# Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking
###### Abstract
Specifically, a set of sigma points is used to approximate the posterior mean and covariance in UKF, and [PERSON] [20] proposes using the orthogonal transformation of the sigma points for improving the approximation accuracy without increasing the computational complexity, thus reducing the high-order error term. However, there exist three adjustable parameters in UKF, and the inappropriate choice of parameters may reduce the filtering accuracy and the numerical stability. The computation amount of GHKF increases exponentially with the system state, which may cause the curse of dimensionality [16]. In order to reduce the computational complexity, the sparse-grid integral rule can be utilized to derive the SGKF, and the number of integral points is increased polynomially [17]. However, the CKF is widely used in engineering [21; 22; 23; 24]. The CKF helps the intractable Gaussian-weighted integral to decompose and is composed of a spherical integral and a radial integral, and a set of cubature points with equal weights are deduced [25]. It can achieve more filtering accuracy and numerical stability compared with UKF. Similar to the IEKF, the principle of iteration can also be expanded to these filters' measurement update processes [26; 27; 28; 29]. Based on the CKF, [PERSON] [30] puts forward the high-degree cubature Kalman filter (HDCKF) by using arbitrary degree symmetric spherical rule and moment matching method, which achieve higher accuracy than the CKF. And then, [PERSON] [31] employs a transformation group of regular simplex to approximate the spherical integral, and proposes the spherical simplex-radial cubature Kalman filter (SSRCKF). Compared with the HDCKF, the SSRCKF has more accurate performance, whereas the moment matching method cannot guarantee optimality when it is used in the abovementioned approaches' radial integrals. Furthermore, [PERSON] [32] uses the high-order Gauss-Laguerre quadrature rule instead of the moment matching method to calculate the radial integral, and proposes the high-degree cubature quadrature Kalman filter (HDCQKF), which could improve the radial integral accuracy effectively, so as to enhance the filtering accuracy. The advantage of the high-degree cubature Kalman filter is that it can increase the filtering accuracy, although it will increase the computational complexity and reduce the numerical stability for the high-dimensional system, and thus limit the application of filters. To further improve the nonlinear filtering accuracy without using the high-degree cubature rule, a new iterated orthogonal simplex cubature Kalman filter (IOSCKF) is proposed in this paper. The way to calculate the intractable integral is to combine the spherical simplex rule with the second-order Gauss-Laguerre quadrature rule. By introducing the orthogonal matrix to reduce the higher-order moments of the Taylor series expansion which appear during the approximation of the mean and covariance, the results in the novel cubature points with corresponding weights are improved. The time update is the method used to reduce the computational complexity for the linear process equation. It is equivalent to a simplification of the time update of the standard linear Kalman filter. The posterior state and covariance matrix, in the measurement update, perform Gaussian Newton iteration until the result is optimal. On target tracking problems, the IOSCKF's performance was tested, and the simulation results verify the effectiveness of the proposed algorithm. The remainder of this paper is organized as follows: a review of Gaussian approximation filter is described in Section 2, the improved spherical simplex-radial cubature rule is deduced in Section 3, the iterated orthogonal simplex cubature Kalman filter is derived in Section 4, the target tracking simulations and results are presented in Section 5, and the conclusions are drawn in Section 6.