Distributed cooperative guidance law without numerical singularity with field-of-view angle constraint

Abstract

A distributed cooperative guidance law without numerical singularities is proposed for the simultaneous attack a stationary target by multiple vehicles with field-of-view constraints. Firstly, the vehicle engagement motion model is transformed into a multi-agent model. Then, based on the state-constrained consensus protocol, a coordination control law with field-of-view (FOV) constraints is proposed. Finally, the cooperative guidance law has been improved to make it more suitable for practical application. Numerical simulations verified the effectiveness and robustness of the proposed guidance law in the presence of acceleration saturation, communication delays and measurement noise.

Nomenclature

${r_i}$

the distance between the vehicle and target

${v_i}$

the velocity of the vehicle

${a_i}$

the acceleration of the vehicle

${u_i}$

the virtual input of the vehicle

$w,B,c$

constant in the virtual control law

PNG

proportional navigation guidance

LOS

line of sight

FOV

field-of-view

Greek symbol

$\sigma $

leading angle

$\lambda $

LOS angle

$\theta $

velocity angle

$\psi $

yaw angle

$\xi $

auxiliary variables

$\zeta $

auxiliary variables

1.0 Introduction

With the development and progress of interception and defense systems, it is becoming increasingly difficult for individual vehicles to break through blockades and avoid interception by defense systems [1]. Cooperative attack by multiple vehicles is an effective strategy that can break through defense systems. As a result, there has been great progress in cooperative guidance and control technologies in recent years. Existing cooperative guidance research can be classified into two strategies [2]. The first strategy is based on the impact time control guidance (ITCG) law [3], its guidance laws are designed for a single vehicle, and a common desired arrival time for each vehicle is preassigned. The second strategy is to obtain information through communication between the vehicles and use it to generate guidance commands [4].

The ITCG-based simultaneous arrival guidance law has been extensively studied. The ITCG law is proposed in Ref. [3] and applied to multi-missile simultaneous arrival tasks. This has resulted in a plethora of studies investigating the simultaneous arrival problem [5–7]. Although the ITCG-based strategy can achieve simultaneous arrival of multiple vehicles, the preassigned expected arrival time must be carefully selected. Considering the constraints of velocity of the vehicles, an unreasonable expected arrival time may cause the vehicle unable to meet the constraints of simultaneous arrival. The strategy based on information exchange is an effective way to overcome the shortcomings of ITCG-based strategy. A cooperative proportional navigation guidance (CPNG) law is proposed in Ref. [8], which enables a simultaneous arrival by multiple vehicles by designing a time-varying control law for the navigation coefficients. An online optimal cooperative guidance law is designed by applying model predictive control (MPC) techniques in Ref. [9].

In recent years, multi-agent consensus protocols have been widely studied [10–14] and applied to the design of cooperative guidance law for multiple vehicles [15–18]. The remaining flight time is usually used as a coordinate variable and then coordinate variable of multiple vehicles is controlled to converge by designing a consistency protocol, thus enabling simultaneous arrival by multiple vehicles [6, 16]. However, this approach requires the ability to accurately estimate the remaining flight time. A cooperative guidance law with FOV constraints is proposed in Ref. [19], which consists of two phases, namely the arrival time synchronisation phase and proportional navigation phases. In Ref. [20], a guidance law that satisfies both the impact time and impact angle constraints is proposed, however, this guidance law requires the vehicle to have the ability to change its own velocity magnitude. A 3D cooperative guidance law for multiple vehicles based on the event-triggered strategy is proposed in Ref. [17], which can reduce the resource utilisation burden while ensuring the good performance of the guidance law.

Although numerous cooperative guidance law have been proposed up to now, there are still some practical issues that require further attention. First, a significant proportion of cooperative guidance laws do not take into account the vehicle’s FOV constraints [21–23]. This can result in target loss of lock during the trajectory adjustment. Second, many cooperative guidance law are afflicted by numerical singularity problem, which could render guidance commands diverge to infinity when the vehicle-target range is close to zero [4, 23]. To circumvent this issue, a two-phase guidance law is frequently employed [19, 24]. However, ensuring the continuity of guidance commands when transitioning between guidance laws remains challenging. Thirdly, the majority of existing cooperative guidance law have not been subjected to rigorous testing to ascertain their effectiveness and robustness in a wide range of scenarios. Nevertheless, these significant issues are seldom discussed in the literature on cooperative guidance due to the highly nonlinear system dynamics involved.

Considering the shortcomings of existing guidance laws, this study proposes a distributed cooperative guidance law with FOV constraints. First, the motion engagement model for vehicle is transformed into a two-integrator model. Then, an arrival time cooperative control law is designed based on the state constraint consensus algorithm. This study’s major contributions and innovations are as follows:

1. (1) A cooperative guidance law considering the FOV constraint is proposed based on the state constraint consensus protocol, which is more feasible and reliable for practical applications.

2. (2) The proposed guidance law has no numerical singularity, and acceleration is bounded. This means that there is no need to switch the guidance law to avoid numerical singularity problems [19, 24].

3. (3) The results of the simulation demonstrate that the proposed guidance law is capable of functioning effectively in a variety of guidance scenarios. Furthermore, the guidance law exhibits robust performance even in the presence of communication delays and measurement noise.

The rest of this paper is organised as follows. Section 2 presents the necessary background and problem statement. Section 3 introduces the main results of the study, including the design of a cooperative guidance law. Then, Section 4 presents the numerical simulations and analyses the results. Finally, Section 5 presents the conclusions and prospects.

Figure 1. Two-dimensional vehicle–target engagement geometry.

2.0 Problem statement and preliminaries

2.1 Problem statement

The following assumptions are first introduced:

Assumption 1. The vehicles and the target are regarded as mass points.

Assumption 2. The speed of vehicles is assumed to be constant, and the target is considered to be stationary.

Considering a scenario in which $N \geqslant 2$ vehicles attack the same static target, Fig. 1 depicts the relative motion between the $i$ th vehicle and the target in the two-dimensional plane, where ${M_i} - xy$ is a Cartesian inertial reference frame, and ${M_i}$ and $T$ represent the $i$ th vehicle and target, respectively. The planar kinematics model of vehicle–target engagement can be described as:

(1) \begin{align}{\dot r_i} = - {v_i}\,{\rm{cos}}\,{\sigma _i}\end{align}

(2) \begin{align}{\dot \lambda _i} = - \frac{{{v_i}\,{\rm{sin}}\,{\sigma _i}}}{{{r_i}}}\end{align}

(3) \begin{align}{\dot \theta _i} = \frac{{{a_i}}}{{{v_i}}}\end{align}

(4) \begin{align}{\theta _i} = {\lambda _i} + {\sigma _i}\end{align}

where ${r_i}$ represents the relative distance between the vehicle and the target; ${v_i}$ represents the speed of the $i$ th vehicle; ${a_i}$ represents the acceleration of the $i$ th vehicle, which is perpendicular to the speed direction; and ${\lambda _i},{\theta _i},{\sigma _i}$ represent the line-of-sight angle, heading angle and leading angle of the $i$ th vehicle, respectively. The positive direction of the angle is counterclockwise.

The purpose of the designed cooperative guidance law is to control $N$ vehicles to reach target $T$ at the same time; that is

(5) \begin{align}\left| {\frac{{{r_i}}}{{{{\dot r}_i}}} - \frac{{{r_j}}}{{{{\dot r}_j}}}} \right| \to 0,\forall i \ne j;\;i,j = 1,2, \cdots, N\end{align}

Consider that the FOV angle of the vehicle should not exceed its maximum allowable value. With the assumption of a small angle-of-attack [9], the FOV constraint can be approximated as

(6) \begin{align}\left| {{\sigma _i}\left( t \right)} \right| \lt {\sigma _{{\rm{max}}}} \le \frac{\pi }{2},t \geqslant {t_0},\forall i = 1,2, \cdots, N\end{align}

where ${\sigma _{{\rm{max}}}} \gt 0$ represents the maximum allowable value of the leading angle of vehicles.

Remark 1. It should be noted that this study considers the isomorphic vehicles; thus, it is assumed that the maximum FOV allowed by each vehicle has the same value; that is, ${\sigma _{{\rm{max}},1}} = {\sigma _{{\rm{max}},2}} = \cdots = {\sigma _{{\rm{max}},N}} = {\sigma _{{\rm{max}}}}$ .

2.2. Preliminaries

2.2.1. Graph theory

The topology of the communication network between the vehicles is described as an undirected weight graph $\mathcal{G}\left( {\mathcal{V},\mathcal{E}} \right)$ , where $\mathcal{V} = \left\{ {1, \cdots, N} \right\}$ represents the set of nodes, where node $i$ corresponds to the $i$ th vehicle. $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ represents the set of edges. The neighbourhood set of node $i$ is defined as ${\mathcal{N}_i} = \{ j \in \mathcal{V}|\left( {i,j} \right) \in \mathcal{E}\} $ . The adjacency matrix of graph $\mathcal{G}$ is expressed as $A = \left[ {{a_{ij}}} \right] \in {\mathbb{R}^{N \times N}}$ , and the unordered pair $\left( {i,j} \right) \in \mathcal{E} \Leftrightarrow {a_{ij}} \gt 0 \Leftrightarrow $ vehicle $i$ and vehicle $j$ can communicate and exchange information; ${a_{ij}} = 0 \Leftrightarrow \left( {i,j} \right) \notin \mathcal{E}$ . In addition, ${a_{ii}} = 0,i = 1,2, \cdots, N$ . Finally, the Laplacian matrix of $\mathcal{G}$ is $L = \left[ {{l_{ij}}} \right] \in {\mathbb{R}^{N \times N}}$ , where ${l_{ii}} = \mathop \sum \nolimits_{j = 1}^N {a_{ij}}$ , and ${l_{ij}} = - {a_{ij}},i \ne j$ .

To satisfy the simultaneous arrival constraint, the communication topology $\mathcal{G}$ among vehicles needs to satisfy the following assumption.

Assumption 3. $\mathcal{G}\left( {\mathcal{V},\mathcal{E}} \right)$ is undirected and connected, which means the $i$ th vehicle and the $j$ th vehicle can get information from each other if $\left( {{v_i},{v_j}} \right) \in \mathcal{E}$ , and there is a communication path that involves all vehicles.

2.2.2 Useful lemmas

Lemma 1. [25] Consider a second-order multi-agent system with the following dynamical equations:

(7) \begin{align}{\dot \xi _i} = {\zeta _i},i = 1, \cdots, N\end{align}

(8) \begin{align}{\dot \zeta _i} = {u_i},i = 1, \cdots, N\end{align}

with the constraint

(9) \begin{align}{\zeta _{{\rm{min}}}} \le {\zeta _i}\left( t \right) \gt {\zeta _{{\rm{max}}}},t \ge {t_0},i = 1,2, \cdots, N,\end{align}

where ${\xi _i},{\zeta _i} \in {\mathbb{R}^m}$ denote the state of the $i$ th agent; ${u_i} \in {\mathbb{R}^m}$ denotes the input of the $i$ th agent; and ${\zeta _{{\rm{min}}}} = \left[ {\zeta _{{\rm{min}}}^1, \cdots, \zeta _{{\rm{min}}}^m{]^T},{\zeta _{{\rm{max}}}} = } \right[\zeta _{{\rm{max}}}^1, \cdots, \zeta _{{\rm{max}}}^m{]^T}$ is a constant value.

If the following conditions are satisfied,

1. (1) the communication topology $\mathcal{G}$ among agents is undirected and connected;

2. (2) (2) ${\zeta _i}\left( {{t_0}} \right) \in \left[ {{\zeta _{{\rm{min}}}},{\zeta _{{\rm{max}}}}} \right],i = 1,2, \cdots, N$ ; and the consensus protocol is designed as

(10) \begin{align}{u_i} = - K\left( {{\zeta _i} - {\zeta _{ir}}} \right)\end{align}

(11) \begin{align}{\zeta _{ir}} = w + Bf\left( {{K_e}\mathop \sum \limits_{j \in {\mathcal{N}_i}} {a_{ij}}\left( {{\xi _i} - {\xi _j}} \right) + c} \right)\end{align}

where $K,{K_e} \in {\mathbb{R}^{m \times m}}$ are a positive definite diagonal matrix; ${a_{ij}}$ is the element of the adjacency matrix $A$ ; and $f\left( \cdot \right)$ is a continuous function that has the following properties:

P1: $f\left( \cdot \right)$ is a strictly increasing function;

P2: $f\left( 0 \right) = 0$ ;

P3: $\mathop {{\rm{lim}}}\limits_{x \to + \infty } f\left( x \right) = 1$ ;

P4: $\mathop {{\rm{lim}}}\limits_{x \to - \infty } f\left( x \right) = - 1$ .

$w,B$ , and $C$ are defined as follows:

(12) \begin{align}w = \frac{{{\zeta _{{\rm{min}}}} + {\zeta _{{\rm{max}}}}}}{2}\end{align}

(13) \begin{align}B = \frac{1}{2}{\rm{diag}}\left\{ {\zeta _{{\rm{max}}}^1 - \zeta _{{\rm{min}}}^1, \cdots, \zeta _{{\rm{max}}}^m - \zeta _{{\rm{min}}}^m} \right\}\end{align}

(14) \begin{align}c = {f^{ - 1}}\left( {{B^{ - 1}}\left( {w - {\zeta _d}} \right)} \right)\end{align}

where ${f^{ - 1}}\left( \cdot \right)$ denotes an inverse function of $f\left( \cdot \right)$ , and ${\zeta _d}$ denotes the reference value of ${\zeta _i},i = 1,2, \cdots, N$ .

Then, the multi-agent system Equations (7) and (8) will reach consensus without violating the constraint Equation (9); that is, ${\xi _i} \to {\xi _j}$ and ${\zeta _i} \to {\zeta _d}$ asymptotically as $t \to \infty $ .

Remark 2. The four properties of function $f\left( \cdot \right)$ mentioned in Lemma 1 are actually not strict. For example, functions such as $tanh\;x = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}$ , $\frac{2}{\pi }\;arctan\;x$ all have these properties.

3.0 Cooperative guidance law design

3.1 Engagement model transformation

Let ${y_{v,i}} = {r_i}\,{\rm{sin}}\,{\sigma _i}$ denote the component of ${r_i}$ in the direction perpendicular to the velocity of the $i$ th vehicle. When ${\sigma _i}$ is small, we take the second derivative of ${y_{v,i}}$ and substitute $\,{\rm{sin}}\,{\sigma _i} \approx {\sigma _i},\,{\rm{cos}}\,{\sigma _i} \approx 1$ . Then,

(15) \begin{align}{\ddot r_i}{\sigma _i} + 2{\dot r_i}{\dot \sigma _i} + {r_i}{\ddot \sigma _i} = {\ddot y_{v,i}} = {a_i}\end{align}

Therefore, we design

(16) \begin{align}{a_i} = {\sigma _i}{u_i} + 2{\dot r_i}{\dot \sigma _i} + {r_i}{\ddot \sigma _i}\end{align}

where ${u_i}$ is a virtual control signal of vehicle $i$ . Substituting Equation (16) into (15) yields

(17) \begin{align}{\ddot r_i} = {u_i}\end{align}

Thus, the nonlinear model has been transformed into a double-integrator model. In addition, substituting Equations (1), (3) and (4) into Equation (16), we have

(18) \begin{align}{a_i} = {\sigma _i}{u_i} + \frac{2}{3}{v_{}}{\dot \lambda _i} + \frac{1}{3}{r_i}{\ddot \sigma _i}{\rm{\;\;}}\end{align}

Remark 3. It should be noted that the Equation (15) was derived with the assumption of a small leading angle. However, this is not a hard constraint that must be strictly satisfied for the subsequent proposed guidance law. The reason is that the small leading angle assumption was not considered in the subsequent proof of the convergence of the guidance law.

To convert the FOV constraints of the vehicles into the state constraint of the agents, two auxiliary variables are introduced; that is,

(19) \begin{align}{\xi _i} = \frac{{{r_i}}}{{{v_i}}}\end{align}

(20) \begin{align}{\zeta _i} = \frac{{{{\dot r}_i}}}{{{v_i}}}\end{align}

Based on Equation (17), we can obtain

(21) \begin{align}{\dot \xi _i} = {\zeta _i}\end{align}

(22) \begin{align}{\dot \zeta _i} = \frac{{{u_i}}}{{{v_i}}}\end{align}

Then, the simultaneous arrival constraint Equation (5) is equivalent to

(23) \begin{align}\left| {{\xi _i} - {\xi _j}\left| { \to 0,} \right|{\zeta _i} - {\zeta _j}} \right| \to 0,\forall i \ne j;\;i,j = 1,2, \cdots, N\end{align}

Furthermore, note that ${\zeta _i} = - \,{\rm{cos}}\,{\sigma _i}$ ; then, the constraint Equation (6) can be converted to

(24) \begin{align} - 1 = {\zeta _{{\rm{min}}}} \le {\zeta _i} \le {\zeta _{{\rm{max}}}} = - \,{\rm{cos}}\,{\sigma _{{\rm{max}}}}\end{align}

Here, we used the fact that $0 \le \left| {{\sigma _i}} \right| \le {\sigma _{{\rm{max}}}} \lt \frac{\pi }{2}$ . As a result, the engagement model is transformed into a multi-agent system, so that we can achieve simultaneous arrivals by controlling the states of multi-agent reach consensus. At the same time, the FOV constraints of the vehicles are transformed into state constraints of the agents.

3.2 Coordinated control of arrival time

Based on Equations (21) and (22) and Lemma 1, ${u_i}$ is designed as

(25) \begin{align}{u_i} = - K{v_i}\left[ {{\zeta _i} - w + B{\rm{tanh}}\left( {{K_e}\mathop \sum \limits_{j = 1}^n {a_{ij}}\left( {{\xi _i} - {\xi _j}} \right) + c} \right)} \right]\end{align}

where $K$ and ${K_e}$ are parameters that need to be designed; ${v_i}$ is the velocity of the $i$ th vehicle; ${\xi _i}$ and ${\zeta _i}$ are defined in Equations (19) and (20), respectively; and ${a_{ij}}$ is an element of the adjacency matrix $A$ corresponding to the communication topology of vehicles. $w$ , $B$ and $c$ are defined as follows:

(26) \begin{align}w = \frac{{ - \,{\rm{cos}}\,{\sigma _{{\rm{max}}}} - 1}}{2},\end{align}

(27) \begin{align}B = \frac{{ - \,{\rm{cos}}\,{\sigma _{{\rm{max}}}} + 1}}{2},\end{align}

(28) \begin{align}c = {\rm{tan}}{{\rm{h}}^{ - 1}}\left( {{B^{ - 1}}\left( {w + \,{\rm{cos}}\,{\sigma _d}} \right)} \right).\end{align}

where ${\sigma _d}$ is the expected value of ${\sigma _i}$ .

Remark 4. From the point of view of the vehicle successfully reaching the target, ${\sigma _d}$ should be set to $0$ . On the other hand, it should be note that $c \to + \infty $ when ${\sigma _d} = 0$ . But the numerical singularity problem still would not occur, because ${u_i} = - K{v_i}\left( {{\zeta _i} - w + B} \right)$ when $c \to + \infty $ , and in practical, we take $c$ to be large enough to ensure sufficient accuracy for convenience of calculation.

Let ${\zeta _{ir}} = w - B{\rm{tanh}}\left( {{K_e}\mathop \sum \nolimits_{j = 1}^n {a_{ij}}\left( {{\xi _i} - {\xi _j}} \right) + c} \right)$ ; then, Equation (25) could be rewritten as

(29) \begin{align}{u_i} = - K{v_i}\left( {{\zeta _i} - {\zeta _{ir}}} \right).\end{align}

Note that ${\zeta _{ir}} \in \left[ {{\zeta _{{\rm{min}}}},{\zeta _{{\rm{max}}}}} \right]$ , then $\left| {{u_i}} \right| \le K{v_i}\left( {{\zeta _{{\rm{max}}}} - {\zeta _{{\rm{min}}}}} \right)$ . Since ${a_i} = {\sigma _i}{u_i} + 2{\dot r_i}{\dot \sigma _i} + {r_i}{\ddot \sigma _i}$ , it is obvious that the second term and the third term are bounded; thus, ${a_i}$ is also bounded.

3.3 Modified cooperative guidance law

Although a cooperative guidance law that takes into account the FOV constraint can be obtained by substituting Equation (25) into (18). However, note that there is a high-order term that includes ${\ddot \sigma _i}$ in Equation (18), and it is difficult to obtain an accurate value of Equation (18), either from actual measurements or from numerical simulations.

To inherit the nonnumerical singularity of Equation (18) and avoid its disadvantage, we modify it and propose a cooperative guidance law considering the FOV constraint as follows:

(30) \begin{align} {{a_i}} & = a_i^{PNG} + a_i^{BT}\nonumber \\[5pt] {} & = {N_p}{v_i}{{\dot \lambda }_i} - K{v_i}{\sigma _i}\left[ {{\zeta _i} - w + B{\rm{tanh}}\left( {{K_e}\mathop \sum \limits_{j = 1}^n {a_{ij}}\left( {{\xi _i} - {\xi _j}} \right) + c} \right)} \right], \end{align}

where $a_i^{PNG} = {N_p}{v_i}{\dot \lambda _i}$ is the PNG term; here, ${N_p} = 3$ is the navigation constant. $a_i^{BT} = {\sigma _i}{u_i} = - K{v_i}{\sigma _i}\left[ {{\zeta _i} - w + B{\rm{tanh}}\left( {{K_e}\mathop \sum \nolimits_{j = 1}^n {a_{ij}}\left( {{\xi _i} - {\xi _j}} \right) + c} \right)} \right]$ is the arrival time coordination term, where the variables have the same definition as in Equation (25).

Compared with Equations (18), (30) has two improvements. First, the navigation constant has been modified to ${N_p} = 3$ . This is because the cooperative guidance law will become a PNG law when the arrival time of each vehicle is synchronised, and it has been shown that ${N_p} = 3$ can lead to stable and efficient guidance performance in practice [19, 24]. Second, the term ${r_i}{\ddot \sigma _i}/3$ is discarded in Equation (30). The reason is that it is difficult to obtain an accurate value of ${\ddot \sigma _i}$ in both practical application and numerical simulation. In addition, even if this term has been discarded, it will not affect guidance performance, which will be shown in the following numerical simulations.

Theorem 1. Under the conditions that assumptions 1, 2, 3 hold, the guidance law Equation (30) will control the simultaneous arrival of all vehicles and will always not violate the FOV constraints.

Proof. According to Equations (3), (4), we have that

(31) \begin{align}{\dot \sigma _i} = \frac{{{a_i}}}{{{v_i}}} - {\dot \lambda _i}\end{align}

Substituting Equations (29), (30) into Equation (31) yields

(32) \begin{align} {{{\dot \sigma }_i}} & = \left( {{N_p} - 1} \right){{\dot \lambda }_i} - K{\sigma _i}\left( {{\zeta _i} - {\zeta _{ir}}} \right)\nonumber \\[5pt] {} & = - K{\sigma _i}\left( {{\zeta _i} - {\zeta _{ir}}} \right) - \left( {{N_p} - 1} \right)\frac{{{v_i}\,{\rm{sin}}\,{\sigma _i}}}{{{r_i}}} \end{align}

Note that ${\zeta _i} = - \,{\rm{cos}}\,{\sigma _i}$ and $ - 1 \le {\zeta _{ir}} \le - \,{\rm{cos}}\,{\sigma _{{\rm{max}}}}$ , so we have that

(33) \begin{align}{\zeta _i} + \,{\rm{cos}}\,{\sigma _{{\rm{max}}}} \le \left( {{\zeta _i} - {\zeta _{ir}}} \right) \le {\zeta _i} + 1\end{align}

when ${\sigma _i} = \pm {\sigma _{{\rm{max}}}}$ , it can be obtained that

(34) \begin{align}0 \le \left( {{\zeta _i} - {\zeta _{ir}}} \right) \le - \,{\rm{cos}}\,{\sigma _{{\rm{max}}}} + 1\end{align}

Therefore, we can have that

(35) \begin{align}{\dot \sigma _i}{|_{{\sigma _i} = {\sigma _{{\rm{max}}}}}} = - K{\sigma _{{\rm{max}}}}\left( {{\zeta _i} - {\zeta _{ir}}} \right) - \left( {{N_p} - 1} \right)\frac{{{v_i}\,{\rm{sin}}\,{\sigma _{{\rm{max}}}}}}{{{r_i}}} \lt 0\end{align}

and

(36) \begin{align}{\dot \sigma _i}{|_{{\sigma _i} = - {\sigma _{{\rm{max}}}}}} = K{\sigma _{{\rm{max}}}}\left( {{\zeta _i} - {\zeta _{ir}}} \right) + \left( {{N_p} - 1} \right)\frac{{{v_i}\,{\rm{sin}}\,{\sigma _{{\rm{max}}}}}}{{{r_i}}} \gt 0\end{align}

So, we can obtain that the $\left[ {0,{\sigma _{{\rm{max}}}}} \right)$ is a positive invariant set of $\left| \sigma \right|$ , which also means that the field-of-view constraints of vehicles Equation (6) are never violated.

Then, substituting Equation (32) into Equation (22) yields

(37) \begin{align}{\dot \zeta _i} = - {{\rm{\Phi }}_i}\left( {{\zeta _i} - {\zeta _{ir}}} \right) - \left( {{N_p} - 1} \right)\frac{{{v_i}{\rm{si}}{{\rm{n}}^2}{\sigma _i}}}{{{r_i}}}\end{align}

where ${{\rm{\Phi }}_i} = K{\sigma _i}\,{\rm{sin}}\,{\sigma _i} \geqslant 0$ . It can be seen that the ${\dot \zeta _i}$ has two parts, then we consider the effect of these two terms on ${\zeta _i}$ and their convergence separately.

The first term is $ - {{\rm{\Phi }}_i}\left( {{\zeta _i} - {\zeta _{ir}}} \right)$ , because ${{\rm{\Phi }}_i} \geqslant 0$ always holds, it is obvious that the first term would control the ${\zeta _i}$ converge to ${\zeta _{ir}}$ gradually. Then based on the Lemma 1, we can obtain that the first term $ - {{\rm{\Phi }}_i}\left( {{\zeta _i} - {\zeta _{ir}}} \right)$ would control the system Equations (21), (22) reach to consensus, and we can have that ${\zeta _i} \to {\zeta _d}$ as $t \to \infty $ , that is ${\sigma _i} \to {\sigma _d} = 0$ . The second term is $ - \left( {{N_p} - 1} \right){v_i}{\rm{si}}{{\rm{n}}^2}{\sigma _i}/{r_i}$ , because $ - \left( {{N_p} - 1} \right){v_i}{\rm{si}}{{\rm{n}}^2}{\sigma _i}/{r_i} \le 0$ always holds, this will control the ${\zeta _i}$ keep decreasing until ${\zeta _i} = - 1$ , that is ${\sigma _i} = 0$ .

Finally, according to the superposition principle, we could obtain that the vehicles will arrive at the target simultaneously and will always not violate the FOV constraints.

Remark 5. It can be observed that the small leading angle assumption was not used in the process of theorem proving, despite its consideration during the design of the guidance law. This implies that the satisfaction or otherwise of the small angle assumption does not affect the convergence of the proposed guidance law. In addition, even if the leading angles do not satisfy the small angle assumption, there is no significant impact on the performance of the guidance law, as will be demonstrated by the subsequent numerical simulations.

Remark 6. The arrival time coordinated control term of the proposed guidance law is ${\sigma _i}{u_i}$ . This means that when ${\sigma _i}$ is small, even if the arrival time synchronisation error is large, $a_i^{BT}$ is still small. However, ${\sigma _i} \to 0$ means the velocity of the vehicle is in line with the line of sight, which helps the vehicle reach the target. This means the guidance law proposed in this study gives the highest priority to reaching the target, while arrival time synchronisation is the secondary priority. After all, successfully reaching the target is the most important task of guidance.

3.4 Extension of the three-dimensional case

Figure 2 depicts the geometry of the relative motion of the $i$ th vehicle–target in 3D space, where ${M_i} - {X_t}{Y_t}{Z_t},{M_i} - {X_m}{Y_m}{Z_m}$ denote the inertial reference frame and the velocity frame, respectively.

Figure 2. Three-dimensional vehicle–target engagement geometry.

First, the model of vehicle–target engagement motion in 3D space [26] is given as follows:

(38) \begin{align}{\dot r_i} = - {v_i}\,{\rm{cos}}\,{\theta _{mi}}\,{\rm{cos}}\,{\psi _{mi}},\end{align}

(39) \begin{align}{\dot \theta _{Li}} = \frac{{ - {v_i}\,{\rm{sin}}\,{\theta _{mi}}}}{{{r_i}}},\end{align}

(40) \begin{align}{\dot \psi _{Li}} = \frac{{ - {v_i}\,{\rm{cos}}\,{\theta _{mi}}\,{\rm{sin}}\,{\psi _{mi}}}}{{{r_i}\,{\rm{cos}}\,{\theta _{Li}}}},\end{align}

(41) \begin{align}{\dot \theta _{mi}} = \frac{{{a_{zi}}}}{{{v_i}}} - {\dot \psi _{Li}}\,{\rm{sin}}\,{\psi _{Li}}\,{\rm{sin}}\,{\psi _{mi}} - {\dot \theta _{Li}}\,{\rm{cos}}\,{\psi _{mi}},\end{align}

(42) \begin{align}{\dot \psi _{mi}} = \frac{{{a_{yi}}}}{{{v_i}\,{\rm{cos}}\,{\theta _{mi}}}} + {\dot \psi _{Li}}\,{\rm{sin}}\,{\psi _{Li}}\,{\rm{cos}}\,{\psi _{mi}}{\rm{tan}}{\theta _{mi}} - {\dot \theta _{Li}}\,{\rm{sin}}\,{\psi _{mi}}{\rm{tan}}{\theta _{mi}} - {\dot \psi _{Li}}\,{\rm{cos}}\,{\theta _{Li}},\end{align}

where ${r_i}$ is the relative distance between the $i$ th vehicle and the target; ${v_i}$ denotes the velocity of the $i$ th vehicle; and ${a_{yi}},{a_{zi}}$ denote the acceleration of the $i$ th vehicle on pitch channel and yaw channel, respectively. The definitions of ${\theta _{Li}},{\psi _{Li}},{\theta _{mi}},{\psi _{mi}}$ are shown in Fig. 2; counterclockwise is positive. In addition, the leading angle ${\sigma _i}$ of the $i$ th vehicle in 3D space is defined as

(43) \begin{align}{\sigma _i} = {\rm{arccos}}\!\left( {{\rm{cos}}\,{\theta _{mi}}\,{\rm{cos}}\,{\psi _{mi}}} \right).\end{align}

The vehicle–target engagement plane is defined as a plane determined by the velocity and line-of-sight direction of the vehicle. In fact, the vehicle–target model in the 3D case can be regarded as a 2D model in the vehicle–target engagement plane; that is, ${\vec a_i},{\vec v_i}$ , and ${\vec r_i}$ are in the same plane. Hence, to extend Equation (30) to the 3D case, we could simply rewrite it in vector form:

(44) \begin{align}{\vec a_i} = \vec a_i^{PNG} + \vec a_i^{BT},\end{align}

where $\vec a_i^{PNG}$ is the PNG term, and $\vec a_i^{BT}$ is the arrival time synchronisation term. Let $a_i^{PN{\rm{G}}},a_i^{BT}$ denote the module of $\vec a_i^{PNG},\vec a_i^{BT}$ , respectively. Then, according to Equation (30), we have

(45) \begin{align}a_i^{PNG} = - \frac{{N{V_i}\,{\rm{sin}}\,{\sigma _i}}}{{{r_i}}},\end{align}

(46) \begin{align}_i^{BT} = - K{V_i}{\sigma _i}\left[ {{\zeta _i} - w + B\,{\rm{tanh}}\left( {{K_e}\mathop \sum \limits_{j = 1}^n {a_{ij}}\left( {{\xi _i} - {\xi _j}} \right) + c} \right)} \right].\end{align}

Let $a_{y,i}^{PNG},a_{z,i}^{PNG}$ denote the components of $\vec a_i^{PNG}$ on the ${M_i}{X_m},{M_i}{Y_m}$ axes, respectively. Following Ref. [26], $a_{y,i}^{PNG},a_{z,i}^{PNG}$ are expressed as follows:

(47) \begin{align}a_{y,i}^{PNG} = - \frac{{NV_i^2}}{{{r_i}}}\,{\rm{sin}}\,{\psi _{mi}},\end{align}

(48) \begin{align}a_{z,i}^{PNG} = - \frac{{NV_i^2}}{{{r_i}}}\,{\rm{sin}}\,{\theta _{mi}}\,{\rm{cos}}\,{\psi _{mi}}.\end{align}

We can easily get that $\vec a_i^{PNG}$ is coplanar with ${\vec v_i}$ and ${\vec r_i}$ from Equations (47) and (48). Then, we need to determine $a_{y,i}^{BT},a_{z,i}^{BT}$ , which are the components of $\vec a_i^{BT}$ on the ${M_i}{X_m},{M_i}{Y_m}$ axes, respectively. Consider that ${\vec a_i}$ must be coplanar with ${\vec v_i}$ and ${\vec r_i}$ , which means $\vec a_i^{BT}$ and $\vec a_i^{PNG}$ are collinear. Therefore, we can get

(49) \begin{align}\frac{{a_{y,i}^{BT}}}{{a_{z,i}^{BT}}} = \frac{{a_{y,i}^{PNG}}}{{a_{z,i}^{PNG}}} = \frac{{{\rm{tan}}{\psi _{mi}}}}{{\,{\rm{sin}}\,{\theta _{mi}}}}.\end{align}

Table 1. Initial states in the 2D scenario

Figure 3. Network topology of vehicles in a 2D scenario.

Figure 4. Guidance results from PNG.

In addition, $a_{y,i}^{BT},a_{z,i}^{BT}$ also need to satisfy the following constraints:

(50) \begin{align}\sqrt {a_{y,i}^{B{T^2}} + a_{z,i}^{B{T^2}}} = a_i^{BT}.\end{align}

Figure 5. Guidance results from the proposed guidance law.

Figure 6. Guidance results with ${\sigma _1}\left( {{t_0}} \right),{\sigma _3}\left( {{t_0}} \right)$ violated constraints.

Figure 7. Guidance results from Equation (18).

Then, combining Equations (49) and (50), we can get

(51) \begin{align}a_{y,i}^{BT} = \frac{{\,{\rm{sin}}\,{\psi _{mi}}}}{{\,{\rm{sin}}\,{\sigma _i}}}a_i^{BT},\end{align}

(52) \begin{align}a_{z,i}^{BT} = \frac{{\,{\rm{sin}}\,{\theta _{mi}}\,{\rm{cos}}\,{\psi _{mi}}}}{{\,{\rm{sin}}\,{\sigma _i}}}a_i^{BT}.\end{align}

Then, combining Equations (44), (47), (48), (51) and (52), the cooperative guidance law with FOV constraints in the 3D case is obtained; that is,

(53) \begin{align}{a_{y,i}} = - \frac{{NV_i^2}}{{{r_i}}}\,{\rm{sin}}\,{\psi _{mi}} + \frac{{\,{\rm{sin}}\,{\psi _{mi}}}}{{\,{\rm{sin}}\,{\sigma _i}}}a_i^{BT},\end{align}

(54) \begin{align}{a_{z,i}} = - \frac{{NV_i^2}}{{{r_i}}}\,{\rm{sin}}\,{\theta _{mi}}\,{\rm{cos}}\,{\psi _{mi}} + \frac{{\,{\rm{sin}}\,{\theta _{mi}}\,{\rm{cos}}\,{\psi _{mi}}}}{{\,{\rm{sin}}\,{\sigma _i}}}a_i^{BT},\end{align}

where ${a_{y,{\rm{i}}}},{a_{z,i}}$ denote the components of ${a_i}$ on the ${M_i}{X_m},{M_i}{Y_m}$ axes, respectively. $a_i^{BT}$ is defined in Equation (46).

Note that $\,{\rm{sin}}\,{\sigma _i}$ appears in the denominator of the expression for $a_{y,i}^{BT},a_{z,i}^{BT}$ , and it follows that $\,{\rm{sin}}\,{\sigma _i} = 0$ when ${\sigma _i} = 0$ or ${\sigma _i} = \pi $ . However, there are still no numerical singularities in the proposed cooperative guidance law in the 3D case.

(1) ${\sigma _i} = 0$ . Then, $\,{\rm{sin}}\,{\sigma _i}$ could be canceled out by ${\sigma _i}$ in $a_i^{BT}$ , because it is always true that $\mathop {{\rm{lim}}}\limits_{{\sigma _i} \to 0} \frac{{{\sigma _i}}}{{\,{\rm{sin}}\,{\sigma _i}}} = 1$ .

(2) ${\sigma _i} = \pi $ . This means ${\vec v_i}$ is opposite to ${\vec r_i}$ ; that is, the vehicle is moving away from the target. Such an extreme situation is unlikely to occur in practice.

3.5 Parameter selection

The proposed cooperative guidance law mainly involves three parameters: $K,{K_e}$ and ${\sigma _d}$ . The value of $K$ determines the importance of the time coordination term $a_i^{BT}$ relative to the PNG term $a_i^{PNG}$ . From the standpoint of improving arrival time synchronisation error convergence speed, we should have $K$ be as large as possible. However, a too-large $K$ value might result in the vehicle not reaching the target accurately. On the other hand, $K$ should not be too small since it will lead to the guidance law being very close to the PNG law. In addition, ${K_e},{\sigma _d}$ should be selected carefully. A larger ${K_e}$ means it is more sensitive to arrival time synchronisation error. However, this might lead to a vibration of the vehicle state, and a smaller ${K_e}$ means the arrival time synchronisation error will be put in a less-important position. In general, ${\sigma _d}$ should be chosen as $0$ , but this will lead to $c \to \infty $ . Furthermore, the arrival time synchronisation error convergence speed will be slowed down when ${\sigma _d}$ is too close to the constraint boundary. Even so, we still need to have ${\sigma _d}$ be as close to $0$ as possible to ensure the vehicle can reach the target.

In conclusion, the above analysis provides a reference for determining the the values of $K,{K_e},{\sigma _d}$ . But for different scenarios, their parameter values still need to be adjusted slightly.

4.0 Numerical simulation

Here, we perform several examples of numerical simulations to demonstrate the effectiveness of the proposed guidance law. The parameters involved in the proposed guidance law are chosen as follows:

(55) \begin{align}K = 2,{\rm{\;\;\;\;}}{K_e} = 8,{\rm{\;\;\;\;}}{\sigma _d} = 0.001{\rm{\;\;}}rad.\end{align}

Although the performance of different simulation scenarios can be improved by carefully tuning the value of parameters, in order to verify the robustness of the proposed guidance law, the same parameter values are used in different simulation examples.

In addition, all simulation examples use the following settings. The simulation step size is fixed as $0.02\,{\rm{s}}$ . The vehicle will stop the simulation when the distance between the vehicle and the target is less than $0.5\,{\rm{m}}$ ; otherwise, it is assumed that it has not successfully reached the target. The FOV limit and the maximum acceleration magnitude of the vehicle are set to ${\sigma _{{\rm{max}}}} = \pi /3$ and ${a_{{\rm{max}}}} = 50\,{\rm{m}}/{{\rm{s}}^2}$ , respectively. The autopilot of the vehicle is considered a first-order lag system; that is,

(56) \begin{align}\frac{{{a_m}}}{{{a_{mc}}}} = \frac{1}{{Ts + 1}},\end{align}

where ${a_m},{a_{mc}}$ denote the real value and commanded value of acceleration, respectively. ${\rm{T}}$ is chosen as $0.2{\rm{s}}$ in the simulation.

Table 2. Initial states in the 3D scenario

Figure 8. Guidance results from the proposed 3D guidance law.

Figure 9. Guidance results with different network topology.

Figure 10. Guidance results with ${\sigma _1}\left( {{t_0}} \right),{\sigma _2}\left( {{t_0}} \right),{\sigma _4}\left( {{t_0}} \right)$ violated constraints.

4.1 Two-dimensional simulation scenarios

Consider a scenario in which three vehicles cooperatively attack one stationary target in the plane. The initial states of the vehicles and the target are listed in Table 1. Figure 3 shows the communication topology of the vehicles, and the corresponding adjacency matrix is given in the following:

(57) \begin{align}A = \left[ {\begin{array}{l@{\quad}l@{\quad}l}0 & 1 & 0\\[5pt] 1 & 0 & 1\\[5pt] 0 & 1 & 0 \end{array}} \right].\end{align}

The PNG law and the proposed guidance law were simulated separately to illustrate the effectiveness of the proposed guidance law. As shown in Fig. 4, the arrival time error for the three vehicles is approximately 5 seconds. However, we can see in Fig. 5 that in this study’s proposed guidance law, three vehicles reaching the target simultaneously, and the FOV constraints are never violated.

To facilitate comparison, it is necessary to revisit the guidelines proposed in Ref. [19]. The guidance law proposed in Ref. [19] is as follows:

(58) \begin{align}{a_i} = - \frac{{v_i^2\,{\rm{sin}}\,{\sigma _i}}}{{{r_i}}} + \frac{{{u_i}}}{{\,{\rm{sin}}\,{\sigma _i}}},\end{align}

where ${u_i}$ denotes a virtual input for synchronising the arrival time of vehicles. Note that ${a_i}$ will diverge to infinity when ${\sigma _i} \to 0$ ; thus, the guidance law has to switch to PNG at the final stage.

According to Lemma 1, our proposed guidance law requires the initial leading angle of each vehicle to meet the FOV constraints. The extreme case in which the initial leading angle of some vehicles violates the constraints is considered, where ${\theta _1}\left( {{t_0}} \right) = 80{\rm{\;\;deg}},{\theta _3}\left( {{t_0}} \right) = 95{\rm{\;\;deg}}$ . We assume, however, that vehicle $1$ and vehicle $3$ can obtain the position of the target by communicating with vehicle $2$ at the start of the guidance. The simulation results are shown in Fig. 6. We can see that although the FOV constraints are violated at the beginning of the guidance, the guidance law does not collapse, and it still works. Furthermore, ${\sigma _1}$ and ${\sigma _3}$ rapidly converge to the constraint region and are maintained until reaching the target.

It should be noted that the term ${r_i}{\ddot \sigma _i}/3$ was discarded. To evaluate its effect, the simulation results from Equation (18) are shown in Fig. 7. Note that ${\ddot \sigma _i} = {\dot a_i}/{v_i} - {\ddot \lambda _i}$ . To avoid an algebraic loop in the simulation process, signal delay and difference are used to compute this term. This approach inevitably introduces numerical error, but it at least shows us the effect of the third term. As shown in Fig. 7, although Equation (18) can perform the guidance task effectively, the leading angle and acceleration of the vehicles show severe oscillation. This is due to the fact although ${\ddot \sigma _i}$ is relatively small, ${r_i}$ is very large, which results in the third term dominating the Equation (18). As the sign of ${\ddot \sigma _i}$ undergoes a series of changes, the resulting acceleration ${a_i}$ oscillates, which in turn causes the value of ${\dot \sigma _i}$ to oscillate, this is because ${\dot \sigma _i} = {a_i}/{v_i} - {\dot \lambda _i}$ . Again, this cause the sign of ${\ddot \sigma _i}$ to keep changing. This process will continue until the values of ${\sigma _i},{\dot \sigma _i}$ and ${r_i}$ gradually approach 0.

4.2 Three-dimensional simulation scenarios

To validate the effectiveness of the proposed guidance law for the 3D case, the following simulations are performed. Considering a scenario with four vehicles and one stationary target whose location is $\left( {0\,{\rm{m}},0\,{\rm{m}},0\,{\rm{m}}} \right)$ . The initial states of the vehicles are listed in Table 2. The adjacency matrix corresponding to the communication topology of the vehicles is as follows:

(59) \begin{align}A = \left[ {\begin{array}{l@{\quad}l@{\quad}l@{\quad}l}0 & 1 & 0 & 1\\[3pt] 1 & 0 & 1 & 0\\[3pt] 0 & 1 & 0 & 1\\[3pt] 1 & 0 & 1 & 0 \end{array}} \right].\end{align}

Figure 8 shows the simulation results from the proposed guidance law. It can be seen that the proposed guidance law works effectively, and the FOV constraints are never violated.

Figure 11. Guidance results with communication delay and measurement noise.

In order to illustrate the robustness of the proposed guidance law, we implemented simulations for the following scenarios: (1) verification of effectiveness under different communication network topologies; (2) the case where the initial conditions do not satisfy the constraints; and (3) the presence of communication delays and measurement noise.

A communication network topology similar to the leader-follower structure was considered, with the corresponding adjacent matrix presented below. The simulation results are shown in Fig. 9. It can be observed that the performance of the proposed guidance law is not sensitive to the communication network topology.

(60) \begin{align}A = \left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0 & 1 & 0 & 0\\[3pt] 1 & 0 & 1 & 1\\[3pt] 0 & 1 & 0 & 0\\[3pt] 0 & 1 & 0 & 0 \end{array}} \right]\end{align}

To demonstrate the robustness of the proposed guidance law in the face of extreme scenarios, we consider a scenario in which the initial condition of a number of vehicles do not satisfy the FOV constraints, where ${\sigma _1}\left( {{t_0}} \right) = 65,{\sigma _2}\left( {{t_0}} \right) = 71,{\sigma _4}\left( {{t_0}} \right) = 67$ , the values are expressed in degrees. The simulation results are shown in Fig. 10. It can be seen that despite the initial conditions not satisfying the requirements of Lemma 1, the proposed guidance law still works and the FOV angle of the vehicle rapidly converging on the constraint region and being maintained until arrived at the target.

To illustrate the robustness of the proposed guidance law in a more realistic setting, we conduct simulations with measurement noise and communication delays. In this case, the measurement noises are added to ${r_i},{v_i},{\theta _{Li}},{\psi _{Li}},{\theta _{mi}},{\psi _{mi}}$ , and the signal-to-noise ratio is set to 50dB. Furthermore, information transmissions between vehicle have a time delay of 0.5 seconds. As shown in Fig. 11, the proposed guiding law has a stable performance despite measurement noise and communication delays.

5.0 Conclusion

A distributed cooperative guidance law considering FOV constraints without numerical singularities is proposed. The proposed guidance law has the following advantages: (1) the FOV constraint of vehicles is taken into account in the design of the guidance law; (2) it has no numerical singularities and does not require accurate time-to-flight estimates; (3) the proposed guidance law is distributed, which means that the vehicles do not need the global information from all the vehicles. Through numerical simulation and comparative study, the effectiveness and robustness of the proposed guidance law are validated.

The next research work can be considered from the following aspects: (1) modifying the proposed guidance law so that it can be suitable for manoeuvering target; (2) design and stability analysis of cooperative guidance law in the case of switched communication topology or time-varying communication topology; (3) event-triggered technology can be introduced into the design of guidance law to reduce the resource utilisation burden between vehicles.