Cooperative trajectory shaping guidance law for multiple anti-ship missiles

Abstract

To enhance the performance of anti-ship missiles cooperative attack, this paper proposes a finite-time trajectory shaping-based cooperative guidance law (TSCGL). Firstly, the cooperative guidance model is established on segmented linearisation of the missile’s heading angle. Then, a trajectory shaping guidance law for a single missile is derived by a weighted optimal energy cost function and Schwarz inequality. On this basis, a finite-time TSCGL is proposed combined with trajectory shaping technology and finite-time theory. The desirable finite-time convergence performance can ensure a simultaneous attack. Through an improved method of time-to-go estimation, it is independent of small-angle assumption and relaxes the launching conditions of the missiles. Additionally, the proposed finite-time TSCGL can achieve better damage performance through energy management. Finally, simulation results demonstrate the effectiveness and superiority of the proposed finite-time TSCGL.

Nomenclature

${a_i}$

guidance command of the $i$ -th missile ( $m/{s^2}$ )

${a_{ci}}$

cooperative term of the ${a_i}$ ( $m/{s^2}$ )

${a_{si}}$

basis shaping guidance term of the ${a_i}$ ( $m/{s^2}$ )

$A$

system matrix

${A_G}$

adjacency matrix of graph $G$

$B$

control matrix

$E$

edge set of graph

$G$

communication network graph

$J$

cost function ( ${(m/{s^2})^2}$ )

$k$

design parameter of the sliding surface

$L$

Laplace matrix of the graph $G$

${M_i}$

$i$ -th missile

${q_i}$

line-of-sight (LOS) angle of the $i$ -th missile ( $deg$ )

${r_i}$

relative distance of the $i$ -th missile ( $m$ )

${s_i}$

sliding surface of the $i$ -th missile ( $s$ )

${t_{goi}}$

time-to-go of the $i$ -th missile ( $s$ )

${t_{0i}}$

initial impact time of the $i$ -th missile ( $s$ )

${t_{fi}}$

final impact time of the $i$ -th missile ( $s$ )

$T$

target

$u$

guidance command ( $m/{s^2}$ )

${u^*}$

optimal guidance command ( $m/{s^2}$ )

$u_i^{nom}$

the consensus protocol

${v_{yi}}$

vertical component of ${V_{mi}}$ ( $m/{s^2}$ )

${v_{y0i}}$

initial vertical component of ${V_{mi}}$ ( $m/{s^2}$ )

${v_{yfi}}$

final vertical component of ${V_{mi}}$ ( $m/{s^2}$ )

$V$

set of vertices

${V_{mi}}$

speed of the $i$ -th missile ( $m/s$ )

${V_{1i}}$ , ${V_{2i}}$

different Lyapunov functions of the $i$ -th missile

$W$

weight function

$X$

state of $i$ -th missile

${X_T}$ , ${Y_T}$

current position of the target

${X_{TP}}$ , ${Y_{TP}}$

predicted interception point of the target

${y_i}$

altitude of the $i$ -th missile ( $m$ )

Greek Symbol

${\beta _i}$

design parameter of the $i$ -th missile’s consistency protocol

${\eta _i}$

lead angle of the $i$ -th missile ( $deg$ )

${\gamma _i}$

heading angle of the $i$ -th missile ( $deg$ )

${\gamma _{0i}}$

segmented linearisation point of the $i$ -th missile’s heading angle ( $deg$ )

$\Delta {\gamma _i}$

a small value, equals to ${\gamma _i} - {\gamma _{0i}}$ ( $deg$ )

${\gamma _{di}}$

desired heading angle of the $i$ -th missile ( $deg$ )

${\gamma _{fi}}$

final heading angle of the $i$ -th missile ( $deg$ )

$\Phi \;$

state transition matrix

${\theta _T}$

heading angle of target

$\varepsilon $

design parameter of the sliding surface

1.0 Introduction

Shaping guidance law as an efficient guidance scheme has been widely used in the research of naval attack and defense [1–4]. The shaping guidance law can ensure a single missile attack the target with optimal energy and achieve the desired accuracy and damage performance. Even though the shaping guidance law is effective in most traditional scenarios, a single missile is difficult to penetrate for a modern warship equipped with missile defense systems [5, 6]. Therefore, cooperative guidance organised by multiple missiles can ensure that more than one missile hits the target, which greatly increases the success rate of the operation [7, 8]. Jeon, I.S. et al. [9] conducted impact time control guidance (ITCG) law, so as to anti-ship missiles can intercept a stationary target with the prescribed impact time. Inspired by this idea, a shaping cooperative guidance law to control impact time and angle was presented in Refs. [10–13].

As a core factor in shaping cooperative guidance law, the estimation of the time-to-go can significantly influence guidance performance. It can be noted that the studies mentioned above are established on the small-angle assumption in estimation [10, 11], which limits the generalisation of the investigations. Therefore, various methods are proposed to improve the accuracy of the estimation in Refs. [14–16]. Liu, S. et al. [16] proposed an improved estimation method of time-to-go, in which the calculation accuracy can be guaranteed with a large initial lead angle. Although enough efforts have been devoted to the accuracy improvement of estimation, without energy management, constraints of energy will limit the execution or even lead to the failure of cooperation. In view of the difficulty in energy management, the optimal shaping cooperative guidance has been widely discussed in Refs. [17–20]. Chen, X. et al. [17] derived an optimal guidance law wherein cooperative impact time and angle constraints can be satisfied. Adding a compensation term to the guidance law, Liu J. et al. [18] derived a multi-missile optimal cooperative guidance law by optimal control theory.

To ensure penetration capability, guidance laws based on consensus theory have been established for cooperation. Multiple anti-ship missiles can perform a saturation attack on the target by exchanging information. Hence, consensus-based cooperative guidance laws have been widely studied to ensure the impact time. However, the asymptotic convergence is insufficient for a transient flight. As an extension, the finite-time shaping cooperative guidance law has become a hotspot in engineering practice [21–23]. Sinha, A. et al. [24] proposed a leader-follower cooperative salvo guidance strategy to intercept stationary targets by exploiting the advantages of supertwisting sliding mode control. Based on powered missiles, some cooperative guidance laws aim to control the axial speed in the direction of LOS to fulfill the cooperation of impact time [22, 25, 26]. Liu, S. et al. [27] designed a finite-time convergence cooperative guidance law from two directions of the LOS, which can easily satisfy the impact time and angle constraint. Although the powered missiles hold better adjustment capability, their cost increases the economic burden for cooperation. Therefore, it is more practical to study the cooperative guidance law for unpowered missiles without speed control [28, 29]. Shin, H.S. et al. [30] designed a finite-time guidance law to nullify the LOS angular rate to address the terminal guidance problem of missiles intercepting targets in three-dimensional space. With the measured initial conditions, the boundary of convergence time can be adjusted by turning parameters. Without energy engagement, however, the execution of guidance command is limited. Hence, fast convergence with optimal energy management is still a challenge for application.

The aforementioned guidance laws have made great progress, but the problem of optimal shaping cooperative guidance law design remains open. (1) The estimation of the time-to-go with the small angle assumption hinders the engineering application of the cooperative guidance law. (2) Optimal energy control needs to be considered to ensure cooperative performance. (3) Fast convergence speed for state variables is necessary in the guidance process.

Inspired by the above discussions, a finite-time trajectory shaping-based cooperative guidance law (TSCGL) is proposed, which can achieve multiple missile cooperation with desired damage and convergence performance. The main contributions are summarised as follows:

1. (1) The proposed finite-time TSCGL can ensure impact time and angle constraints, so as to achieve simultaneous attack with desired damage performance. Different from most existing studies in Ref. [31], independent of small-angle assumption, the proposed finite-time TSCGL holds higher accuracy of impact time estimation, even in the case of a large initial leading angle.

2. (2) Compared with existing optimal shaping guidance law in Ref. [2], the proposed TSCGL can guarantee optimal energy management without simplifying terminal boundary conditions. Based on the weighted optimal energy cost function, the stability of the optimal TSCGL is derived by the Schwarz inequality.

3. (3) The proposed finite-time TSCGL can achieve better convergence performance. Compared with asymptotic cooperative guidance law in Ref. [32], the final impact time can converge within a finite time.

The rest of the paper is organised as follows. Section 2.0 gives problem descriptions and relative models. Section 3.0 mainly derives the optimal guidance command of the shaping guidance law and the determination of cooperative guidance law. In Section 4.0, numerical simulations for four cases are conducted to demonstrate the effectiveness and advantages of the cooperative shaping guidance law. Section 5.0 gives the conclusions of this paper.

2.0 Problem formulation and model derivation

Multiple missiles-target engagement geometry in vertical plane is shown in Fig. 1. The notations ${M_i}$ and $T$ represent the $i$ -th missile and target. ${V_{mi}}$ , ${a_i}$ , ${r_i}$ , ${y_i}$ , ${q_i}$ , ${\eta _i}$ and ${\gamma _i}$ denote speed, guidance command, relative distance, altitude, line of sight (LOS) angle, lead angle and heading angle of the $i$ -th missile, $i = 1,2, \cdots ,m$ .

Figure 1. Multiple missiles-target engagement geometry.

The relative motion model of the $i$ -th missile are given as

(1) \begin{align}{\dot r_i} = - {V_{mi}}\cos\!({\eta _i})\end{align}

(2) \begin{align}{\dot q_i} = {V_{mi}}\sin\! ({\eta _i})/{r_i}\end{align}

(3) \begin{align}{\dot \gamma _i} = {a_{_i}}/{V_{mi}}\end{align}

(4) \begin{align}{\eta _i} = {q_i} - {\gamma _i}\end{align}

The guidance command ${a_i}$ is constructed as

(5) \begin{align}{a_i} = {a_{si}} + {a_{ci}}\end{align}

where ${a_{si}}$ and ${a_{ci}}$ are the basis shaping guidance term and cooperative term.

The core of multiple missiles cooperatively attacking the target is that they can satisfy the constraints of both time convergence and interception accuracy. In addition, in order to improve damage performance on the target, impact angle and energy consumption should be taken into consideration. Generally, the impact angle constraint translates into controlling the heading angle at interception [2, 4]. Thus, the design objective of the finite-time TSCGL can be described as follows:

(6) \begin{align}\left\{ {\begin{array}{l}{{\lim\limits_{t \to {t_f}}} {\kern 1pt} ({t_{goi}} - {t_{goj}}) \to 0}\\[6pt]{{\lim\limits_{t \to {t_f}}} {\kern 1pt} {r_i} \to 0}\\[6pt]{{\lim\limits_{t \to {t_f}}} {\kern 1pt} {\gamma _i} \to {\gamma _{di}}}\\[6pt]{{\lim\limits_{t \to {t_f}}} {\kern 1pt} \min J}\end{array}} \right.\end{align}

where ${t_{go}}$ is the time-to-go, ${\gamma _{di}}$ denotes desired heading angle and $J$ represents cost function.

3.0 Guidance law design

Before designing the finite-time TSCGL, some preliminary assumptions need to be put forward:

Assumption 1. Anti-ship missiles and ship are viewed as mass points.

Assumption 2. Compared to missiles, the ship can be considered a stationary target.

Assumption 3. All the velocities of unpowered missiles are assumed to be fixed with adjustable perpendicular acceleration throughout the engagement.

3.1 Derivation of shaping guidance law

The cost function is definited as

(7) \begin{align}\min J = \frac{1}{2}\int_t^{{t_{fi}}} W(\tau ){u^2}(\tau )d\tau \end{align}

where $W(t) \gt 0,t \in \left( {{t_{0i}},{t_{fi}}} \right)$ . ${t_{0i}}$ and ${t_{fi}}$ are the initial time and the final impact time of the $i$ -th missile.

Considering that the kinematics of the $i$ -th missile are established by

(8) \begin{align}{\dot y_i} = {V_{mi}}\sin {\gamma _i}\end{align}

Linearisation at point ${\gamma _{0i}}$ of segment $k$ are as follows

(9) \begin{align}{\dot y_i} = {V_{mi}}\sin\! ({\gamma _{0i}}) + {V_{mi}}\Delta {\gamma _i}\cos\!({\gamma _{0i}}) = {v_{yi}}\end{align}

where ${\gamma _i} = {\gamma _{0i}} + \Delta {\gamma _i}$ . $\Delta {\gamma _i}$ is a small value and ${\gamma _{0i}}$ is a constant in the $k$ segment. ${v_{yi}}$ is the vertical component of ${V_{mi}}$ . Derivations are derived as

(10) \begin{align}{\ddot y_i} = {V_{mi}}\dot{\Delta \gamma _i}\cos\!({\gamma _{0i}}) = {\dot v_{yi}}\end{align}

(11) \begin{align}{\dot \gamma _i} = \dot{\Delta \gamma _i}\end{align}

Then, based on Equation (3), one can be obtained as

(12) \begin{align}{\ddot y_i} = {a_i}\cos\!({\gamma _{0i}}) = {\dot v_{yi}}\end{align}

The linearised kinematic Equations (9) and (10) can be rewritten in the matrix form as

(13) \begin{align}\dot x = Ax + Bu\end{align}

where

(14) \begin{align}\begin{array}{*{20}{l}}{x \buildrel \Delta \over = {{\left[ {\begin{array}{*{20}{l}}{{y_i}}\quad {{v_{yi}}}\end{array}} \right]}^T},u \buildrel \Delta \over = {a_i}}\\[8pt]{A \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}}0\quad 1\\[5pt]0\quad 0\end{array}} \right],B \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}}0\\[5pt]{cos({\gamma _{0i}})}\end{array}} \right]}\end{array}\end{align}

Based on the linear control theory, the solution of Equation (13) can be expressed as

(15) \begin{align}x\!\left( {{t_{fi}}} \right) = \Phi \!\left( {{t_{fi}} - t} \right)x\!\left( t \right) + \int_t^{{t_{fi}}} \Phi \!\left( {{t_{fi}} - \tau } \right)Bu\!\left( \tau \right)d\tau \end{align}

(16) \begin{align}\Phi \!\left( {{t_{fi}} - t} \right) = {e^{A({t_{fi}} - t)}}\end{align}

where $\Phi \!\left( t \right)$ is the state transition matrix of the linear system as Equation (13). Expand the right side of Equation (16) as follow

(17) \begin{align}{e^{A\!\left( {{t_{fi}} - t} \right)}} = I + A\!\left( {{t_{fi}} - t} \right) + \frac{{{A^2}}}{{2!}}{\left( {{t_{fi}} - t} \right)^2} + \cdots \end{align}

Calculating the right side of Equation (15) and omitting the high order terms, we can obtain

(18) \begin{align}x\!\left( {{t_{fi}}} \right) = \left[ \begin{array}{c@{\quad}c}1 & {{t_{fi}} - t}\\[6pt]0 & 1\end{array} \right]x\!\left( t \right) + \int_t^{{t_{fi}}} \left[ \begin{array}{c@{\quad}c}1 & {{t_{fi}} - t}\\[6pt]0 & 1\end{array} \right]\left[ {\begin{array}{*{20}{l}}0\\[6pt]{cos({\gamma _{0i}})}\end{array}} \right]u\!\left( \tau \right)d\tau \end{align}

Using shorthand notations as

(19) \begin{align}\begin{array}{*{20}{l}}{{f_1} \buildrel \Delta \over = {x_1}(t) + \left( {{t_{fi}} - t} \right){x_2}(t) - {x_1}\!\left( {{t_{fi}}} \right),}\, {{h_1}(\tau ) \buildrel \Delta \over = - cos({\gamma _{0i}})\left( {{t_{fi}} - t} \right)}\\[6pt]{{f_2} \buildrel \Delta \over = {x_2}(t) - {x_2}\!\left( {{t_{fi}}} \right),}\qquad\qquad\qquad\ {{h_2}(\tau ) \buildrel \Delta \over = - \cos\!({\gamma _{0i}})}\end{array}\end{align}

Equation (15) can be rewritten as

(20) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{f_1} = \int_t^{{t_{fi}}} {h_1}(\tau )u(\tau )d\tau }\\[6pt]{{f_2} = \int_t^{{t_{fi}}} {h_2}(\tau )u(\tau )d\tau }\end{array}} \right.\end{align}

By introducing a new variable denoted as $\lambda $ , then by Equation (20) we can obtain

(21) \begin{align}f \buildrel \Delta \over = {f_1} - \lambda {f_2} = \int_t^{{t_{fi}}} \left[ {{h_1}(\tau ) - \lambda {h_2}(\tau )} \right]u(\tau )d\tau \end{align}

Furthermore, Equation (21) can be transformed as

(22) \begin{align}f = \int_t^{{t_{fi}}} \left( {{h_1}(\tau ) - \lambda {h_2}(\tau )} \right){W^{1/2}}(\tau ){W^{ - 1/2}}(\tau )u(\tau ){\rm{d}}\tau \end{align}

Based on the Schwarz inequality, the following inequality can be obtained as

(23) \begin{align}\frac{{{f^2}}}{{2\int_t^{{t_{fi}}} {{\left[ {{h_1}(\tau ) - \lambda {h_2}(\tau )} \right]}^2}{W^{ - 1}}(\tau ){\rm{d}}\tau }} \le \frac{1}{2}\int_t^{{t_{fi}}} W(\tau ){u^2}(\tau ){\rm{d}}\tau \end{align}

It is obvious that the right side of Equation (23) equals the cost function defined in Equation (12). In other words, it provides a lower bound for the minimum cost function. According to the Schwarz inequality, the condition of inequality Equation (23) becoming the equation is that the guidance command must be as

(24) \begin{align}u(t) = K\!\left[ {{h_1}(t) - \lambda {h_2}(t)} \right]{W^{ - 1}}(t)\end{align}

where $K$ is a constant to be determined. Substituting Equation (24) into Equation (19), the constant $K$ can be obtained as

(25) \begin{align}K = \frac{{{f_1}}}{{\int_t^{{t_{fi}}} h_1^2(\tau ){W^{ - 1}}(\tau )d\tau - \lambda \int_t^{{t_{fi}}} {h_1}(\tau ){h_2}(\tau ){W^{ - 1}}(\tau )d\tau }}\end{align}

To facilitate the derivation, some notations are given as follows

(26) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{g_{1i}} \buildrel \Delta \over = \int_t^{{t_{fi}}} h_1^2(\tau ){W^{ - 1}}(\tau )d\tau }\\[6pt]{{g_{2i}} \buildrel \Delta \over = \int_t^{{t_{fi}}} {h_1}(\tau ){h_2}(\tau ){W^{ - 1}}(\tau )d\tau }\\[6pt]{{g_{3i}} \buildrel \Delta \over = \int_t^{{t_{fi}}} h_2^2(\tau ){W^{ - 1}}(\tau )d\tau }\end{array}} \right.\end{align}

Thus Equation (25) can be rewritten as

(27) \begin{align}K = \frac{{{f_1}}}{{{g_{1i}} - \lambda {g_{2i}}}}\end{align}

Substituting Equation (27) into Equation (24), the guidance command can be obtained as

(28) \begin{align}u(t) = \frac{{{f_1}\left[ {{h_1}(t) - \lambda {h_2}(t)} \right]{W^{ - 1}}(t)}}{{{g_{1i}} - \lambda {g_{2i}}}}\end{align}

From Equation (23), the minimum value of cost function can be expressed as

(29) \begin{align}J = \frac{{{{\left( {{f_1} - \lambda {f_2}} \right)}^2}}}{{2\!\left( {{g_{1i}} - 2\lambda {g_{2i}} + {\lambda ^2}{g_{3i}}} \right)}}\end{align}

Taking the derivative of $J$ with respect to $\lambda $ and imposing $dJ/d\lambda = 0$ , the optimal solution of $\lambda $ can be expressed as

(30) \begin{align}{\lambda ^*} = \frac{{{f_1}{g_{2i}} - {f_2}{g_{1i}}}}{{{f_1}{g_{3i}} - {f_2}{g_{2i}}}}\end{align}

Substituting Equation (30) into Equation (28), the optimal guidance command can be obtained as

(31) \begin{align}{u^*}(\tau ) = \frac{{\left[ {{f_1}{h_1}(\tau ){g_{3i}} - {g_{2i}}\!\left( {{f_2}{h_1}(\tau ) + {f_1}{h_2}(\tau )} \right) + {f_2}{h_2}(\tau ){g_{1i}}} \right]{W^{ - 1}}(\tau )}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}\end{align}

Substituting Equations (14) and (19) into Equation (31), the optimal trajectory shaping guidance law is expressed as

(32) \begin{align}{a_{si}} = {u^*}(t) = {k_{1i}}{y_i} + {k_{2i}}{v_{yi}} + {k_{3i}}\end{align}

where the guidance gains ${k_{1i}}$ , ${k_{2i}}$ and ${k_{3i}}$ are given as follows

(33) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{k_{1i}} = \dfrac{{ - {t_{goi}}{g_{3i}} + {g_{2i}}}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}{W^{ - 1}}\left( t \right)}\\[15pt]{{k_{2i}} = \dfrac{{ - t_{goi}^2{g_{3i}} + 2{g_{2i}}{t_{goi}} - {g_{1i}}}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}{W^{ - 1}}\left( t \right)}\\[15pt]{{k_{3i}} = \dfrac{{\left( {{t_{goi}}{g_{3i}} - {g_{2i}}} \right){y_i}\!\left( {{t_{fi}}} \right) + \left( {{g_{1i}} - {t_{goi}}{g_{2i}}} \right){v_{yi}}\!\left( {{t_{fi}}} \right)}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}{W^{ - 1}}\!\left( t \right)}\end{array}} \right.\end{align}

where ${t_{goi}} = \left( {{t_{fi}} - t} \right)$ is the time-to-go of the $i$ -th missile, and ${g_{1i}}$ , ${g_{2i}}$ , ${g_{3i}}$ are defined as

(34) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{g_{1i}} = \int_t^{{t_{fi}}} {{\left( {{t_{fi}} - \tau } \right)}^2}{W^{ - 1}}(\tau )d\tau }\\[10pt]{{g_{2i}} = \int_t^{{t_{fi}}} \left( {{t_{fi}} - \tau } \right)\cos\!({\gamma _{0i}}){W^{ - 1}}(\tau )d\tau }\\[10pt]{{g_{3i}} = \int_t^{{t_{fi}}} {{\cos }^2}({\gamma _{0i}}){W^{ - 1}}(\tau )d\tau }\end{array}} \right.\end{align}

From Equation (34), the analytical solution of ${g_{1i}}$ , ${g_{2i}}$ , ${g_{3i}}$ depend on the prescribed the final impact time ${t_{fi}}$ and the weighting function ${W^{ - 1}}(t)$ which is a time-varying function. In order to achieve time-cooperative guidance for multiple missiles and remove constraint of the final impact time ${t_{fi}}$ , we make the assumption as follows

(35) \begin{align}{W^{ - 1}}\left( \tau \right) = 1\end{align}

(36) \begin{align}{\gamma _{0i}} \equiv 0\end{align}

After simplification, the shaping guidance command in Equation (32) can be transformed as

(37) \begin{align}{a_{si}} = - 6\frac{{{y_i}}}{{t_{goi}^2}} - 4\frac{{{v_{yi}}}}{{{t_{goi}}}} + 6\frac{{{y_i}({t_{fi}})}}{{t_{goi}^2}} - 2\frac{{{v_{yi}}({t_{fi}})}}{{{t_{goi}}}}\end{align}

Note that these results are identical to the optimal control guidance law as studied in Ref. [2]. Combining Equation (4) with Equation (9), the following equation can be obtained as

(38) \begin{align}{\dot v_{yi}} = {V_{mi}}{\dot \gamma _i}\cos {\gamma _{0i}}\end{align}

The integral of Equation (38) from $0$ to ${t_{fi}}$ can be expressed as

(39) \begin{align}\int_0^{{t_{fi}}} {\dot v_{yi}}dt = {V_{mi}}\cos {\gamma _{0i}}\int_0^{{t_{fi}}} {\dot \gamma _i}dt\end{align}

Then, the final speed constraint ${v_{yi}}({t_{fi}})$ can be transformed into the heading angle constraint ${\gamma _i}({t_{fi}})$ by

(40) \begin{align}{v_{yi}}({t_{fi}}) = {v_{y0i}} + {V_{mi}}\left( {{\gamma _i}({t_{fi}}) - {\gamma _{0i}}} \right)\cos {\gamma _{0i}}\end{align}

Remark 1. $\Delta {\gamma _i}$ is considered as a small value in some ${\gamma _{0i}}(k)$ segment (at point ${\gamma _{0i}}$ of segment $k$ ). Once the absolute value of $\Delta {\gamma _i}$ is bigger than 5 degrees ( $\left| {\Delta {\gamma _i}} \right|{ \gt 5^ \circ }$ ), the linearised kinematics is switched in next ${\gamma _{0i}}(k + 1)$ segment. Furthermore, their relationship is satisfied by

(41) \begin{align}{\gamma _{0i}}(k + 1) = {\gamma _{0i}}(k) + \Delta {\gamma _i}\end{align}

where $k = 1,2, \cdots ,n$ , $\Delta {\gamma _i} = \pm {5^ \circ }$ .

3.2 Derivation of cooperative shaping guidance law

In this subsection, the concept of algebraic graph theory is introduced by Ref. [27]. Suppose that communication network between missiles can be expressed as $G = \left\{ {V,E} \right\}$ , where $V = \left\{ {1,2, \cdots ,n} \right\}$ denotes the set of vertices, $E \subseteq V \times V$ denotes the edge set of graph, the subscript $i$ denotes the $i$ -th missile, ${e_{ij}}$ is the edge of graph $G$ , and ${e_{ij}} \subseteq E$ indicates that the agents $i$ and $j$ can receive message from each other in an undirected graph. If there is a connection between any two missiles in the graph, then the graph $G$ is connected. In the directed graph, ${e_{ij}} \subseteq E$ means the missile $i$ can receive message from missile $j$ . In addition, ${A_G} = ({a_{ij}}) \in {R^{n \times n}}$ is the adjacency matrix of graph $G$ . If ${e_{ij}} \subseteq E$ , then ${a_{ij}} \gt 0$ ; ${a_{ij}} = 0$ , otherwise. It is worth noting that ${a_{ij}} = {a_{ji}}$ when the communication topology is an undirected graph. The Laplace matrix of the graph $G$ is defined as $L = ({L_{ij}}) \in {R^{n \times n}}$ , which can be expressed as

(42) \begin{align}{L_{ij}} = \left\{ {\begin{array}{*{20}{l}}{ - {a_{ij}}\;\;\;i \ne j}\\[6pt]{\sum\limits_{j = 1,j \ne i}^n {a_{ij}}\;\;\;i = j}\end{array}} \right.\end{align}

The final impact time of the $i$ th missile can be defined as

(43) \begin{align}{t_{fi}} = {t_{goi}} + t\end{align}

Let ${X_i} = {t_{fi}}$ , a multiple missiles system is described as

(44) \begin{align}{\dot X_i} = u_i^{nom}\end{align}

where $u_i^{nom}$ is the consensus protocol which should be designed with the information of the missile itself and its neighbours.

In order to verify the finite-time stability of the system, the following Lemma is given.

Lemma 1. [33] Suppose that there exists a continuous positive function ${V_t}$ , positive real numbers $\alpha $ , $\beta $ and $0 \lt \gamma \lt 1$ , satisfying ${\dot V_t} \le - \alpha V(t) - \beta {V^\gamma }(t)$ . Then, the system in Equation (44) is stable in a finite time $T \le (1/\alpha (1 - \gamma ))\ln \left( {\left( {\alpha {V^{1 - \gamma }}\left( {{x_0}} \right) + \beta } \right)/\beta } \right)$ .

Lemma 2. [34] Suppose that there exists a continuous positive function ${V_t}$ , positive real numbers $c$ and $0 \lt \alpha \lt 1$ , satisfying ${\dot V_t} \le - c{(V(t))^\alpha }$ . Then, the system in Equation (44) is stable in a finite time $T \le (1/c(1 - \alpha ))V{({X_0})^{1 - \alpha }}$ .

Lemma 3. If the undirected graph is connected. The consensus protocol $u_i^{nom}$ shown in Equation (45) can ensure that the state of multi-missile converges in finite time.

(45) \begin{align}u_i^{nom} = sgn\!\left( {\sum\limits_{i,j = 1}^n {a_{ij}}({X_j} - {X_i})} \right){\left| {\sum\limits_{i,j = 1}^n {a_{ij}}({X_j} - {X_i})} \right|^{{\beta _i}}}\end{align}

where ${a_{ij}}$ are the elements of the weight coefficient matrix. And ${\beta _i}$ is a constant, $0 \lt {\beta _i} \lt 1$ .

Lemma 4. [35] If the undirected graph is connected and $1_N^TX = 0$ , then ${X^T}LX \geq {\lambda _2}(L){X^T}X$ , where ${\lambda _2}(L)$ denotes the second smallest eigenvalue of Laplacian matrix $L$ .

Lemma 5. [36] Suppose that there exists a series of ${x_1},{x_2}, \cdots ,{x_n} \geq 0$ and $0 \lt p \le 1$ . Then, the inequality $\sum\limits_{i = 1}^n x_i^p \geq {\left( {\sum\limits_{i = 1}^n {x_i}} \right)^p}$ holds.

Theorem 1. Subject to the system in Equation (44), in the existence of $k,\varepsilon \gt 0$ , under the action of the guidance command in Equation (46), multiple missiles can achieve consistent convergence of the cooperative variable ${t_{fi}}$ in a finite time ( $T \le {T_1} + {T_2}$ ), thereby ensuring that multiple missiles simultaneously arrive at the target.

(46) \begin{align}{a_i} = (u_i^{nom} - k{s_i} - \varepsilon sgn({s_i}) - {k_c})/{k_a}\end{align}

where $u_i^{nom} = sgn\!\left( {\sum\limits_{i,j = 1}^n {a_{ij}}({t_{fj}} - {t_{fi}})} \right){\left| {\sum\limits_{i,j = 1}^n {a_{ij}}({t_{fj}} - {t_{fi}})} \right|^{{\beta _i}}}$ , ${k_c} = 1 - \cos\!({\eta _i})\left( {1 + {{\sin }^2}({\eta _i})/10} \right) + {\sin ^2}({\eta _i}) \cos\!({\eta _i})/5$ , ${k_a} = - {r_i}\sin\! ({\eta _i})\cos\!({\eta _i})/5V_{mi}^2$ , ${\dot s_i} = - k{s_i} - \varepsilon sgn({s_i})$ .

Proof.

1. (1) System transformation

   Before proving the theorem, some preparations are made as follows.

   The derivative of ${t_{fi}}$ can be presented as

   (47) \begin{align}{\dot t_{fi}} = {\dot t_{goi}} + 1\end{align}
   From Ref. [16], in order to remove the small-angle assumption of the ${\eta _i}$ , an improved estimation method of ${t_{goi}}$ is chosen as follows
   (48) \begin{align}{t_{goi}} = {r_i}\!\left( {1 + {{\sin }^2}({\eta _i})/10} \right)/{V_{mi}}\end{align}
   The derivation of ${t_{goi}}$ is derived as
   (49) \begin{align}{\dot t_{goi}} = {\dot r_i}\left( {1 + {{\sin }^2}({\eta _i})/10} \right)/{V_{mi}} + {r_i}\sin\! ({\eta _i})\cos\!({\eta _i}){\dot \eta _i}/5{V_{mi}}\end{align}
   Substituting Equations (1–4) into Equation (49), then we can obtain
   (50) \begin{align}{{\dot t}_{goi}} & = { - \cos\!({\eta _i})\left( {1 + {{\sin }^2}({\eta _i})/10} \right) + {{\sin }^2}({\eta _i})\cos\!({\eta _i})/5}\nonumber\\[6pt] & \quad - {a_i}{r_i}\sin\! ({\eta _i})\cos\!({\eta _i})/5V_{mi}^2 \end{align}
   Substituting Equation (50) into Equation (47) and using shorthand notations, Equation (47) can be rewritten as
   (51) \begin{align}{\dot t_{fi}} = {k_c} + {k_a}{a_i}\end{align}
   where
   (52) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{k_c} = 1 - \cos\!({\eta _i})\left( {1 + {{\sin }^2}({\eta _i})/10} \right) + {{\sin }^2}({\eta _i})\cos\!({\eta _i})/5}\\[6pt]{{k_a} = - {r_i}\sin\! ({\eta _i})\cos\!({\eta _i})/5V_{mi}^2}\end{array}} \right.\end{align}

2. (2) Convergence of the sliding surface

   In order to achieve time-cooperative guidance for multiple missiles, considering the final impact time constraint, a sliding surface is defined as

   (53) \begin{align}{s_i} = {t_{fi}} - {t_{fi}}(0) - \int_0^t u_i^{nom}dt\end{align}
   Take the derivative of sliding surface ${s_i}$ as
   (54) \begin{align}{\dot s_i} = {\dot t_{fi}} - u_i^{nom}\end{align}
   Apply the reaching law by choosing
   (55) \begin{align}{\dot s_i} = - k{s_i} - \varepsilon sgn({s_i})\end{align}
   Take the Lyapunov function ${V_{1i}}$ as
   (56) \begin{align}{V_{1i}} = \frac{1}{2}s_i^2\end{align}
   Taking the derivative of ${V_{1i}}$ with respect to time $t$ , and substituting Equation (55) into it, we can get
   (57) \begin{align}{\dot V_{1i}} = {s_i}{\dot s_i} = {s_i}({-} k{s_i} - \varepsilon sgn({s_i})) \le - ks_i^2 - \varepsilon \left| {{s_i}} \right| \le 0\end{align}
   Based on Equation (56), the following equation is obtained
   (58) \begin{align}\left| {{s_i}} \right| = \sqrt {2{V_{1i}}} \end{align}
   Substituting Equation (58) into Equation (57), we have
   (59) \begin{align}{\dot V_{1i}} \le - ks_i^2 - \varepsilon \!\left| {{s_i}} \right| = - 2k{V_{1i}} - \sqrt 2 \varepsilon V_{1i}^{\frac{1}{2}}\end{align}
   According to Lemma 1 , the sliding surface ${s_i}$ can converge to zero in a finite time ${T_1}$ , where
   (60) \begin{align}{T_1} \le (1/k)\ln \!\left( {\left( {2kV_{1i}^{\frac{1}{2}}\left( {{x_0}} \right) + \sqrt 2 \varepsilon } \right)/\sqrt 2 \varepsilon } \right)\end{align}

3. (3) Convergence of finite-time cooperative errors

   When ${s_i} = 0$ , from Equation (54), the following equation can be obtained as

   (61) \begin{align}{\dot t_{fi}} = u_i^{nom}\end{align}
   Take the Lyapunov function ${V_{2i}}$ as
   (62) \begin{align}{V_{2i}} = \frac{1}{4}\sum\limits_{i,j = 1}^n {a_{ij}}{\left( {{t_{fj}} - {t_{fi}}} \right)^2} = \frac{1}{2}T_f^TL{T_f}\end{align}
   where ${T_f} = {\left[ {{t_{f1}},{t_{f2}}, \cdots ,{t_{fn}}} \right]^T}$ is the vector consisting of the final impact time of each missile and $L$ is the Laplace matrix. Based on the symmetry $L$ due to the assumption of undirected graph, the partial derivative of ${V_{2i}}$ with respect to ${t_{fi}}$ can be presented as
   (63) \begin{align}\frac{{\partial {V_{2i}}}}{{\partial {t_{fi}}}} = - \sum\limits_{j = 1}^n {a_{ij}}\left( {{t_{fj}} - {t_{fi}}} \right)\end{align}
   Taking the derivative of ${V_{2i}}$ with respect to $t$ , combining it with Equations (63), (61) and Lemma 5 , the following equation can be obtained as
   (64) \begin{align}{\frac{{d{V_{2i}}}}{{dt}}} & = \sum\limits_{i = 1}^n \frac{{\partial {V_{2i}}}}{{\partial {t_{fi}}}}{{\dot t}_{fi}}\nonumber\\[6pt]& = - \sum\limits_{i = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right) \cdot {{\left( {sgn\!\left( {\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^{{\beta _i}}}\\[6pt]& = - \sum\limits_{i = 1}^n {{\left( {{{\left( {\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}} \right)}^{\frac{{1 + {\beta _i}}}{2}}}\nonumber\\[6pt]& \le - {{\left( {\sum\limits_{i = 1}^n {{\left( {\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}} \right)}^{\frac{{1 + {\beta _i}}}{2}}}\nonumber\end{align}
   when ${V_{2i}} \ne 0$ , we obtain
   (65) \begin{align}\frac{{\sum\limits_{i = 1}^n {{\left( {\sum\limits_{j = 1}^n {a_{ij}}\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}}}{{{V_{2i}}}} = \frac{{T_f^T{L^T}L{T_f}}}{{\frac{1}{2}T_f^TL{T_f}}}\end{align}
   According to Lemma 4 , Equation (65) can be transformed as
   (66) \begin{align}\frac{{T_f^T{L^T}L{T_f}}}{{\frac{1}{2}T_f^TL{T_f}}} \geq 2{\lambda _2}(L)\end{align}
   Substituting Equation (66) into Equation (64), the following inequality is obtained
   (67) \begin{align}{\frac{{d{V_{2i}}}}{{dt}}} & \le - {{\left( {\frac{{\sum\limits_{i = 1}^n {{\left( {\sum\limits_{j = 1}^n {a_{ij}}\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}}}{{{V_{2i}}}}{V_{2i}}} \right)}^{\frac{{1 + {\beta _i}}}{2}}}\nonumber\\[3pt]& \le - {{\left( {2{\lambda _2}(L)} \right)}^{\frac{{1 + {\beta _i}}}{2}}}V_{2i}^{\frac{{1 + {\beta _i}}}{2}}\end{align}
   According to Lemma 2 , the final impact time of the missiles can converge to zero in a finite time ${T_2}$ , where
   (68) \begin{align}{T_2} \le \left( {\frac{1}{{{\lambda _2}{{(L)}^{\frac{{1 + {\beta _i}}}{2}}}(1 - {\beta _i})}}} \right)V_{2i}^{\frac{{1 - {\beta _i}}}{2}}({x_0})\end{align}

Remark 2. The sliding surface can converge to zero within a finite ${T_1}$ , $\lim\limits_{t \to {T_1}} {\kern 1pt} \left| {{s_i}} \right| = 0$ , which guarantees the establishment of the system in Equation (44). Then, the difference of final impact times can converge to zero in a finite $T \le {T_1} + {T_2}$ , $\lim\limits_{t \to T} {\kern 1pt} \left| {{t_{fi}} - {t_{fj}}} \right| = 0$ , which implies multiple missiles can complete a cooperative attack. Since Equation (43), $\lim\limits_{t \to T} {\kern 1pt} \left| {{t_{goi}} - {t_{goj}}} \right| = 0$ also holds.

Table 1. The initial conditions of missiles

Figure 2. Communication topology for three missiles.

4.0 Simulation

In this section, simulations were conducted to demonstrate the effectiveness and advantages of the finite-time TSCGL in Equation (48). In all simulations, the initial position of target is located at $(20km,0km)$ . The initial conditions of missiles are shown in Table 1. The communication topology of multiple missiles is given in Fig. 2. The consistency parameter in Equation (45) is selected with ${\beta _i} = 0.86$ . The parameters in sliding mode convergence law shown in Equation (55) are chosen as $k = 0.1$ , $\varepsilon = 0.001$ . The limitation of guidance command is ${a_{\max }} = 50m/{s^2}$ . To mitigate the chattering caused by the sign function in Equation (55), the following function is adopted as a replacement

(69) \begin{align}{sgmf(x) = 2\!\left( {\frac{1}{{1 + {e^{ - \tau x}}}} - \frac{1}{2}} \right),}\, {\tau \gt 0}\end{align}

where $\tau $ is chosen as 20.

4.1 Multiple missiles cooperative attack

In this subsection, we compared with PNG algorithm to further demonstrate the superiority of the finite-time TSCGL in Equation (46). The results are presented in Fig. 3, and the attack results are given in Table 2.

Figure 3. Simulation results of case 1: (a) Trajectory of missiles, (b) Heading angle, (c) Guidance command, (d) Relative distance, (e) Time-to-go.

Table 2. Attack results of case 1

From Fig. 3 and Table 2, both the finite-time TSCGL and PNG can attack the target. The miss-distances of the finite-time TSCGL are less than $0.36m$ . The miss-distances of PNG are at least $1.05m$ larger than finite-time TSCGL. Compared with PNG, finite-time TSCGL has higher attack accuracy. The trajectories of finite-time TSCGL are more curved than that of PNG. The final heading angle errors between the finite-time TSCGL and PNG are bigger than at least $16.59deg$ , which implies that finite-time TSCGL has better damage performance. At the beginning of the guidance process, a larger guidance command is required to adjust the posture of the missiles to be consistent. The time-to-go of the finite-time TSCGL can converge within a short time. Compared with PNG, the finite-time TSCGL has faster convergence speed. The final impact time of finite-time TSCGL are $72s$ with their errors less than $0.1s$ . The final impact time of PNG are $66.68s$ , $70.58s$ , $68.58s$ and their differences are approximately $4s$ . In conclusion, the finite-time TSCGL can achieve cooperative attack with higher attack accuracy, better damage performance and faster convergence speed.

4.2 Compared with other methods of estimating time-to-go

In this subsection, another method of estimating ${t_{go}}$ in Ref. [31] is provided to compare that in Equation (48), which is expressed as

(70) \begin{align}{t_{goi}} = {r_i}\!\left( {1 + \eta _i^2/10} \right)/{V_{mi}}\end{align}

The results are illustrated in Fig. 4, and the miss-distance are given in Table 3. As can be seen in Fig. 4, multiple missiles can attack the target cooperatively. A striking difference was noted when the method of estimating ${t_{go}}$ was considered individually in Equation (48) and Equation (70). From Table 3, the miss-distances of Equation (48) are less than $0.36m$ . The miss-distances of Equation (70) are at least $1.95m$ larger than Equation (48). By comparison, Equation (48) demonstrates the superiority in attack accuracy. The time-to-go of the two methods can converge within a short time, which indicates that the proposed finite-time TSCGL has fast convergence speed. Similarly, their final impact time are approximately the same, which shows that the proposed finite-time TSCGL can achieve simultaneous attack. Consequently, the different methods of estimating time-to-go mainly affect the attack accuracy rather than the convergence speed and final impact time, further indicating that the proposed finite-time TSCGL has good expandability.

4.3 Cooperative attack with Monte Carlo simulations

In order to further verify the robustness of the finite-time TSCGL in Equation (46), 300 Monte Carlo simulations were performed with different conditions, and the bias are listed in Table 4. The initial conditions of the missiles and the target are the same as in Table 1. The simulation results are illustrated in Figs 5 and 6.

It can be seen from Fig. 5 that multiple missiles can cooperatively attack the target in all Monte Carlo bias scenarios. The distributions of miss distance in Fig. 6 are less than $1.83m$ . In Table 5, the average mean and standard deviation of the miss-distances are $0.87m$ , $0.39m$ . The maximum and minimum are $1.80m$ , $0.11m$ . All the statistics can meet the accuracy requirement. Consequently, finite-time TSCGL has strong robustness.

Table 3. Miss-distance of case 2

Figure 4. Simulation results of case 2: (a) Trajectory of missiles, (b) Heading angle, (c) Guidance command, (d) Relative distance, (e) Leading angle, (f) Time-to-go.

4.4 Expansion to a target with constant speed

Inspired by the concept of predicted interception point (PIP) in Ref. [16], a target with a constant speed is considered to verify the generality of the proposed finite-time TSCGL. The position of virtual stationary target can be expressed as follows:

(71) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{X_{TP}} = {X_T} + {V_T}\cos {\theta _T}{t_{{\rm{go}}}}}\\[6pt]{{Y_{TP}} = {Y_T} + {V_T}\sin {\theta _T}{t_{{\rm{go}}}}}\end{array}} \right.\end{align}

where ( ${X_{TP}}$ , ${Y_{TP}}$ ) is the PIP, ( ${X_T}$ , ${Y_T}$ ) is current position of the target, and ${\theta _T}$ is the heading angle of the target.

Table 4. Bias of Monte Carlo simulations

Table 5. Monte Carlo results of the miss-distance

Figure 5. Trajectory of missiles.

Figure 6. Miss-distance of case 3: (a) Missile 1, (b) Missile 2, (c) Missile 3.

The target moves with the heading angle of $0deg$ and speed of $20m/s$ . It can be seen from Fig. 7 that finite-time TSCGL can simultaneously attack the target with a constant speed under the PIP theory. The miss-distance are $1.31m$ , $1.37m$ and $1.39m$ , which implies that finite-time TSCGL has high attack accuracy. The time-to-go can converge within a short time, which shows that finite-time TSCGL has fast convergence speed. Therefore, the finite-time TSCGL demonstrates good expandability.

Figure 7. Simulation results of case 4: (a) Trajectory of missiles, (b) Heading angle, (c) Guidance command, (d) Relative distance, (e) Time-to-go.

5.0 Conclusion

In this paper, we proposed the finite-time TSCGL law, which can ensure that multiple missiles cooperative attack with better damage and convergence performance. The main contributions can be drawn as:

1. (1) The proposed finite-time TSCGL performs higher attack accuracy and better convergence performance. It can cooperatively attack the target, wherein the miss-distances, convergence time and final errors of impact time are less than $0.36m$ , $3s$ and $0.1s$ , separately. Compared to traditional PNG, the proposed finite-time TSCGL holds better damage performance with obviously higher final heading angles.

2. (2) The Monte Carlo simulations show that the proposed finite-time TSCGL has strong robustness. It can achieve simultaneous attack with fast convergence speed. The average mean and standard deviation of the miss-distances are $0.87m$ and $0.39m$ , respectively.

3. (3) The finite-time TSCGL has good expandability. The estimation methods of time-to-go can be replaced, and the proposed method in this paper shows better attack accuracy. Through the PIP theory, it can be expanded against a target with a constant speed.

In the future, the guidance law will be promoted to the three-dimensional space and the complex manoeuver of targets.