2.1 Problem statement

The following assumptions are first introduced:

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 [Reference Kang, Wang, Li, Shan and Petersen9], 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.

2.2. Preliminaries

2.2.2 Useful lemmas

Lemma 1. [Reference Jesus, Pimenta, Tôrres and Mendes25] 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.

3.1 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}

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 ${\sigma _i}$ .

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 [Reference Chen, Wang, Wang, Shan and Xin19, Reference He and Lee24]. 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.

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.

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 [Reference Song and Ha26] 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. [Reference Song and Ha26], $a_{y,i}^{PNG},a_{z,i}^{PNG}$ 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.