# Student's t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter for GPS Navigation
## 2 Review on Extended Kalman Filter with Maximum Correntropy Criterion
### MCCEKF Algorithm 2: MCCEKF-2
 \tag{18}\]  
where  
\[\mathbf{\Lambda}_{\textbf{x}}=\text{diag}(G_{\sigma}(e_{k,1}),\ldots,G_{\sigma}(e_ {k,n}))=\text{diag}(\exp(-\frac{e_{k,1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac {e_{k,n}^{2}}{2\sigma^{2}})) \tag{19}\]\[\Lambda_{z}=\text{diag}(G_{\sigma}(e_{k,n+1}),\ldots,G_{\sigma}(e_{k,n+m}))=\text{ diag}(\exp(-\frac{e_{k,n+1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac{e_{k,n+m}^{2}}{2 \sigma^{2}})) \tag{20}\]  
After derivation, we have  
\[\mathbf{B}_{\mathbf{P}}^{-T}\Lambda_{\mathbf{x}}\mathbf{B}_{\mathbf{P}}^{-1}( \mathbf{x}_{k}-\overset{\Lambda}{\mathbf{x}}_{k|k-1})-\mathbf{H}_{k}^{T} \mathbf{B}_{\mathbf{R}}^{-T}\Lambda_{\mathbf{z}}\mathbf{B}_{\mathbf{R}}^{-1}( \mathbf{z}_{k}-h(\mathbf{x}_{k}))=0 \tag{21}\]  
Express \(\mathbf{\tilde{P}}_{k|k-1}\) and \(\mathbf{\tilde{R}}_{k}\) as  
\[\mathbf{\tilde{P}}_{k|k-1}=\mathbf{B}_{\mathbf{P}}\Lambda_{x}^{-1}\mathbf{B} _{\mathbf{P}}^{T};\ \mathbf{\tilde{R}}_{k}=\mathbf{B}_{\mathbf{R}}\Lambda_{z}^{-1}\mathbf{B}_{ \mathbf{R}}^{T} \tag{22}\]  
and thus,  
\[\mathbf{\tilde{P}}_{k|k-1}^{-1}=\mathbf{B}_{\mathbf{P}}^{-T}\Lambda_{\mathbf{ x}}\mathbf{B}_{\mathbf{P}}^{-1};\ \mathbf{\tilde{R}}_{k}^{-1}=\mathbf{B}_{\mathbf{R}}^{-T}\Lambda_{\mathbf{z}} \mathbf{B}_{\mathbf{R}}^{-1} \tag{23}\]  
and we have  
\[\mathbf{\tilde{P}}_{k|k-1}^{-1}(\mathbf{x}_{k}-\overset{\Lambda}{\mathbf{x}} _{k|k-1})-\mathbf{H}_{k}^{T}\mathbf{\tilde{R}}_{k}^{-1}(\mathbf{z}_{k}-h( \mathbf{x}_{k}))=0 \tag{24}\]  
Since the measurement model is given by  
\[\mathbf{z}_{k}=h(\mathbf{x}_{k})+\mathbf{v}_{k}=h(\overset{\Lambda}{\mathbf{ x}}_{k|k-1})+\mathbf{H}_{k}(\mathbf{x}_{k}-\overset{\Lambda}{\mathbf{x}}_{k|k-1})+ \mathbf{v}_{k}\]  
we have  
\[\mathbf{\tilde{P}}_{k|k-1}^{-1}(\mathbf{x}_{k}-\overset{\Lambda}{\mathbf{x}} _{k|k-1})-\mathbf{H}_{k}^{T}\mathbf{\tilde{R}}_{k}^{-1}(\mathbf{z}_{k}-h( \overset{\Lambda}{\mathbf{x}}_{k|k-1})-\mathbf{H}_{k}(\mathbf{x}_{k}- \overset{\Lambda}{\mathbf{x}}_{k|k-1}))=0\]  
and thus,  
\[(\mathbf{\tilde{P}}_{k|k-1}^{-1}+\mathbf{H}_{k}^{T}\mathbf{\tilde{R}}_{k}^{-1 }\mathbf{H}_{k})\mathbf{x}_{k}=(\mathbf{\tilde{P}}_{k|k-1}^{-1}+\mathbf{H}_{k }^{T}\mathbf{\tilde{R}}_{k}^{-1}\mathbf{H}_{k})\overset{\Lambda}{\mathbf{x}} _{k|k-1}+\mathbf{H}_{k}^{T}\mathbf{\tilde{R}}_{k}^{-1}(\mathbf{z}_{k}-h( \overset{\Lambda}{\mathbf{x}}_{k|k-1})) \tag{25}\]  
Finally, we obtain  
\[\mathbf{x}_{k}=\overset{\Lambda}{\mathbf{x}}_{k|k-1}+(\mathbf{\tilde{P}}_{k |k-1}^{-1}+\mathbf{H}_{k}^{T}\mathbf{\tilde{R}}_{k}^{-1}\mathbf{H}_{k})^{-1} \mathbf{H}_{k}^{T}\mathbf{\tilde{R}}_{k}^{-1}(\mathbf{z}_{k}-h(\overset{ \Lambda}{\mathbf{x}}_{k|k-1})) \tag{26}\]  
Let the Kalman gain  
\[\Lambda_{z}=\text{diag}(G_{\sigma}(e_{k,n+1}),\ldots,G_{\sigma}(e_{k,n+m}))=\text{ diag}(\exp(-\frac{e_{k,n+1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac{e_{k,n+m}^{2}}{2 \sigma^{2}})) \tag{20}\]0  
we obtain  
\[\Lambda_{z}=\text{diag}(G_{\sigma}(e_{k,n+1}),\ldots,G_{\sigma}(e_{k,n+m}))=\text{ diag}(\exp(-\frac{e_{k,n+1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac{e_{k,n+m}^{2}}{2 \sigma^{2}})) \tag{20}\]1  
According to the fixed-point iteration algorithm, the filter estimates state using standard propagation equations and updates posterior estimates, which is given by  
\[\Lambda_{z}=\text{diag}(G_{\sigma}(e_{k,n+1}),\ldots,G_{\sigma}(e_{k,n+m}))=\text{ diag}(\exp(-\frac{e_{k,n+1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac{e_{k,n+m}^{2}}{2 \sigma^{2}})) \tag{20}\]2  
where  
\[\Lambda_{z}=\text{diag}(G_{\sigma}(e_{k,n+1}),\ldots,G_{\sigma}(e_{k,n+m}))=\text{ diag}(\exp(-\frac{e_{k,n+1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac{e_{k,n+m}^{2}}{2 \sigma^{2}})) \tag{20}\]3  
Finally, the state covariance matrix in its updated form can be written as  
\[\Lambda_{z}=\text{diag}(G_{\sigma}(e_{k,n+1}),\ldots,G_{\sigma}(e_{k,n+m}))=\text{ diag}(\exp(-\frac{e_{k,n+1}^{2}}{2\sigma^{2}}),\ldots,\exp(-\frac{e_{k,n+m}^{2}}{2 \sigma^{2}})) \tag{20}\]4