Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3921608, 11 pages

http://dx.doi.org/10.1155/2016/3921608

## Tracking Air-to-Air Missile Using Proportional Navigation Model with Genetic Algorithm Particle Filter

Aeronautics and Astronautics Engineering College, Air Force Engineering University, Xi’an 710038, China

Received 17 November 2015; Accepted 25 February 2016

Academic Editor: Vladimir Turetsky

Copyright © 2016 Hongqiang Liu 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

The purpose of this paper is to track the air-to-air missile. Here we put forward the PN-GAPF (Proportional Navigation motion model and Genetic Algorithm Particle Filter) method to solve the problem. The main jobs we have done can be listed as follows: firstly, we establish the missile state space model named as the Proportional Navigation (PN) motion model to simulate the real motion of the air-to-air missile; secondly, the PN-EKF and PN-PF methods are proposed to track the missile, through combining PN motion model with EKF and PF; thirdly, in order to solve the particle degeneracy and diversity loss, we introduce the intercross and variation in GA to the particles resampling step and then the PN-GAPF method is put forward. The simulation results show that the PN motion model is better than the CV and CA motion models for tracking the air-to-air missile and that the PN-GAPF method is more efficient than the PN-EKF and PN-PF.

#### 1. Introduction

The air-to-air missile is the main weapon in the air combat. The aircraft been chased should maneuver to avoid the attacking missile, after the air-to-air missile was fired by an opposition fighter. The aircraft guidance method for maneuver evasion is based on knowing the missile location and the real-time track [1]. However, we cannot acquire the exact location information of the missile because the measurement for the missile location has great error. We need an on-line filter method to eliminate the error and track the air-to-air missile. How to effectively track the missile is the research content in this paper.

This problem is the domain of the single target tracking. The common target motion models such as the CV, CA, and Current Statistical (CS) models cannot well and truly describe the air-to-air missile maneuver because the missile has a good maneuverability and a supersonic speed [2]. However, we can research from the navigation law point to establish a smarter motion model because the missile maneuver obeys some navigation law [3]. The Proportional Navigation (PN) law is the most common in the air-to-air missiles [4]. Therefore we establish a new PN motion model in 3d Cartesian coordinate system by analysis of the PN mechanism.

The state space model for tracking the air-to-air missile is obtained further, through combining with the nonlinear measurement equation in the radar spherical coordinate system. The standard Kalman Filter (KF) cannot be used to track the missile because of the nonlinear measurement equation. The Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are used to solve the nonlinear problem, but the methods are not efficient enough in practice [5]. Recently, the Particle Filter (PF) or Monte Carlo (MC) method is widely applied in the nonlinear and non-Gaussian filter problem because PF is to be able to represent the required unknown probability density function by a scatter of particles sampled from a known probability density function [6]. However, the resampling of standard PF has a disadvantage of the loss of particle diversity [7]. We are inspired by the evolution idea of Genetic Algorithm (GA) to improve the performance of the PF at resampling step [8]. We take the particles and weights in the PF as the chromosomes and adaptability in the GA, respectively. Then the intercross and variation steps in the GA can be adopted to deal with the particles in resampling step for avoiding the particle degeneracy and loss of diversity. In this paper, Genetic Algorithm Particle Filter (GAPF) combined with the PN motion model (PN-GAPF) is used to track the air-to-air missile. PN-GAPF has better the estimate accuracy and the tracking stability, in comparison with other algorithms and models in computer simulation.

This paper is divided into five sections. Section 1 is introduction. Section 2 presents the state space model for tracking the air-to-air missile and the method of establishing the PN motion model. Section 3 describes the nonlinear filter algorithms such as the EKF, PF, and GAPF proposed. Section 4 discusses the performance of the PN motion model and GAPF algorithm on the base of the simulation results. Section 5 summarizes the main research content.

#### 2. Problem Formulation

##### 2.1. The State Space Model

The state space model for tracking the air-to-air missile consists of the missile state equation and the measurement equation. In order to simplify the problem, the missile state equation is usually modeled linearly in Cartesian coordinate system and the measurement equation is expressed nonlinearly in spherical coordinate system because of the air-to-air missile measurements given in the radar spherical coordinate system [9].

###### 2.1.1. The Linear Missile State Equation

In general, we estimate the state of the moving air-to-air missile with the discrete-time linear state space in 3d Cartesian coordinate system [10]:where and are the state vector at time and respectively and is a sampling period. is the state transition matrix. is the process noise which is modeled as a zero-mean white Gaussian process with covariance matrix .

The CV model and the CA model are the most common motion models for tracking air-to-air missile [11]. The state vector and the state transition matrix of CV model in 3d Cartesian coordinate system areAnd the state vector and the state transition matrix of CA model arewhere , , and are the missile positions along each axis of coordinates. Correspondingly, , , and are the velocities. , , and are the accelerators. If sampling the measurement in a very short time, we can consider the missiles maneuvering as an approximate CV model or CA model [11].

###### 2.1.2. The Nonlinear Measurement Equation

In the real air combat, the measurement of air-to-air missile is as follows in the radar spherical coordinate system [12]:where is the measurement vector. is the distance between the aircraft attacked and the air-to-air missile which can be detected by the airborne laser range finder. and are the pitching and azimuth angles, respectively, which can be detected by the aircraft radar warning device [13].

The nonlinear measurement equation iswhere is the nonlinear measurement function which will be given at the next section. is the measurement error which is zero-mean white Gaussian noise with covariance matrix [14].

##### 2.2. The Proportional Navigation Motion Model

If the air-to-air missile movement is modeled just simply as the CV or CA motion model, it will produce great errors of the missile state estimation [4]. It is known that the air-to-air missile is designed to maneuver by the Proportional Navigation law. Therefore, the motion model based on the Proportional Navigation law is a smarter model to reflect the states in the attack process. We describe the relative movement between the aircraft attacked and the air-to-air missile in Figure 1.