# Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking
## 3 The Improved Spherical Simplex-Radial Cubature Rule  
In this section, an improved spherical simplex-radial cubature rule is proven to be more accurate than the traditional conventional spherical-radial rule. Before introducing the improved cubature rule, the definition of the \(d\)th-degree rule is given first for the integral  
**Definition 1** ([30]).: \[\int_{\mathbb{R}^{n}}\mathbf{g}(\mathbf{x})\omega_{g}(\mathbf{x})d\mathbf{x}\approx\sum \limits_{i}\omega_{i}\mathbf{g}(\gamma_{i})\] (8)  
_where \(\mathbf{x}=\big{(}x_{1}\quad x_{2}\quad\cdots\quad x_{n}\big{)}^{\text{T}}\in \mathbb{R}^{n}\) and \(\omega_{g}(\mathbf{x})\) is a given weighing function. Equation (8) is a \(d\)th-degree rule if it is accurate for \(\mathbf{g}(\mathbf{x})\), whose components are linear combinations of monomials \(x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{n}^{\alpha_{n}}\) with a total degree up to \(d\). \(\alpha_{1}\), \(\alpha_{2}\), \(\cdots\), \(\alpha_{n}\) are nonnegative integers and \(\alpha_{1}+\alpha_{2}+\cdots\alpha_{n}\leq d\), and there is at least one monomial of degree \(d+1\) for which Equation (8) is not exact._  
Specifically, an intractable integral of the form \(I_{g}=\int_{\mathbb{R}^{n}}\mathbf{g}(\mathbf{x})e^{-x^{\text{T}}x}d\mathbf{x}\) is considered first in the cubature rule. Let \(\mathbf{x}=\mathbf{r}\mathbf{y}\) with \(\mathbf{y}^{\text{T}}\mathbf{y}=1\), where \(\mathbf{y}\) means the direction vector and the radius \(r\geq 0\). As a consequence, \(\mathbf{x}^{\text{T}}\mathbf{x}=r^{2}\); then, the integral \(I_{g}\) can be written anew as in a spherical-radial coordinate system as follows [12]:  
\[I_{g}=\int_{0}^{\infty}\int_{\mathbb{U}_{n}}\mathbf{g}(\mathbf{r}\mathbf{y})\mathbf{r}^{n-1}e^ {-r^{2}}d\sigma(\mathbf{y})d\mathbf{r} \tag{9}\]where \(U_{n}\) is the sphere surface defined by \(U_{n}=\left\{\mathbf{y}\in\mathbf{\mathrm{R}}^{n}:\mathbf{y}^{\mathrm{T}}\mathbf{y}=1\right\}\), and \(\sigma(\cdot)\) is the area element on \(U_{n}\). The integral \(I_{g}\) is decomposed into the spherical integral \(S(r)=\int_{U_{n}}\mathbf{g}(\mathbf{r}\mathbf{y})d\sigma(\mathbf{y})\) and radial integral \(R=\int_{0}^{\infty}S(r)r^{n-1}e^{-r^{2}}dr\), respectively.  
Because of the difficulty in achieving the analytical solutions, the approximate solution method of \(I_{g}\) requires numerical integration, which should be approximated using the numerical integration method. From the third-degree spherical-radial cubature rule proposed in [17], we obtain  
\[S(r) \approx\sum_{i=1}^{N_{\mathrm{s}}}\omega_{s,i}\mathbf{g}(\mathbf{r}\mathbf{y}_ {i}) \tag{10}\] \[R \approx\sum_{j=1}^{N_{r}}\omega_{r,j}S(r_{j}) \tag{11}\]  
and \(I_{g}\) is approximated using the following formula:  
\[I_{g}=\int_{0}^{\infty}r^{n-1}e^{-r^{2}}dr\int_{U_{n}}\mathbf{g}(\mathbf{r}\mathbf{y})d \sigma(\mathbf{y})\approx\sum_{j=1}^{N_{r}}\sum_{i=1}^{N_{\mathrm{s}}}\omega_{r,j }\omega_{s,i}\mathbf{g}(r_{j}\mathbf{y}_{i}) \tag{12}\]  
where \(\mathbf{y}_{i}\) and \(\omega_{s,i}\) denote the points and weights to calculate the spherical integral, and the number of points is \(N_{\mathrm{s}}\). \(r_{j}\) and \(\omega_{r,j}\) represent the points and weights to calculate the radial integral, and the number of points is \(N_{r}\). The total number of points is \(N_{r}N_{\mathrm{s}}\) if \(r_{j}\
eq 0\), or \((N_{r}-1)N_{\mathrm{s}}+1\) if one of \(r_{j}\) is zero.