# Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking
## 2 Brief Review of Gaussian Approximation Filter  
The following discrete nonlinear system model is defined:  
\[\left\{ \begin{array}{l}\mathbf{x}_{k}=\mathbf{f}(\mathbf{x}_{k-1})+\mathbf{w}_{k-1}\\ \mathbf{z}_{k}=\mathbf{h}(\mathbf{x}_{k})+\mathbf{v}_{k}\end{array} \right. \tag{1}\]  
where \(k\) is the time step, \(\mathbf{x}_{k}\in\mathbb{R}^{n}\) denotes the state vector at time step \(k\), \(\mathbf{z}_{k}\in\mathbb{R}^{p}\) represents the measurement vector at time step \(k\), and \(\mathbf{f}(\cdot)\) and \(\mathbf{h}(\cdot)\) are the known nonlinear state function and measurement function, respectively. The process noise \(\mathbf{w}_{k}\) and measurement noise \(\mathbf{v}_{k}\) are uncorrelated zero-mean Gaussian white noise, with the covariance matrices being \(\mathbf{Q}_{k}\) and \(\mathbf{R}_{k}\), respectively. The key problem in the nonlinear Kalman filter is calculating the multi-dimensional vector integral of the form \"nonlinear function \(\times\) Gaussian pdf\" [8], that is, calculating the integral \(I_{N}=\int_{\mathbb{R}^{n}}\mathbf{g}(\mathbf{x})N(\mathbf{x};\overline{\mathbf{x}},\mathbf{P}_{x} )d\mathbf{x}\) in the Cartesian coordinate system, where \(\mathbf{g}(\mathbf{x})\) denotes the arbitrary nonlinear function, and \(N(\mathbf{x};\overline{\mathbf{x}},\mathbf{P}_{x})\) represents the Gaussian pdf, with the mean and covariance being \(\overline{\mathbf{x}}\) and \(\mathbf{P}_{x}\), respectively. This Gaussian-weighted integral \(I_{N}\) can be approximated using various quadrature rules, and the integral with respect to \(N(\mathbf{x};\mathbf{0},\mathbf{I})\) can be approximated by the following quadrature rule:  
\[I_{N}=\int_{\mathbb{R}^{n}}\mathbf{g}(\mathbf{x})N(\mathbf{x};\mathbf{0},\mathbf{I})d\mathbf{x}\approx \sum_{i=1}^{N_{p}}\omega_{i}\mathbf{g}(\sigma_{i}) \tag{2}\]  
where \(\mathbf{I}\) is the identity matrix, \(N_{p}\) denotes the total number of points, and \(\sigma_{i}\) and \(\omega_{i}\) represent the quadrature points and weights, respectively. The integral with respect to the more general Gaussian distribution \(N(\mathbf{x};\overline{\mathbf{x}},\mathbf{P}_{x})\) can be obtained using the equation below:  
\[ \begin{array}{l}I_{N}\quad=\int_{\mathbb{R}^{n}}\mathbf{g}(\mathbf{x})N(\mathbf{x}; \overline{\mathbf{x}},\mathbf{P}_{x})d\mathbf{x}=\int_{\mathbb{R}^{n}}\mathbf{g}\big{(} \overline{\mathbf{x}}+\sqrt{\mathbf{P}_{x}}\mathbf{x}\big{)}N(\mathbf{x};\mathbf{0},\mathbf{I})d\mathbf{x} \\ \quad\approx\sum\limits_{i=1}^{N_{p}}\omega_{i}\mathbf{g}\big{(}\overline{\mathbf{x}} +\sqrt{\mathbf{P}_{x}}\sigma_{i}\big{)}\end{array}  \tag{3}\]  
where \(\sqrt{\mathbf{P}_{x}}\) denotes the square root of \(\mathbf{P}_{x}\), which can be solved by the Cholesky decomposition, and \(\overline{\mathbf{x}}^{(i)}=\overline{\mathbf{x}}+\sqrt{\mathbf{P}_{x}}\sigma_{i}\) represents the integral point. The Gaussian approximation filter, which is deduced using the numerical integration method in Equation (3), is summarized below [30]. **Time Update:**  
The predicted state \(\hat{\mathbf{x}}_{k}^{-}\) and error covariance \(\mathbf{P}_{k}^{-}\) are calculated, respectively, as follows:  
\[\hat{\mathbf{x}}_{k}^{-}=\sum\limits_{i=1}^{N_{p}}\omega_{i}\mathbf{f}\Big{(}\hat{\mathbf{x} }_{k-1}^{(i)}\Big{)} \tag{4}\]  
\[\mathbf{P}_{k}^{-}=\sum\limits_{i=1}^{N_{p}}\omega_{i}\Big{(}\mathbf{f}\Big{(}\hat\mathbf{{x}}_{k-1}^{(i)}\Big{)}-\hat{\mathbf{x}}_{k}^{-}\Big{)}\Big{(}\mathbf{f}\Big{(}\hat\mathbf{{x}}_{k-1}^{(i)}\Big{)}-\hat{\mathbf{x}}_{k}^{-}\Big{)}^{\text{T}}+\mathbf{Q}_{k-1} \tag{5}\]  
where \(\hat{\mathbf{x}}_{k-1}^{(i)}=\hat{\mathbf{x}}_{k-1}^{+}+\sqrt{\mathbf{P}_{k-1}^{+}}\mathbf{ \sigma}_{i}\) stands for the integral points generated using the numerical rule.