Wireless Communications and Mobile Computing

Volume 2018, Article ID 1476426, 12 pages

https://doi.org/10.1155/2018/1476426

## Simplex Cubature Kalman-Consensus Filter for Distributed Space Target Tracking

^{1}Graduate School, Space Engineering University, Beijing 101416, China^{2}Department of Electrical and Optical Engineering, Space Engineering University, Beijing 101416, China

Correspondence should be addressed to Dan Ding; moc.361@rjndd

Received 11 January 2018; Accepted 15 April 2018; Published 24 June 2018

Academic Editor: Pavlos I. Lazaridis

Copyright © 2018 Zhaoming Li 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

A simplex cubature Kalman-consensus filter, which is suitable for distributed space target tracking using multiple radars, is proposed to improve the target tracking accuracy. The detailed orbital dynamics model and radar measurement model are given as the system filtering models. The intractable nonlinear Gaussian weighted integral in the filter is decomposed into the spherical integral and radial integral, which are calculated using the spherical simplex rule and the second-order Gauss-Laguerre quadrature rule, respectively. In this way, a new simplex cubature rule is derived. By means of the statistical linear regression method, the posterior mean, covariance, and the transmitted messages in the extended Kalman-consensus filter are approximated using the deduced simplex cubature rule, which results in the proposed simplex cubature Kalman-consensus filter. No data fusion center exists in the filter, and each radar only needs to exchange the information with its neighbors to reach a consensus estimate. The simulation results show that the proposed filter can achieve more accurate results compared with the cubature Kalman-consensus filter.

#### 1. Introduction

In recent years, the orbital resources are becoming increasingly strained on account of the increase in the number of space targets, which should be monitored to improve the utilization efficiency of the space resources. Ground-based radar is a significant sensor in the space surveillance system, it provides round the clock working capability, and it is a key technology to use its measurement data for space target tracking [1].

The space target tracking can be considered as a nonlinear system state estimation problem, and two methods, including the particle filter and the nonlinear Kalman filter, are mainly taken. Particle filter is a type of Monte Carlo based filter, which is theoretically applicable to arbitrary nonlinear non-Gaussian system without any assumptions on the posterior probability density function (PDF) [2]. However, in practical applications, there appear some problems containing particle degradation and depletion, large amount of calculation, and the choice of the importance function that will affect the filtering accuracy as well as the computational efficiency [3]. Therefore it is not suitable for application in the occasion that requires high real-time performance [4].

In the nonlinear Kalman filter, the posterior PDF is assumed to be Gaussian distribution, and the suboptimal estimate of the nonlinear system state is obtained. The extended Kalman filter (EKF) [5] is the most widely used nonlinear Kalman filter in the past several decades, it uses the multidimensional Taylor series expansion to linearize the nonlinear function, and then the standard Kalman filter is applied. For arbitrary nonlinear transformation, it can be seen using the Taylor series that the linearized mean can only match to the first order of the true one; hence the EKF is considered to achieve the first-order filtering accuracy. Moreover, the accuracy and the numerical stability of EKF are reduced for the strong nonlinear system, and the calculation of Jacobian matrix imposes further restrictions on the system models.

Based on the assumption that the approximation to the posterior PDF is easier than that of arbitrary nonlinear function, Julier [6] proposed the unscented Kalman filter (UKF). UKF is a deterministic sampling filter; that is, the posterior mean and covariance are calculated using the sampling points, which are generated using the nonlinear propagation of sigma points selected as certain criterion, and the third-order accuracy can be achieved [7]. UKF is a derivation-free filter, which can commendably overcome the defects of EKF. However, there exist some tunable parameters in sigma points, the selection of which lacks rigorous mathematical basis, and the negative weight on the center point may reduce the numerical stability for high-dimensional system [8, 9].

By means of the coordinate transformation, Arasaratnam [10] proposed the cubature Kalman filter (CKF), in which the key intractable integral is decomposed into the spherical integral and radial integral. These two integrals are approximated using different numerical methods, respectively, and result in the spherical-radial cubature rule, which is used to calculate the posterior mean and covariance in the nonlinear Kalman filter framework. CKF can be regarded as a special case of UKF with the parameter [11], whereas CKF gives the rigorous mathematical reason why should be chosen zero for the first time. Furthermore, CKF has higher numerical stability than UKF and has a wide range of applications in engineering [12, 13].

In order to further improve the estimation accuracy of CKF, Wang [14] introduced the transformation group of the regular simplex to compute the spherical integral and proposed the spherical simplex-radial cubature Kalman filter (SSRCKF). The simulation results show that SSRCKF can achieve more accurate results compared with CKF for high-dimensional system. Bhaumik [15, 16] adopts -order generalized Gauss-Laguerre quadrature rule to calculate the radial integral and puts forward the cubature quadrature Kalman filter (CQKF). It is pointed out that CQKF is the generalized form of CKF; that is, CKF is the simplified form of CQKF with the first-order Gauss-Laguerre quadrature rule. Jia [17] proposes the higher-order cubature Kalman filter; it improves the filtering accuracy with much more points needed.

However, the space target tracking accuracy obtained by a single radar cannot satisfy the demands in some practical applications; thus the information fusion by multiple radars should be considered. In tradition, the information is centralized fused; namely, all radars send their measurement data to a data fusion center, on which the centralized filter is carried out. In this mode, the whole system will collapse once the center failure occurs. With the development of distributed sensor network technology, distributed filter has become a subject undergoing intense study [18, 19]. Olfati [20, 21] gives a computational framework of the Kalman-consensus filter (KCF) for linear system, in which multiple sensors synchronously observe the same target, and the estimation errors among local nodes are eliminated through information collaborative interaction, such that the state estimates of various sensors reach a consensus. Furthermore, a consensus-based distributed Kalman filter for state estimation in a sensor network considering the random failures of the connections is proposed in [22], and a Kalman filter type consensus+innovations distributed linear estimator of the linear time-invariant dynamical systems is developed in [23]. These two distributed filters have shown better performance compared with KCF in some cases; however, they are designed for linear systems. For nonlinear system, Pellett [24] proposes the extended Kalman-consensus filter; nevertheless, the filter holds the inherent defects of EKF, that is, to calculate the Jacobian matrix and achieve the first-order accuracy.

In this paper, a simplex cubature Kalman-consensus filter (SCKCF) is put forward to improve the distributed space target tracking accuracy by multiple radars. First, for the intractable nonlinear Gaussian weighted integral, the spherical simplex rule and the second-order Gauss-Laguerre quadrature rule are utilized to calculate the spherical integral and radial integral, respectively, and a new simplex cubature rule is derived. Then, combined with the statistical linear regression method, the SCKCF is proposed in the extended Kalman-consensus filter framework. Different from the centralized filter, there is no data fusion center in the SCKCF, and each radar only exchanges information with its neighbors, which can effectively improve the system's fault tolerance and scalability. The simulation results show that the proposed SCKCF can achieve higher orbit determination accuracy than cubature Kalman-consensus filter.

The rest of this paper is organized as follows: the mathematical models for space target tracking are provided in Section 2. The new simplex cubature rule is derived in Section 3. The simplex cubature Kalman-consensus filter is proposed in Section 4. The simulation results and analysis are presented in Section 5. The conclusion is given in Section 6.

#### 2. Mathematical Models for Space Target Tracking

In this section, the space target orbital dynamics model and the radar measurement model, which are considered as the state equation and measurement equation in the filter, respectively, are given below.

##### 2.1. The Orbital Dynamics Model

The space target orbit is described in the J2000 earth inertial coordinate system (, shown in blue in Figure 1), and the orbital dynamics model with nonspherical gravitational perturbation is given as follows:where and denote the position and velocity of the space target, respectively. The parameters , , and represent the harmonic coefficient, the earth gravitational constant, and the radius of the earth, respectively. The perturbation is the sum of the high-order nonspherical perturbation, three-body gravitational perturbation, and solar radiation pressure perturbation, which can be considered as the zero mean white Gaussian noise in this study.