## 3 Orthogonal Simplex Chebyshev-Laguerre Cubature Kalman Filter
### Orthogonal Spherical Simplex Chebyshev-Laguerre Quadrature Rule  
In Section 3.1, integral (13) is transformed into two kinds of integrals, which are dealt with in Sections 3.1 and 3.2, respectively. Substituting Equations (18) and (23) into Equation (13), the orthogonal spherical simplex Chebyshev-Laguerre rule for \(N(\mathbf{x};0,\mathbf{I})\) is as follows:  
\[\int_{\mathbb{R}^{n_{x}}}f(\mathbf{x})N(\mathbf{x};0,\mathbf{I})d\mathbf{x}\approx\sum_{i=1}^{ n_{x}}\sum_{j=1}^{n_{c}}w_{i}^{s}w_{j}^{c}f(\cdot), \tag{28}\]That is,  
\[\begin{array}{l}I(f)\approx\sum\limits_{i=1}^{n_{c}}\sum\limits_{j=1}^{n_{c}}w_ {ij}^{a}f(\cdot)\\ =\sum\limits_{i=1}^{n_{c}}\sum\limits_{j=1}^{n_{c}}\frac{\Lambda_{n_{c}}}{n_{c}} \cdot\frac{-n_{c}!\Gamma(a+n_{c}+1)}{\sqrt{\pi^{n_{2}}}!\mu_{j}!\left[\hat{L}_{n _{c}}^{(a)}(t_{j})\right]^{2}}f\Big{(}\sqrt{2t_{j}}\mathbf{B}\times\left[\mathbf{a} \right.-\mathbf{\alpha}\big{]}_{i}\Big{)}\\ =\sum\limits_{i=1}^{n_{c}}\sum\limits_{j=1}^{n_{c}}\frac{2\sqrt{\pi^{n_{2}}}/ \Gamma(n_{x}/2)}{2(n_{x}+1)}\cdot\frac{n_{c}!\Gamma(a+n_{c}+1)}{\sqrt{\pi^{n_{ 2}}}!\mu_{j}!\left[\hat{L}_{n_{c}}^{(a)}(t_{j})\right]^{2}}f\Big{(}\sqrt{2t_{ j}}\mathbf{\gamma}_{i}\Big{)}\end{array} \tag{29}\]  
Therefore,  
\[I(f)=\sum\limits_{i=1}^{2(n_{x}+1)}\sum\limits_{j=1}^{n_{c}}\frac{n_{c}!\Gamma (n_{x}/2+n_{c})}{2t_{j}(n_{x}+1)\Gamma(n_{x}/2)\left[\hat{L}_{n_{c}}^{(n_{x}/2 -1)}(t_{j})\right]^{2}}f\Big{(}\sqrt{2t_{j}}\mathbf{\gamma}_{i}\Big{)} \tag{30}\]  
The cubature formula is easy to extend to an arbitrary Gaussian distribution \(N(\mathbf{x};\hat{\mathbf{x}},\mathbf{P}_{x})\). Computing Equations (5)-(7) using the formula, the new nonlinear filter, called the orthogonal simplex Chebyshev-Laguerre cubature Kalman filter, can be obtained.  
The new filter uses \(2(n_{x}+1)n_{c}\) sample points to approximate the Gaussian-weighted integral. The number of sample points is linear to that of the state dimension \(n_{x}\) and the polynomials' order \(n_{c}\), respectively. It is more efficient than high-degree CKF and high-degree SCKF from the computational point of view. The accuracy of the filter depends on the order of the Chebyshev-Laguerre polynomials. In theory, the higher the order is, the more accurate the filter would be. Moreover, the high order moments of the OSCL transformation are tuned according to the designed orthogonal matrix, which can alleviate the nonlocal sampling effects efficiently.  
When \(n_{c}=1\) and \(\mathbf{\gamma}=\left[\mathbf{\alpha}\right.-\mathbf{\alpha}\big{]}\), Equation (30) can be rewritten as follows:  
\[\begin{array}{lcl}I(f)&=\sum\limits_{i=1}^{2(n_{x}+1)}\frac{1!\Gamma(n_{x}/ 2+1)}{2t_{1}(n_{x}+1)\Gamma(n_{x}/2)\left[\hat{L}_{1}^{(n_{x}/2-1)}(t_{1}) \right]^{2}}f\Big{(}\sqrt{2t_{1}}\left[\mathbf{\alpha}\right.-\mathbf{\alpha}\big{]}_ {i}\Big{)}\\ &=\sum\limits_{i=1}^{2(n_{x}+1)}\frac{\Gamma(3)}{4(n_{x}+1)\Gamma(2)\left[ \hat{L}_{1}^{(1)}(2)\right]^{2}}f\Big{(}2\left[\mathbf{\alpha}\right.-\mathbf{\alpha} \big{]}_{i}\Big{)}\\ &=\frac{1}{2(n_{x}+1)}\sum\limits_{i=1}^{2(n_{x}+1)}f\big{(}2\left[\mathbf{\alpha} -\mathbf{\alpha}\big{]}_{i}\Big{)}\end{array}, \tag{31}\]  
which means that the third-degree SCKF is a specific form of the proposed filter, where the orthogonal matrix is an identity matrix and the order of the Chebyshev-Laguerre polynomials is \(n_{c}=1\). The accuracy can be further improved by increasing \(n_{c}\). On the other hand, the advantages of the SCKF are also retained in the proposed filter, such as derivative-free and better numerical stability compared with the UKF.