# Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking
###### Abstract  
In order to increase a nonlinear system's state estimate precision, an iterated orthogonal simplex cubature Kalman filter (IOSCKF) is presented in this study for target tracking. The Gaussian-weighted integral is decomposed into a spherical integral and a radial integral, which are approximated using the spherical simplex-radial rule and second-order Gauss-Laguerre quadrature rule, respectively, and result in the novel simplex cubature rule. To decrease the high-order error terms, cubature points with appropriate weights are taken from the cubature rule and processed using the provided orthogonal matrix. The structure supporting the nonlinear Kalman filter incorporates the altered points and weights and the calculation steps; from this, the updated time and measurement can be inferred. The Gauss-Newton iteration is employed repeatedly to adjust the measurement update until the termination condition is met and the IOSCKF is attained. The proposed algorithms are applied in target tracking, including CV target tracking and spacecraft orbit tracking, and the simulation results reveal that the IOSCKF can achieve higher accuracy compared to the CKF, SCKF, and OSCKF. In spacecraft orbit tracking simulation, compared with the SCKF, the position tracking accuracy and velocity tracking accuracy of the OSCKF are increased by 2.21% and 1.94%, respectively, which indicates that the orthogonal transformation can improve the tracking accuracy. Furthermore, compared with the OSCKF, the position tracking accuracy and velocity tracking accuracy of the IOSCKF are increased by 2.71% and 2.97%, respectively, which indicates that the tracking accuracy can be effectively improved by introducing iterative calculation into the measurement equation, thus verifying the effectiveness of the method presented in this paper. c 2023 2023 2  
mainly includes a particle filter [9] and makes no assumptions on the pdf, which leads to a large amount of computation and is not suitable for applications in engineering. For the latter method, the posterior pdf is assumed to have a Gaussian distribution, and the typical representation, the nonlinear Kalman filter, is primarily studied in practical applications [10]. In a nonlinear Kalman filter, the recursive estimation of the state vector is divided into two processes consisting of a time update and a measurement update, and a unified filtering calculation framework is given. The key problem in the filtering framework is in the approximation of the Gaussian-weighted integral of multi-dimensional nonlinear function [8], and two approximate approaches, including the approximation of the nonlinear function and that of the Gaussian pdf, are mainly taken. The extended Kalman filter (EKF) [11] belongs to the former, and the basic idea of EKF is to use the first-order Taylor expansion to linearize the nonlinear system equations, and then, the conventional Kalman filter is used to estimate the state [12]. It is required to calculate the Jacobian matrix of the system, which demands that the model be smooth and differentiable. The EKF filtering accuracy is reduced or even divergent for a highly nonlinear system, and an improved algorithm, named iterated EKF (IEKF) [13], is proposed. The latest measurements are fully utilized through iterative calculation in the measurement update of IEKF to improve the estimation accuracy; however, the inherent disadvantage of EKF is still present in IEKF, that is, the differential operation of the Jacobian matrix. In general, the approximation of pdf is easier than that of the nonlinear function, so the approximation is performed using the deterministic sampling points in the latter method, and results in the unscented Kalman filter (UKF) [14], Gauss-Hermite-Kalman filter (GHKF) [15], sparse-grid Kalman filter (SGKF) [16], and cubature Kalman filter (CKF) [17; 18]. These filters are all derivative-free filters, that is, there is no need to calculate the Jacobian matrix to linearize the nonlinear equation [19].