# A Novel SINS/SRS/CNS Multi-Information Fusion Global Autonomous Navigation Method
## 4 Parameter Conversion of the SINS/SRS/CNS Multi-Information Fusion Autonomous Navigation System
### Filter Parameter Conversion
\begin{bmatrix}-\sin\sigma \cdot v_{E}-\cos\sigma\cdot v_{N}\\ \cos\sigma\cdot v_{E}-\sin\sigma\cdot v_{N}\\ 0\end{bmatrix} \delta\sigma\] \[=\mathbf{C}_{g}^{G}\delta\mathbf{v}^{g}+ \begin{bmatrix}-\sin\sigma\cdot v _{E}-\cos\sigma\cdot v_{N}\\ \cos\sigma\cdot v_{E}-\sin\sigma\cdot v_{N}\\ 0\end{bmatrix} \Big{[} \begin{array}{ccc}\frac{\sin\sigma\cos\sigma\cos L}{ \sin L}&\frac{1-\cos^{2}\sigma\cos^{2}L}{\sin L}&0\end{array} \Big{]}\Bigg{[} \begin{array}{ccc}\delta L\\ \delta\lambda\\ \delta h\end{bmatrix}\]The position in the middle-low latitudes is expressed by latitude, longitude, and altitude, with the error of \(\delta p^{\text{g}}= \begin{bmatrix}\delta L&\delta\lambda&\delta h\end{bmatrix} ^ {\text{T}}\). Meanwhile, the position in the high latitudes is expressed by the coordinates in the \(e\) frame, with the error of \(\delta p^{e}= \begin{bmatrix}\delta x&\delta y&\delta z\end{bmatrix} ^{\text{T}}\). Depending on the relation between latitude, longitude, altitude and the coordinates in the \(e\) frame described in Equation (23), it can be obtained that:  
\[ \begin{cases}\delta x=\delta[(R_{N}+h)\cos L\cos\lambda]=(R_{N}+h)(-\sin L\cos \lambda\delta L-\cos L\sin\lambda\delta\lambda)+\cos L\cos\lambda\delta h\\ \delta y=\delta[(R_{N}+h)\cos L\sin\lambda]=(R_{N}+h)(-\sin L\sin\lambda\delta L +\cos L\cos\lambda\delta\lambda)+\cos L\sin\lambda\delta h\\ \delta z=\delta\big{[}\begin{bmatrix}R_{N}\big{(}1-f^{2}\big{)}+h\end{bmatrix} \sin L\big{]}=\begin{bmatrix}R_{N}\big{(}1-f^{2}\big{)}+h\end{bmatrix}\cos L \delta L+\sin L\delta h\end{cases}  \tag{34}\]  
Equation (34) can be written as:  
\[\delta p^{e}= \begin{bmatrix}-(R_{N}+h)\sin L\cos\lambda&-(R_{N}+h)\cos L\sin \lambda&\cos L\cos\lambda\\ -(R_{N}+h)\sin L\sin\lambda&(R_{N}+h)\cos L\cos\lambda&\cos L\sin\lambda\\ \big{[}R_{N}\big{(}1-f^{2}\big{)}+h\big{]}\cos L&0&\sin L\end{bmatrix} \delta p =T_{3}\delta p^{g} \tag{35}\]  
When the UAV enters the polar regions, the navigation coordinate frame switches from the \(g\) frame to the \(G\) frame. Combining Equations (32), (33), and (35), the polar state vector can be obtained as:  
\[\mathbf{X}^{G}(t)= \begin{bmatrix}\mathbf{C}_{g}^{G}&0&-\mathbf{T}_{1}&0&0\\ 0&\mathbf{C}_{g}^{G}&\mathbf{T}_{2}&0&0\\ 0&0&\mathbf{T}_{3}&0&0\\ 0&0&0&\mathbf{I}_{3\times 3}&0\\ 0&0&0&0&\mathbf{I}_{3\times 3}\end{bmatrix} \mathbf{X}^{\text{g}}(t)=\mathbf{\Phi}\mathbf{X}^{ \text{g}}(t) \tag{36}\]  
According to the definition of the variance matrix and Equation (36):  
\[ \begin{split}\mathbf{P}^{G}(t)&=\text{E}\Bigg{\{}\bigg{(}\mathbf{X}^{ G}(t)-\overset{\mathbf{\Lambda}}{\mathbf{X}^{\text{G}}}(t)\bigg{)}\bigg{(}\mathbf{X}^{ \text{G}}(t)-\overset{\mathbf{\Lambda}}{\mathbf{X}^{\text{G}}}(t)\bigg{)}^{\text{T}} \Bigg{\}}&=\text{E}\Bigg{\{}\bigg{(}\mathbf{\Phi}\mathbf{X}^{\text{g}}(t)-\mathbf{\Phi} \mathbf{X}^{\text{g}}(t)\bigg{)}\bigg{(}\mathbf{\Phi}\mathbf{X}^{\text{g}}(t)-\overset{ \mathbf{\Lambda}}{\mathbf{X}^{\text{g}}}(t)\bigg{)}^{\text{T}}\Bigg{\}}\\ &=\text{E}\Bigg{\{}\mathbf{\Phi}\bigg{(}\mathbf{X}^{\text{g}}(t)-\overset{ \mathbf{\Lambda}}{\mathbf{X}^{\text{g}}}(t)\bigg{)}\bigg{(}\mathbf{X}^{\text{g}}(t)- \overset{\mathbf{\Lambda}}{\mathbf{X}^{\text{g}}}(t)\bigg{)}^{\text{T}}\mathbf{\Phi}^{ \text{T}}\Bigg{\}}&=\mathbf{\Phi}\text{E}\Bigg{\{}\bigg{(}\mathbf{X}^{\text{g}}(t)- \overset{\mathbf{\Lambda}}{\mathbf{X}^{\text{g}}}(t)\bigg{)}\bigg{(}\mathbf{X}^{\text{g}}(t )-\overset{\mathbf{\Lambda}}{\mathbf{X}^{\text{g}}}(t)\bigg{)}^{\text{T}}\Bigg{\}}\mathbf{ \Phi}^{\text{T}}\\ &=\mathbf{\Phi}\mathbf{P}^{\text{g}}(t)\mathbf{\Phi}^{\text{T}}\end{split}  \tag{37}\]  
where \(\mathbf{P}^{\text{g}}(t)\) is the covariance matrix in the \(g\) frame and \(\mathbf{P}^{\text{g}}(t)\) is the covariance matrix in the \(G\) frame. Similarly, when the UAV leaves the polar regions, the state vector and covariance matrix in the \(g\) frame can be obtained as:  
\[