to equation (41), it can be verified that $μ_{i} \to μ_{d}$ as $t \to \infty$ , which means that the consensus with the desired state $μ_{d}$ will be reached asymptotically, then, $μ_{i} = μ_{j} \to μ_{d}$ as $t \to \infty$ is proved.

This completes the proof.

Remark 1

Define a virtual control input

μ_{vi} = A + B \frac{2}{π} \arctan (K_{e} \sum_{j = 1}^{n} a_{ij} γ_{ij} + C)

(42)

Substituting equation (42) into equation (19), one has

u_{i} = - K [μ_{i} - μ_{vi}]

(43)

From equation (42), it can be shown that $- 1 \leq (2 / π) \arctan (.) \leq 1$ . Therefore, when $(2 / π) \arctan (.) = 1$ , $μ_{vi}$ takes the maximum value; similarly, when $(2 / π) \arctan (.) = - 1$ , the value of $μ_{vi}$ is the minimum. Notice that the function $(2 / π) \tan (π x)$ in C is a strictly increasing function, if the initial state satisfies the constraints, which means that $μ_{max} \leq μ_{i} (0) \leq μ_{max}$ hold, then the state $μ_{i}$ (which is $- \cos σ_{i}$ ) will always satisfy the FOV constraint in the consensus stage.

Remark 2

The parameters K and $K_{e}$ in the consensus controller equation (14) determine the convergence rate of the consensus algorithm. The larger the K and $K_{e}$ is designed, the faster rate of convergence in the controller, and the UAV formation can rapidly reach the same state in the coordination stage; nevertheless, a large K and $K_{e}$ will lead to the system oscillate at the beginning. for the reason, a reasonable K and is $K_{e}$ design guarantee better system.

It is clear that the trajectory of the UAVs is determined by $a_{y}$ and $a_{z}$ , so it is necessary to find the relationship between the guidance command parameter $a_{y}$ and $a_{z}$ with the consensus control u. From the governing equation of relative motion about 3D homing guidance equations (6–10) and second-order system equation (14), it can be verified that

u_{i} - υ_{i} = g_{1, i} a_{y, i} + g_{2, i} a_{z, i}

(44)

where $υ = (V_{i} / R_{i}) \sin^{2} θ_{m, i} + (V_{i} / R_{i}) \cos^{2} θ_{m, i} \sin^{2} φ_{m, i}$ , $g_{1, i} = \sin φ_{m, i} / V_{i}$ and $g_{2, i} = \sin θ_{m, i} \sin φ_{m, i} / V_{i}$ .

It can be shown that there are two unknowns (a_y and a_z) in equation (48). For the reason that there are countless solutions for this equation. In order to obtain the optimal solution, define a matrix composed of acceleration vectors as $a_{i} = [\begin{matrix} a_{x, i} & a_{y, i} & a_{z, i} \end{matrix}]$ , when the $| | a_{i} | |_{2}$ is the smallest, it means the acceleration energy is the smallest. Then the virtual control $u_{i}$ is the most efficient solution under the premise, which is the optimal solution.

Let $| | a_{i} | |_{2}$ is the smallest, the corresponding optimal solution can be obtained as $a_{y, i} = g_{1, i} (u_{i} - ν_{i}) / (g_{1, i}^{2} + g_{2, i}^{2})$ and $a_{z, i} = g_{2, i} (u_{i} - ν_{i}) / (g_{1, i}^{2} + g_{2, i}^{2})$ . According to the above analysis, it can be verified that

{\begin{matrix} a_{y, i} = - \frac{V_{i}^{2}}{R_{i}} \sin φ_{m, i} + \frac{u_{i} V_{i}}{2 \sin φ_{m, i}} \\ a_{z, i} = - \frac{V_{i}^{2}}{R_{i}} \sin θ_{m, i} \cos φ_{m, i} + \frac{u_{i} V_{i}}{2 \sin θ_{m, i} \cos φ_{m, i}} \end{matrix}

(45)

where $u_{i}$ is the virtual command in Definition 1.

Based on the above results, the cooperative guidance law equation (40) in the cooperative stage is designed.

Remark 3

Define $| | a_{i} | |_{2}$ as total acceleration $a_{total}$ , it can be shown that $a_{total} = V_{i} | u_{i} - ν_{i} | / \sin σ_{i}$ . It is easy to see that the singularity occurs when the line of sight (LOS) $σ = 0$ , that is, the cooperative guidance law equation (45) in the cooperative stage cannot guarantee that the group of UAVs hit the target, the reason is that $σ = 0$ when UAV swarm against the target. Therefore, it is necessary to design a PNG controller for cooperative homing guidance when the UAV approaches the target.

Theorem 1

Consider the UAV formation system against a stationary target, the swarm system modeled as a second-order system designed as equation (14). Suppose the communication graph G defined on the formation system is undirected and connected. Then under the consensus controller equation (19) and the cooperative guidance law equation (40), the UAV formation satisfying the state $R_{i} / V_{i}$ and $\cos σ_{i}$ achieve consensus with the desired state $\cos σ_{d}$ , which guarantees of convergence without violating the constraints in the cooperative stage.

Proof: By taking equation (45) into (14) gives

{\overset{\cdot}{μ}}_{i} = \frac{u_{i} k_{1, i} \sin φ_{m, i} + u_{i} k_{2, i} \sin θ_{m, i} \cos φ_{m, i}}{k_{1, i} \sin φ_{m, i} + k_{2, i} \sin θ_{m, i} \cos φ_{m, i}} = u_{i}

(46)

It can be seen from equation (46) that the cooperative guidance law is transformed into a second-order integral form, which satisfies the conditions of the model in equation (14) mentioned above. On the contrary, it has been proved that the virtual control $u_{i}$ , which is designed in Definition 1 satisfies the condition of consensus. Therefore, it can be explained that the cooperative guidance law in equation (45) guarantees of convergence without violating the constraints, that is, $lim_{t \to \infty} η_{i} = lim_{t \to \infty} η_{j}$ and $μ_{i} = μ_{j} \to μ_{d}$ is achieved as $t \to \infty$ , $μ_{min} \leq - \cos σ_{i} \leq μ_{max}$ is guaranteed simultaneously.

The proof of Theorem 1 is completed.

Cooperative homing guidance laws

Consider a UAV swarm system against a target, if each of these UAV hit the target in coordination, it is necessary to ensure that the line of sight (LOS) angle is equal to zero. According to Remark 5, the cooperative guidance law in equation (45) is singular when $σ = 0$ occurs. Therefore, in order to avoid singularity, the cooperative guidance control law is supposed to be switched in the PNG stage, which guarantee to precisely against the target. As shown in Figure 3, since the UAV swarm is close enough to the target, the control strategy is switched from the cooperative stage to PNG stage.

The traditional PNG guidance law is design to against the target in the second stage (Song and Ha, 1994)

{\begin{matrix} a_{y, i} = - N V_{i} {\overset{\cdot}{λ}}_{y, i} \sin θ_{m, i} \sin φ_{m, i} + N V_{i} {\overset{\cdot}{λ}}_{z, i} \cos θ_{m, i} \\ a_{z, i} = - N V_{i} {\overset{\cdot}{λ}}_{y, i} \cos φ_{m, i} \end{matrix}

(47)

where Ni denotes the effective navigation constant, ${\overset{\cdot}{λ}}_{y, i}$ and ${\overset{\cdot}{λ}}_{z, i}$ are the rate of line-of-sight (LOS) angle components in the inertial frame, according to the relationship between the velocity frame and the inertial frame, one has

{\begin{matrix} λ_{y, i} = \frac{V_{m, i}}{R_{i}} \sin θ_{m, i} \\ λ_{z, i} = - \frac{V_{m, i}}{R_{i}} \cos θ_{m, i} \sin φ_{m, i} \end{matrix}

(48)

Substituting equation (54) into (53), the guidance command in PNG stage can be expressed as

{\begin{matrix} a_{y, i} = - \frac{{NV}_{i}^{2}}{R_{i}} \sin φ_{m, i} \\ a_{z, i} = - \frac{{NV}_{i}^{2}}{R_{i}} \sin θ_{m, i} \cos φ_{m, i} \end{matrix}

(49)

Remark 4

Once the UAV swarm is close enough to the target, the control strategy is switched from the cooperative stage to PNG stage. Define the switch point as $γ_{s}$ , afterwards, the control strategy switch to PNG stage as $γ_{i} < γ_{s}$ . In other words, since the UAV state achieves consensus in the cooperative stage, the control strategy is switched simultaneously.

Since the value of $γ_{s}$ is large, the time of cooperative stage is short, which lead to a larger rate of change in the leading angel. As a result, the overload of UAVs exceeded the limit at the PNG stage; on the contrary, as the value of $γ_{s}$ is small, the time of PNG stage will be shorter, then, the overload of UAVs over limited at the cooperative stage. Therefore, the value of $γ_{s}$ must be in a suitable range to ensure that the overload of UAVs does not exceed the maximum value. As shown in Figures 4 and 5, since the switching point $γ_{s}$ is between 5.4 and 20 s, the overload can be guaranteed in the range of $- 100 to 100 m / s^{2}$ . For this reason, the switching point $γ_{s}$ is selected as 14 s in the paper.

Remark 5

Since the cooperative stage is completed, all of the UAV’s states are identical in equations (11) and (12). As a result, once the control strategy switch to PNG stage, the initial states of UAVs are all the same, that is, there are identical $γ_{i}$ and $σ_{i}$ at the beginning in PNG stage.

Theorem 2

Consider the UAV formation system against a stationary target, the swarm system modeled as a second-order system designed as equation (14). Suppose the communication graph G defined on the formation system is undirected and connected. Then under the consensus controller equation (19) and the cooperative guidance law equation (40) in the cooperative stage, and the PNG guidance law equation (46) in the PNG stage, respectively, the UAV formation satisfying the state $R_{i} = 0$ with the time-to-go values $t_{go, i} = t_{go, j}$ , which guarantees of hitting the target simultaneously without violating the constraints.

Proof: Based on the PNG laws in equation (49), it can be verified that

{\begin{matrix} {\overset{\cdot}{γ}}_{i} = - \cos σ_{i} \\ {\overset{\cdot}{σ}}_{i} = - (N - 1) γ_{i} \sin σ_{i} \end{matrix}

(50)

By equation (50), it can be shown that the variation of $γ_{i}$ and $σ_{i}$ is determined by the effective navigation constant N and the initial conditions. Based on the Remark 5, it is known that the initial conditions $γ_{i}$ and $σ_{i}$ are identical in the PNG stage. As a result, it can be guaranteed that $lim_{t \to \infty} γ_{i} = lim_{t \to \infty} γ_{j}$ and $lim_{t \to \infty} R_{i} / V_{i} = lim_{t \to \infty} R_{j} / V_{j}$ since $t \to \infty$ , that is, the UAV swarm against the target simultaneously without violating the constraints, which guarantees of convergence.

Theorem 2 is proved.

The schematic diagrams of the consensus control scheme and algorithm procedure are provided in Figures 6 and 7, respectively.

As shown in Figure 6, the overall control scheme consists of the HRV model, communication relationship designer, and Piecewise guidance strategy consensus controller. Especially, the Piecewise guidance strategy consensus controller is composed of switch point trigger, cooperative strategy controller, and PNG controller. At the beginning, the time-to-go state $γ_{i} \geq γ_{s}$ , the consensus controller is in the cooperative strategy. When the UAV swarm is close enough to the target, the consensus controller is switched to the PNG controller from switch point trigger as $γ_{i} < γ_{s}$ . until hitting the target simultaneously.

Simulation results

Simulation results of different scenarios

To verify the accuracy and effectiveness of the cooperative homing guidance control algorithm considering FOV constraints in this paper, a series of combat scenarios are designed, including different formation of five UAV to against stationary targets cooperatively, which are represented by Case 1–3.

Different formations are formed based on diversiform operational scenarios and mission requirements. Case 1: horizontal formation is mainly used for large patrol areas; Case 2: vertical formation is appropriate for airdrop bombing and regional obstacle avoidance; Case 3: diamond formation applies to fuel conservation through prominent aerodynamic performance. The three different formations in application are shown in Figure 8.

Therefore, a reasonable and effective formation is capable of tremendously improving the security of formation and mission completion rate. Then the numerical simulations are conducted from above three cases.

In both of the three cases, in order to facilitate simulation analysis, the target is assumed to be the origin of coordinate system. Based on the actual flight dynamics conditions, the maximum and minimum angles of FOV are defined as $60 °$ and $0 °$ , respectively, and the maximum acceleration of UAV is $100 m / s^{2}$ , the desired leading angel $σ_{d}$ is designed as $25 °$ . According to the definition in Chapter 2, the communication topology of UAV is connected and undirected simultaneously.

The communication topology for each UAV is shown in Figure 9. Consider the practical reality, the network topology among UAVs is undirected and connected.

Based on the above results, one can see that the interconnection adjacency matrix A can be expressed as

A = [\begin{matrix} 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{matrix}]

(51)

Case 1: horizontal formation

The initial states of each UAV for horizontal formation are selected in Table 1. Generally, the performance of the cooperative guidance controller is delicately dependent on the parameters, which is K and $K_{e}$ . According to Remark 3, the controller is designed with $K = 1$ and $K_{e} = 4.4$ . The simulation results are shown in Figures 10 –15.

Table 1.

Initial states of UAVs for horizontal formation.

	V, m/s	R, km	$θ_{L}$ , °	$φ_{L}$ , °	$θ_{m}$ , °	$φ_{m}$ , °
UAV1	330	16029	−29.94	36	−30	−40
UAV2	325	16007	−29.98	38	−25	−30
UAV3	320	16000	−30	40	10	20
UAV4	315	16007	−29.98	42	15	25
UAV5	310	16029	−29.94	44	20	30

UAV: unmanned aerial vehicle.

As shown in Figure 10, the UAV swarm initially in horizontal formation, the trajectories of the five UAVs are capable of hitting the targets precisely and simultaneously from Figures 10, 11, and 13. According to Figure 11, it is clear that the leading angel converge to $25 °$ in the cooperative stage at 40.9 s and finally converge to zero in the guidance stage simultaneously, which demonstrates an effective cooperation under the controller; on the contrary, it can be seen that the leading angles are limited between $0 °$ and $60 °$ , which indicates the FOV constraints are satisfied all the time. From Figures 14 and 15, it is shown that the accelerations are all within the limited among the two stages.

Above all, it is indicated that the UAV swarm for horizontal formation against the target simultaneously without violating the constraints, which achieves of convergence.

Case 2: vertical formation

The initial states of each UAV for vertical formation are shown in Table 2. According to aforementioned study, the parameters K and $K_{e}$ determine the performance of the cooperative guidance control law, that is, $K = 1$ and $K_{e} = 4.4$ . The simulation results are shown in Figures 16 –21.

Table 2.

Initial states of UAVs for vertical formation.

	V, m/s	R, km	$θ_{L}$ , °	$φ_{L}$ , °	$θ_{m}$ , °	$φ_{m}$ , °
UAV1	330	16,348	−29.30	36	−30	−40
UAV2	325	16,174	−29.65	38	−25	−30
UAV3	320	16,000	−30	40	10	20
UAV4	315	15,827	−30.36	42	15	25
UAV5	310	15,655	−30.73	44	20	30

UAV: unmanned aerial vehicle.

From Figure 16, the vertical formation and horizontal formation rotate 90° from the UAV 3. It is clear that the trajectories of the five UAVs are able to proceed against the targets precisely and simultaneously from, Figures 16, 17, and 19. As shown in Figure 18, the leading angel converges to $25 °$ in the cooperative stage at 39.9 s and finally converge to zero in the guidance stage simultaneously. Compared to Case 1, the convergence speed and response rate of the leading angel for the vertical formation in Case 2 are a little faster than the horizontal formation. On the other side, the leading angles are limited between $0 °$ and $60 °$ , which indicates the FOV constraints are satisfied all the time. From Figures 20 and 21, it is shown that the accelerations are all within the limited among the two stages.

On conclusion, it is indicated that the UAV swarm for vertical formation against the target simultaneously without violating the constraints, which confirms the convergence of the coordinated control algorithm.

Case 3: diamond formation

The initial states of each UAV for diamond formation are selected in Table 3. It is easy to see that the performance of the cooperative guidance control law is delicately dependent on the parameters, which is K and $K_{e}$ . According to Remark 3, the controller is designed with $K = 1$ and $K_{e} = 4.4$ . The simulation results are shown in Figures 22 –27.

Table 3.

Initial states of UAVs for diamond formation.

	V, m/s	R, km	$θ_{L}$ , °	$φ_{L}$ , °	$θ_{m}$ , °	$φ_{m}$ , °
UAV1	330	16,029	−29.94	36	−30	−40
UAV2	325	16,348	−29.29	40	−25	−30
UAV3	320	16,000	−30	40	10	20
UAV4	315	15,655	−30.73	40	30	40
UAV5	310	16,029	−29.94	44	20	30

UAV: unmanned aerial vehicle.

According to Figure 22, the UAV swarm initially in diamond formation, which combines with the horizontal formation in Case 2 and the vertical formation in Case 1. The trajectories of the five UAVs are capable of hitting the targets precisely and simultaneously from, Figures 22, 23, and 25. As shown in Figure 24, one has that the leading angel converges to $25 °$ in the cooperative stage at 40.8 s and finally converge to zero in the guidance stage simultaneously. Comparing the diamonds formation with Cases 1 and 2, the convergence speed of the leading angel is almost the same as Case 1, and the response rate is faster than Case 2. After all, the accuracy and effectiveness of the controller is demonstrated. On the other side, the leading angles are limited between $0 °$ and $60 °$ , which indicates the FOV constraints are satisfied all the time. From Figures 26 and 27, it is shown that the accelerations are all within the limited among the two stages.

In conclusion, it is indicated that the UAV swarm for diamond formation against the target simultaneously without violating the constraints, which improves the efficiency and accuracy of the cooperative interception of the target.

Comparisons with traditional guidance law and decentralized-cooperative guidance law

In the subsection, to demonstrate the innovation of the proposed approaches to multi-UAVs cooperative guidance law, the proposed method in the paper is compared with traditional and decentralized-cooperative guidance law(D-CPN). in order to facilitate simulation analysis, the target is assumed to be the origin of coordinate system.

Case 1: comparison with traditional guidance law

The methods used in the traditional PNG guidance method (Song and Ha, 1994) have been compared with the work presented in this paper. The PNG guidance results are shown in Figures 28 and 29.

Simulations show that the trajectories of the five UAVs are capable of hitting the targets precisely; however, according to Figure 29, it can be seen that the UAV formation cannot achieve simultaneous hitting of the target, the control parameters need to be precisely and repeatedly adjusted to meet the FOV constraint. Moreover, the PNG guidance law does not theoretically guarantee that the FOV meet the constraints.

Case 2: comparison with D-CPN

For other methods of cooperative guidance, this paper used in the D-CPN cooperative guidance method (Zhao and Zhou 2015) have been compared with the work presented in this paper. The D-CPN cooperative guidance results are shown in Figures 30, 31 and 32:

As can be seen from the above figures, compared with the traditional PNG guidance law, the D-CPN cooperative guidance law can ensure that multiple-UAV formations hit the target at the same time, and the accelerations during the entire flight are less than the overload limit of the UAV. However, it can be seen from Figure 12 comparing Figure 32 based on the guidance law of this paper, the parameter response speed of the D-CPN cooperative guidance law is slow, and the control parameters need to be precisely and repeatedly adjusted to meet the FOV constraint.

In conclusion, it is indicated that the proposed cooperative guidance law in this paper achieved simultaneous target hits without violating the constraints, which improves the efficiency and accuracy of the cooperative interception of the target.

Conclusion and future research

This paper studies a 3D piecewise guidance scheme for multiple UAVs guaranteeing simultaneous arrival even under the FOV constraint. In order to consider the FOV constraint, the two-stage (cooperative stage and PNG stage) cooperative control strategy is taken into account.

For the cooperative stage, algebraic graph theory is adopted to communicate with neighbors’ new auxiliary states, in which time-to-go ( $γ$ ) and trigonometric of leading angle ( $μ$ ) are mentioned. A distributed cooperative control law has been proposed to make UAV swarms consistent and convergent in finite time. In addition, the proposed cooperative controller can be designed without depending on the global states. For the FNG stage, by applying the PNG guidance control law, the leading angle reduced to zero to ensure that the UAVs hit the target simultaneously.

Finally, to verify the performance of the cooperative guidance control law considering FOV constraint, a series of combat scenarios are designed, including different formation (horizontal formation, vertical formation, and diamond formation) of five UAV to against stationary. Simulation results show that the designed controller performs accuracy and effectiveness, which confirms the convergence of the coordinated control algorithm without violating the constraints.

The future work will attempt to consider the extension of formation control such as leader–follower consensus. Consider actual operational requirements, it is necessary to take into consideration the UAV formation transformation before attacking the target. Besides, the application of this guidance control law based on the real UAV model should be improved.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported in part by the National Natural Science Foundation of China (grant nos. 61803308 and 61973254), and in part by the Natural Science Basic Research Program of Shaanxi (Program No. 2020JQ-221).

ORCID iDs

Liu Zequn

Guo Jianguo

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