## 4 State Estimation Based on Nonlinear/Non-Gaussian Systems
### Mogcp  
In this paper, we propose a modified algorithm, MOGCPF, based on the CPF algorithm. CPF leverages the CKF to obtain a Gaussian distribution, which is used as the importance density function. However, in practice, the importance density function cannot always be accurately approximated by a Gaussian distribution, which limits the accuracy of CPF. To address this issue, we leverage MOG, which can approximate any posterior probability density. By selecting appropriate mixture components, we obtain an improved approximation for the importance density function, which enhances the accuracy of the MOGCPF algorithm.  
In the framework of CPF, using the MOG approximation, the posterior density of states and the densities of process and measurement noise at moment k are assumed to be approximated as follows:  
\[\begin{split}& p_{G}(x_{k}|z_{k})=\sum_{i=1}^{G}w_{k}^{i}N(x_{k} ;u_{k}^{i},P_{k}^{i}),\\ & p_{G}(v_{k})=\sum_{i=1}^{T}r_{k}^{\text{k}}N(v_{k};u_{v,k}^{a },Q_{k}^{a}),\\ & p_{G}(q_{k+1})=\sum_{m=1}^{M}\gamma_{k+1}^{m}N(q_{k+1};u_{r,k+ 1}^{m},R_{k+1}^{m}),\end{split} \tag{34}\]where \(G\) is the number of mixed components; \(\omega_{i}\) is mixed weights. The MOG-approximated predicted density \(p_{G}(x_{k}|z_{k})\) of states at time \(k+1\) is  
\[\begin{split} p_{G}(x_{k+1}|z_{k})&=\int p(x_{k+1}|x_ {k})p(x_{k}|z_{k})\,dx_{k},\\ &\approx\sum\limits_{r=1}^{K}\sum\limits_{i=1}^{G}w_{k}^{i}r_{k} ^{i}\times\int N(x_{k+1};f(x_{k})+u_{v,k}^{b},Q_{k}^{b})\times N(x_{k+1};f(x_{ k})+u_{k}^{i},p_{k}^{i})\,dx_{k},\end{split} \tag{35}\]  
where \(G^{\prime}=GK,i=1,2,\ldots,G,r=1,2,\ldots,K,i^{\prime}=1,\ldots,G^{\prime}\), with mixed weights \(w_{k+1}^{i^{\prime}}\)  
\[w_{k+1}^{i^{\prime}}=\frac{w_{k}^{i}r_{k}^{i}}{\sum\limits_{i=1}^{C}\sum \limits_{r=1}^{K}w_{k}^{i}r_{k}^{i}}, \tag{36}\]  
During the CKF time update,  
\[\begin{split} x_{k}^{i^{\prime}}&=u_{k}^{i}\,\\ S_{k}^{i^{\prime}}&=\sqrt{P_{k}^{i}}\,\end{split} \tag{37}\]  
Calculate the cubature point (\(l\) is the number of dimensions):  
\[\begin{split} X_{l,k}^{i^{\prime}}&=S_{k}^{i^{ \prime}}\sqrt{n[l]_{l}}+x_{l,k}^{i^{\prime}},\\ X_{l,k+1}^{i^{\prime}}&=f(X_{l,k}^{i^{\prime}})+u_ {v,k}^{i},\\ \hat{x}_{k+1|k}^{i^{\prime}}&=\frac{1}{2n}\sum\limits _{i=1}^{2n}X_{l,k+1}^{i^{\prime}},\\ P_{k+1|k}^{i}&=\frac{1}{2n}\sum\limits_{i=1}^{2n}X_ {l,k+1}^{i^{\prime}}(X_{l,k+1}^{i^{\prime}})^{\mathrm{T}}-\hat{x}_{k+1|k}^{i^ {\prime}}(\hat{x}_{k+1|k}^{i^{\prime}})^{\mathrm{T}}+Q_{k}^{b},\end{split} \tag{38}\]  
and let \(u_{k+1}^{i^{\prime}}=x_{k|k+1^{\prime}}^{i^{\prime}},P_{k+1}^{i^{\prime}}=P_{ k+1|k}^{i^{\prime}}\); thus, the MOG of the prior density function is  
\[p_{G}(x_{k+1}|z_{k})=\sum\limits_{i=1}^{G}w_{k+1}^{i}N(x_{k};u_{k+1}^{i^{ \prime}},P_{k+1}^{i^{\prime}}), \tag{39}\]  
Similarly, in the CKF measurement update phase, the posterior density function  
\[p_{G}(x_{k+1}|z_{k})=\sum\limits_{i^{\prime\prime}=1}^{G^{\prime\prime}}w_{k+1 }^{i^{\prime\prime\prime}}N(x_{k};u_{k+1}^{i^{\prime\prime}},P_{k+1}^{i^{ \prime\prime}}), \tag{40}\]  
It can be seen that in the time update, the Gaussian component increases from \(G\) to \(G^{\prime}\), and the Gaussian component increases from \(G^{\prime}\) to \(G^{\prime\prime}\) in the measurement update; as the estimation process proceeds, it leads to an increasing amount of computation, so it is necessary to reduce \(G^{\prime}\) to \(G\) during resampling.