# Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking
## 3 The Improved Spherical Simplex-Radial Cubature Rule
### Radial Rule  
Then, the calculation of the radial integral \(R\) is considered. Two methods, including the moment matching method and Gauss-Laguerre quadrature rule, may be used. To construct the \((2m+1)\)th-degree radial rule, the two methods lead to identical points and weights when \(m\) is odd and \((m+1)/2\) points are used. However, they may be different when \(m\) is even and \((m/2+1)\) points are used [30]. In the CKF and SSRCKF, the radial integral is calculated using the former one, and results in the following third-degree radial rule with only one point (that is, \(m=1\), \(N_{r}=1\)).  
\[R=\frac{1}{2}\Gamma\Big{(}\frac{n}{2}\Big{)}S\bigg{(}\sqrt{\frac{n}{2}}\bigg{)} \tag{16}\]  
In this paper, the second-order Gauss-Laguerre quadrature rule is adopted to approximate the radial integral. This method improves the accuracy of radial integration.  
For radial integral \(R=\int_{0}^{\infty}S(r)r^{m-1}e^{-r^{2}}dr\), let \(r^{2}=t\) to obtain \(r=\sqrt{t}\), which takes the place of the radial integral, and for \(R=\frac{1}{2}\int_{0}^{\infty}S\Big{(}\sqrt{t}\Big{)}t^{\frac{n-2}{2}}e^{-t}dt\), let \(q(t)=S\Big{(}\sqrt{t}\Big{)}\), and \(\beta=\frac{n}{2}-1\). Then, \(R\) is transformed into \(R=\frac{1}{2}\int_{0}^{\infty}q(t)t^{\beta}e^{-t}dt\), and this form can be handily approximated using the Gauss-Laguerre quadrature [32] rule as follows:  
\[R=\frac{1}{2}\int_{0}^{\infty}q(t)t^{\beta}e^{-t}dt\approx\frac{1}{2}\sum_{j= 1}^{m}A_{j}q\big{(}t_{j}\big{)} \tag{17}\]  
where \(t_{j}\) denotes the quadrature points, with \(A_{j}\) being the corresponding weights.  
The approximation accuracy depends on the order \(m\). The \(m\)-order Chebyshev-Laguerre polynomial's solutions are the quadrature points, as shown in the following:  
\[L_{m}^{\beta}(t)=(-1)^{m}t^{-\beta}e^{t}\frac{d^{m}}{dt^{m}}\Big{(}t^{\beta+m }e^{-t}\Big{)}=0 \tag{18}\]  
and the weights are calculated using the following equation.  
\[A_{j}=\frac{m!\Gamma(\beta+m+1)}{t_{j}\bigg{[}\hat{L}_{m}^{\beta}\big{(}t_{j} \big{)}\bigg{]}^{2}} \tag{19}\]  
On account of \(m\) being set at 2, the solutions are obtained are \(t_{1}=n/2+1+\sqrt{n/2+1}\), \(t_{2}=n/2+1-\sqrt{n/2+1}\), \(A_{1}=n\Gamma(n/2)/\big{(}2n+4+2\sqrt{2n+4}\big{)}\), and \(A_{2}=n\Gamma(n/2)/\big{(}2n+4-2\sqrt{2n+4}\big{)}\), which are substituted into Equation (17) to deduce the following radial rule.  
\[R=\frac{n\Gamma\big{(}\frac{n}{2}\big{)}}{4n+8+4\sqrt{2n+4}}S\Bigg{(}\sqrt{ \frac{n}{2}+1+\sqrt{\frac{n}{2}+1}}\Bigg{)}+\frac{n\Gamma\big{(}\frac{n}{2} \big{)}}{4n+8-4\sqrt{2n+4}}S\Bigg{(}\sqrt{\frac{n}{2}+1-\sqrt{\frac{n}{2}+1}} \Bigg{)} \tag{20}\]  
From Equation (20) we see that the number of points is \(N_{r}=2\), which leads to the radial rule to achieve the fifth-degree accuracy. Compared with third-degree radial rule used in the CKF and SSRCKF Equation (16), the fifth-degree accuracy is more accurate. It is noted that the two points can be solved using the moment matching method as well, with one of the points set to zero. However, it is pointed out that the two points generated by the Gauss-Laguerre quadrature rule have higher radial accuracy than those generated using the moment matching method [30].