International Journal of Aerospace Engineering

Volume 2015, Article ID 479427, 8 pages

http://dx.doi.org/10.1155/2015/479427

## A Three-Dimensional Cooperative Guidance Law of Multimissile System

Key Laboratory of Ministry of Education for Image Processing and Intelligent Control, School of Automation, Huazhong University of Science and Technology (HUST), Wuhan 430074, China

Received 15 May 2015; Accepted 13 September 2015

Academic Editor: Christopher J. Damaren

Copyright © 2015 Xing Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to conduct saturation attacks on a static target, the cooperative guidance problem of multimissile system is researched. A three-dimensional guidance model is built using vector calculation and the classic proportional navigation guidance (PNG) law is extended to three dimensions. Based on this guidance law, a distributed cooperative guidance strategy is proposed and a consensus protocol is designed to coordinate the time-to-go commands of all missiles. Then an expert system, which contains two extreme learning machines (ELM), is developed to regulate the local proportional coefficient of each missile according to the command. All missiles can arrive at the target simultaneously under the assumption that the multimissile network is connected. A simulation scenario is given to demonstrate the validity of the proposed method.

#### 1. Introduction

Saturation attack, which involves simultaneous attack from different missiles in a communication network, is an important combat manner to penetrate the missile defence system. In fact, a group of well-organized missiles of low cost and poor performance may yield better results than a single excellent one. The key to cooperative guidance of multimissile system is that all missiles reach the target at the same time in the case of saturation attacking.

Cooperative control theories have been researched broadly with respect to different agents, such as unmanned aerial vehicle [1–3], satellite [4], and some abstract objects [5, 6]. A cooperative control strategy for achieving cooperative timing among teams of vehicles which is based on coordination variables and functions was developed [7]. However, there are only a few existing literatures considering the cooperative timing problem of missiles. In [8], with a combination of the PNG law and the feedback of the impact time error, an impact time control guidance (ITCG) law for salvo attack of antiship missiles was presented and could be used to guide multiple missiles to hit a stationary target simultaneously at a desirable impact time. Based on this law, a cooperative PNG law was proposed by introducing a new concept of the variance of time-to-go of multiple missiles in [9]. With the weighting average consensus algorithm [10], a cooperative guidance scheme [11] was developed based on the ITCG law.

Different from the unmanned aerial vehicle or satellite, the maneuver flight of missile is mainly based on aerodynamic force and its guidance does not include task assignment, cooperative path planning, and path tracking. This makes it particularity hard to design a cooperative strategy for the multimissile system. In the case of a group of missiles intercepting a single maneuver target, an optimal cooperative guidance law [12] based on comprehensive cost function of missiles was derived with the constraint of a relative intercept angle. A time-cooperative guidance architecture which is a centralized coordination control form was proposed based on leader-follower strategy [13]. Using the dynamic surface control theory and disturbance observation technology, [14] developed a novel integrated guidance and control law, which can realize cooperation of impact time and flight position for multiple missiles during their cooperative attack. All these methods adopt a centralized control manner in which one missile must exchange information with all the other missiles.

Many classical literatures [15–18] described and analysed the PNG law in two-dimensional plane, and most of the three-dimensional trajectory simulations [19] generally adopted the dimensionality reduction method to decompose the movement of missile into longitudinal and lateral plane. Then PNG law was utilized to obtain the trajectory in two planes, respectively, and thereby compounded the desired three-dimensional trajectory. However, this method needs to solve some antitrigonometric functions [20] on account of the direction of coordinate axis; meanwhile the coupled problems have adverse effects on the precision. Furthermore, geometric methods mentioned above are on a necessary condition that the velocities of all missiles are constant. In order to settle these problems, this paper introduces the space vector method to generate the PNG law in three dimensions.

In order to achieve the simultaneous attack of missiles, the consensus of time-to-go (remaining flight time, i.e., arriving time) is considered in this paper. From the previous literature, it is known that the time-to-go of missiles can be changed by adjusting proportional coefficient of the PNG law. However, the relationship between the command and the local proportional guidance law is nonlinear and hard to obtain directly, especially in three dimensions, and then the fitting methods are considered. An artificial neural network (ANN), with a fixed number of numeric inputs and outputs, can be regarded as a complex nonlinear function. By learning iteratively large numbers of samples, the ANN stores the mapping relation between inputs and outputs. Due to this ability of self-learning and self-adaption, ANN is widely used in expert systems. Although ANN is of high precision, the learning speed of traditional ANN is so slow that it cannot be used in online cooperative guidance. ANN with extreme learning machine (ELM) is introduced in the paper to overcome this problem. It is notable that ELM is of strong generalization ability [21], which can learn the rules more accurately from training data; meanwhile it can obtain the appropriate output even if the input data is out of training set. Therefore, the problem of the inaccurate results caused by some special inputs of ANN is solved.

In this paper, we design a cooperative guidance strategy to achieve simultaneous attack based on expert system using ELM, which just requires that the communication network is connected. Considering that the communication between missiles might be incomplete on account of the disturbances from defense system and other environment factors, the centralized cooperative guidance strategy is liable to fail and the distributed strategy is more effective. Via a distributed protocol through the connected network, which aims at asymptotical consensus of time-to-go commands, the commands of all missiles are coordinated. Then the local proportional guidance law of each missile is regulated by the expert system according to the commands. Through the consensus algorithm of distributed cooperative guidance strategy, the proportional coefficients of all missiles are adjusted to achieve saturation attack.

The remainder of this paper is organized as follows. Section 2 describes the cooperative guidance problem of multiple missiles on three dimensions. Section 3 proposes a new distributed cooperative guidance strategy based on the expert system with ANNs using ELM. Section 4 shows the simulation results to illustrate the validity of the proposed algorithm. Finally, some concluding remarks to this paper are presented in Section 5.

#### 2. Cooperative Guidance Problem Formulation

Consider the cooperative guidance of similar missiles which are denoted by . The objective is to make all missiles arrive at a static target at the same time. First, the coordinate system which is used to describe the relative movement of missile and target in space is built, and the three-dimensional guidance model is obtained. Then, the PNG law is extended to three dimensions. At last, the cooperative guidance problem of multiple missiles is described.

##### 2.1. Movement Coordinate System

Set as the relative velocity vector of missile to the target and as the relative position vector (sight vector). Assume that the plane composed of and is instantaneous sight transfer plane at any time. and are the projection elements of on direction and vertical direction of , respectively. The relative movement coordinate system is depicted in Figure 1. The origin is located on the target. The plane is instantaneous sight transfer plane and axis is on the horizontal plane of target which points to the missile. Axis is on the plumb line of target which points up and axis is perpendicular to the plane. is the line of sight (LOS) angle and is the included angle of and . is the LOS angular velocity vector and is the command acceleration vector of missiles.