Journal of Control Science and Engineering

Volume 2018, Article ID 6432485, 6 pages

https://doi.org/10.1155/2018/6432485

## Performance Analysis and Comparison for High Maneuver Target Track Based on Different Jerk Models

^{1}PLA 91550, Dalian 116024, China^{2}College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China^{3}Beijing Institute of Spacecraft System Engineering, China Academy of Space Technology, Beijing 100086, China

Correspondence should be addressed to Jiongqi Wang; moc.361@dkfg_qjw

Received 29 March 2018; Accepted 14 May 2018; Published 11 June 2018

Academic Editor: Darong Huang

Copyright © 2018 Qinghai Meng 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 Jerk model is widely used for the track of the maneuvering targets. Different Jerk model has its own state expression and is suitable to different track situation. In this paper, four Jerk models commonly used in the maneuvering target track are advanced. The performances of different Jerk models for target track with the state variables and the characters are compared. The corresponding limit conditions in the practical applications are also analyzed. Besides, the filter track is designed with UKF algorithm based on the four different models for the high-maneuvering target. The simplified dynamic model is used to gain the standard trajectory with Runge-Kutta numerical integration method. The mathematical simulations show that Jerk model with self-adaptive noise variance has the best robustness while other models may diverge when the initial error is much larger. If the process noise level is much lower, the track accuracy for four Jerk models is similar and stationary in the steady track situation, but it will be descended greatly in the much highly maneuvering situation.

#### 1. Introduction

Target tracking refers to estimating the motion parameters, such as the position and the velocity for a target through the noise-containing measurement data acquired by the measurement device in real time. Because of the uncertainty of the maneuvering target motion, the measurement process uncertainty, and the difficulty of estimating the nonlinear system, the maneuvering target tracking has always been a research focus for more than half a century and there are a lot of proposed algorithms. To sum up, the track algorithms study has been focused on two parts,* i.e.*, maneuver target modeling and nonlinear filter design [1, 2].

For the nonlinear filter design, there are three main methods, including Extended Kalman filter (EKF), Unscented Kalman filter (UKF), and Particle filter (PF) [3, 4]. Many articles have sufficiently studied the three algorithms in the nonlinear fitness, filter accuracy, filter stability, computational complexity, and other aspects [5, 6]. The conclusion is that UKF has the best performance in much practical applications [7].

For the maneuver target modeling, there are lots of motion model describing the maneuver process of the target. The accuracy of the target modeling directly affects the tracking performance of maneuvering target, detection of the target [8, 9], and the fault diagnosis of the target [10, 11]. According to their modeling state dimensions, they can be divided into second-order model, three-order model and four-order model. Two-order models contain CV (Constant Velocity) model and CT (Constant Turn) model; three-order models contain CA (Constant Acceleration) model, Singer model, CS (Current Statistic) model, semi-Markov model, and so on [12, 13]. Four-order models contain Jerk (Jerk denotes the rate of the acceleration change) model and its corresponding improved version. The higher the order of maneuvering model is, the higher the order of the target is described. Jerk model extends the target maneuvering form via estimating the acceleration changing rate in real time. Theoretically, it can be applied on the highly maneuvering target tracking much better [14, 15].

Therefore, in this paper, we mainly focused on the maneuvering target track with Jerk model and UKF filter algorithm.

The references related to Jerk model has shown the good simulation results. However, due to the simple and the special simulation background, the trajectory is different from the true target. Therefore, the results cannot be convincing. This paper summarizes the various Jerk models in the references. Firstly, the advantages and disadvantages are analyzed in theory. Then every model is applied on the near-space high-speed maneuvering target, and the simulation results are compared. The track trajectory is generated through the integral of the simplified dynamic equation, which is close to the true target. Therefore, it is convictive to some extent.

#### 2. Description for Different Jerk Model

Jerk denotes the acceleration changing rate. Jerk model is a model which describes the target Jerk mathematically. The models in the references mainly contain the following different kinds.

##### 2.1. SJ Model

The earliest Jerk model is proposed by Mehrota, etc. [16]. They used Singer model for reference and modeled the Jerk model of the target as a zero-mean and first-order time related process. To distinguish other Jerk models, it is marked as SJ (Singer Jerk) model. Taking a one-dimension linear motion as an example, SJ model is expressed aswhere denotes the target Jerk, the denotes “Jerk” frequency (the reciprocal of the “Jerk” constant), denotes zero-mean Gaussian white noise, and the covariance is , denotes the covariance of target Jerk.

##### 2.2. CSJ Model

Qiao [15] used the analyzing method which is also applied on Singer model tracking accuracy in [17] to analyze the SJ model. He proposed that SJ model shows a steady-state deterministic error in the tracking step Jerk signals. Therefore, the Jerk model with nonzero mean and first-order time correlation is built using CS model for reference. According to the same analyzing process, the new model has eliminated the steady-state deterministic error. The model is marked as CSJ (Current Statistic Jerk) model. Taking the one-dimension model as an example, CSJ model is expressed aswhere denotes the nonzero mean of Jerk. denotes the covariance of the zero-mean colored Jerk noise, and denotes zero-mean Gaussian white noise. Other parameters are defined as SJ model.

The model should set and previously before the practical application. Therefore, Pan proposed a novel CSJ algorithm to describe the probability density of Jerk according to truncated normal distribution and construct the connection between and current Jerk estimation. It isIn this way, the probable extreme Jerk can be predefined as . That is the covariance of the state noise can be self-adapted according to the Jerk estimation during the filter process to adapt to different maneuvering situations. The model is marked as MCSJ (Modified CSJ) model.

##### 2.3. *α*J Model

The Jerk models above cannot avoid the problem that the Jerk frequency should be predefined. However, cannot be directly measured and it is constantly changing in the target practical motion process. For this reason, Luo [7] considered in SJ model as an estimated parameter and took it as the extension variable. Therefore, it can be estimated in real time during the filter process. is modeled as nonzero-mean Gaussian white noise and the derivative of it is zero mean Gaussian white noise which can be considered aswhere denotes the zero-mean input noise, whose variance is . This is called J (Alpha Jerk) model. J model can estimate in real time, but and need to be predefined carefully. If the values of them are not proper, the estimation accuracy of will be decreased severely and even cause the divergence of the filter.

Take the one-dimension motion as an example. (Three-dimension situation has the same principle as the one-dimension motion.) The state variable, state equation, and the characteristic of the four models are shown in Table 1. The specific form of the matrices in Table 1 can be found in the related references [14, 16].