Mathematical Problems in Engineering

Volume 2018, Article ID 5062184, 17 pages

https://doi.org/10.1155/2018/5062184

## Maneuvering Detection Using Multiple Parallel CUSUM Detector

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

Correspondence should be addressed to Zhongliang Zhou; moc.uhos@lzzuotum

Received 23 October 2017; Accepted 5 February 2018; Published 12 March 2018

Academic Editor: Paolo Addesso

Copyright © 2018 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 switching model tracking algorithm based on hard decisions is an important method to solve the maneuvering target tracking problem. The use of Doppler velocity not only helps shorten the delay time of maneuvering detection but also provides information about the target motion model. A novel target maneuvering detection method named Multiple Parallel Cumulative Sum (M-CUSUM) for target multiple maneuvering models is proposed in this paper based on Doppler velocity. The main scheme of the proposed approach consists of the following: firstly, the problem framework of multiple model maneuvering detection is put forward; secondly, the statistic of acceleration is obtained through modeling the mapping relationship between Doppler velocity and the normal/tangential acceleration according to the geometry and kinematics; thirdly, the joint empirical distribution of the normal/tangential acceleration is obtained by the statistical experiment method and then the approximate joint probability distribution function of the normal/tangential acceleration is acquired by use of Gaussian Mixture Model (GMM) with Expectation Maximization (EM) algorithm; fourthly, it is taken as the prior information of target maneuvering which is introduced to the likelihood ratio of prediction measurement residual by the marginalization method; finally, the standard Cumulative Sum (CUSUM) detector is extended as Multiple Parallel CUSUM detector. Simulation results show that M-CUSUM detector has a smaller maneuver onset detection delay time compared with similar detectors and has the ability of pattern recognition of target maneuvers.

#### 1. Introduction

Tracking the manned maneuvering aircraft by use of radar is a maneuvering target tracking (MTT) problem [1–3]. It is hard to accurately estimate the motion state of maneuvering target by using only single motion model, because the current radar technology cannot obtain the observation about accelerations directly supporting the selection of the target motion model. The switching model tracking (SMT) algorithm [4] based on (the) hard decision and the interacting multiple model (IMM) tracking algorithm [5–7] based on the soft decision are two kinds of basic methods to solve MTT. The SMT algorithms use a maneuvering detector to judge the target maneuvering onset. When the target maneuvering is successfully detected, the motion model of tracking algorithm is immediately switched. The timely and correct maneuvering detection is the key to the SMT algorithm. The most maneuver detectors [8] only use the position measurement conditioned on the linear Gauss hypothesis. As the position measurement cannot directly reflect the maneuver state, however, radar can obtain not only the target position measurement but also Doppler velocity measurement directly affected by the maneuvering [9]. The maneuvering state of target can be obtained by use of Doppler velocity. Some scholars obtain the turn rate estimation from Doppler velocity [10–12]. As the acceleration is directly related to the target maneuvering type, other scholars obtain the statistics of the target acceleration directly through Doppler velocity. Bizup and Brown [13] propose a method of maneuvering target detection using Doppler velocity. By assuming the constant turn rate motion, a statistic of acceleration is deduced from Doppler velocity. Ru et al. [14] combine the statistic with the tangential acceleration to obtain the statistic of total acceleration, , and . The main difference between and is whether Doppler velocity measurement is applied in the filtering. It turns out that the statistic or using Cumulative Sum (CUSUM) detector has a shorter detection onset delay time than [15]. Lu et al. [16] use Mahalanobis distance and Euclidean distance optimization method to give two statistics of the total acceleration and , respectively, which overcome the problem that the acceleration has no solution when the noise is higher. Using the statistics in the above references to detect and track the target, there are the following problems.

*Assuming a Constant Turn Rate Motion. *The statistics , , and are obtained by assuming a constant turn rate motion. However, in addition to the normal acceleration, the target maneuver usually has a tangential acceleration and the trajectory exhibits a curvilinear shape. There is an inherent system bias when the statistics , , and are used in the maneuver target detection.

*One-to-Many Mapping Relationship. *There is a one-to-many mapping relationship between Doppler velocity and the normal acceleration. It is assumed that the target velocity direction change is less than in the radar scanning period, so the one-to-many mapping is restricted to a pair of four mappings in the statistics , , and . In practice, the tracking radar scanning period is less than s and the maximum instantaneous turn rate of the manned maneuvering aircraft is not more than [17]. Therefore, if one-to-four mapping is applied to track such target, it will cause the dispersion of probability weight and affect the sensitivity of detector as some unlikely turns are considered.

*Unknown Motion Model. *The detector of total acceleration based on Doppler velocity can detect target maneuver rapidly and distinguish the maneuvering and nonmaneuvering state. After the maneuvering is detected, the motion model with larger acceleration noise is usually chosen to filter. However, the motion model mismatch problem cannot be completely solved by increasing the acceleration noise. If a maneuver detector can not only detect whether the target is making a maneuvering or not but also recognize the target maneuvering model, we could select the matched motion model to improve the tracking accuracy with a lower acceleration noise.

To solve the above problems, the mapping relationship between Doppler velocity and the normal/tangential acceleration is deduced without assuming a constant turn rate motion. Meanwhile, it is assumed that the target velocity direction change is less than during the radar scanning period, and one-to-four mapping relationship is restricted to one-to-two which reduces the dispersion degree of probability weight. In this paper, two parameters, namely, normal/tangential acceleration, are used to describe the target maneuver, and the single total acceleration is not adopted. The joint empirical distribution of the normal/tangential acceleration is obtained by the statistical experiment method [15] and approximated by use of Gaussian Mixture Model (GMM) with Expectation Maximization (EM) algorithm [18]. Then the approximate joint probability distribution function (PDF) of the normal/tangential accelerations is obtained. The approximate joint PDF is taken as the a priori information of target maneuvering which is introduced to calculate the likelihood ratio of prediction measurement residual by the marginalization method. A Multiple Parallel Cumulative Sum (M-CUSUM) detector is proposed by extending the standard CUSUM detector. In addition, it has been proposed that the use of Doppler velocity measurement can improve tracking accuracy [19–21]. In this paper, Doppler velocity is used separately in the maneuvering detector and filtering. As the measurement equation with Doppler velocity is highly nonlinear, the traditional EKF algorithm [22] does not work well. We use the measurement conversion algorithm proposed by Jifeng and Huimin to construct the linear KF filter.

The content of this paper is designed as follows. We describe the tracking and detection problem and propose a concept framework of multiple maneuver model detection and recognition in Section 2. We present the method of using Doppler velocity to deduce the normal/tangential acceleration in Section 3. We present how to obtain the empirical distribution of the normal/tangential acceleration and the approximated GMM with EM algorithm in Section 4. We describe M-CUSUM detector for multiple maneuver model detection and recognition in Section 5. In Section 6, firstly the results of the empirical distribution of the normal/tangential acceleration and GMM fitting are shown in five scenes; secondly, M-CUSUM proposed in this paper is compared with other maneuver detection methods; finally, it is applied to the SMT algorithm and is compared with IMM algorithm and CSAF algorithm. We summarize the research content of this paper in Section 7. In Appendix A, the modified measurement conversion algorithm and filter algorithm are given. In Appendix B, The EM algorithm is given to learn the parameters in GMM. In Appendix C, the state space equation of CSAF algorithm is given in two-dimensional space.

#### 2. Problem Descriptions

##### 2.1. Target Maneuver Model and Measurement Model

An aircraft is moving in a two-dimensional plane whose acceleration can be decomposed into a normal acceleration perpendicular to the velocity direction and a tangential acceleration along the velocity direction. The normal/tangential accelerations are used as the input vector of the motion model and then the target maneuver model [18] is expressed aswhere the state vector is in Cartesian coordinate system; are the position component; are the velocity component; is the acceleration vector; is the tangential acceleration; is the normal acceleration; is a white Gaussian noise vector and its covariance matrix is :

and are as follows:where is the radar scanning period; is the target velocity direction.

Assume that the radar is located at the origin of coordinate . The measurement equation of radar which includes Doppler velocity iswhere is the observation vector; is the range; is the azimuth; is Doppler velocity; is a white Gaussian noise and its covariance matrix is ; , and are variances of the range, azimuth, and Doppler velocity, respectively; is the nonlinear measurement function vector:where “” is “+” if and “” is “−” if .

##### 2.2. The Problem Framework of Multiple Maneuvering Detection

We define the nonmaneuvering target model as a constant velocity motion model; that is, . Then the maneuvering target model is denoted as . The traditional maneuvering detection problem is expressed as follows:The target does not maneuver; for , where is the radar sample time.The target starts to maneuver at unknown time ; for .

For the above maneuvering detection problem, the traditional maneuver detector can accomplish the judgment between the maneuver and nonmaneuver, that is to say, to solve a simple binary hypothesis [23] test task. However, decision-makers often want to get the information of the target maneuver type. Then hypothesis can be decomposed into multiple branches and the maneuver detection problem can be described as a new form:The target does not maneuver; for .The target starts to maneuver at unknown time ; for and .

represents the th maneuvering model; is the total number of maneuvering target models. In particular, the nonmaneuver is , which means that and . The maneuvering is , which can represent the different maneuvering model if is the different nonzero vector.

#### 3. The Normal/Tangential Acceleration

The key of reasonable multiple maneuvering detector is to obtain the information of normal/tangential acceleration. In the aspect of radar measurements, Doppler velocity contains the target maneuvering information. And [13] establishes the mapping relationship between Doppler velocity and the normal acceleration on the assumption of a constant turn rate motion. This paper renews to build a mapping relationship between Doppler velocity and the normal/tangential acceleration without the assumption.

We assume that the target moves in 2D plane of coordinate , as shown in Figure 1.