# 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  
The navigation coordinate frame switches while the UAV enters/leaves the polar regions, and the filter structure and parameters also change. The system equations in the polar/nonpolar regions were established (Section 3), and the coefficient matrices in different coordinate frames were also determined. The system noise and measurement noise are only related to the sensor accuracy, not to the navigation coordinate frame. Beyond that, to switch between the different system equations, it is necessary to convert the filter state vector and the covariance matrix. By comparing Equations (13) and (19), it can be known that the inertial sensor errors, \(\mathbf{\varepsilon}^{\text{b}}\) and \(\
abla^{\text{b}}\), are the same in the \(g\) frame and the \(G\) frame, so no conversion is required. The conversion relationship between \(\mathbf{\Phi}^{g}\) and \(\mathbf{\Phi}^{G}\) can be derived as follows. Due to the calculation error, the navigation coordinate frame described by SINS is the \(c\) frame, containing the platform error angle. The platform error angle, \(\mathbf{\Phi}^{G}\), relative to the \(G\) frame is usually a slight angle, so the first-order approximation is feasible. It can be deduced that:  
\[\mathbf{C}_{\text{G}}^{c}=I-(\mathbf{\Phi}^{G}\times) \tag{28}\]  
Then,  
\[\delta\mathbf{C}_{b}^{G}=\mathbf{C}_{b}^{c}-\mathbf{C}_{b}^{G}=\mathbf{C}_{\text{G}}^{c}\mathbf{C} _{b}^{G}-\mathbf{C}_{b}^{G}=-(\mathbf{\Phi}^{G}\times)\mathbf{C}_{b}^{G} \tag{29}\]  
Derive both sides of \(\mathbf{C}_{b}^{G}=\mathbf{C}_{S}^{G}\mathbf{C}_{b}^{g}\)  
\[\delta\mathbf{C}_{b}^{G}=\delta\mathbf{C}_{S}^{G}\mathbf{C}_{b}^{g}+\mathbf{C}_{S}^{G}\delta \mathbf{C}_{b}^{g}=-\Big{(}\mathbf{\Phi}_{S\text{G}}^{G}\times\Big{)}\mathbf{C}_{S}^{G} \mathbf{C}_{b}^{g}-\mathbf{C}_{S}^{G}(\mathbf{\Phi}^{g}\times)\mathbf{C}_{b}^{g}=-\Big{(}\mathbf{ \Phi}_{S\text{G}}^{G}\times\Big{)}\mathbf{C}_{b}^{G}-\mathbf{C}_{S}^{G}(\mathbf{\Phi}^{g} \times)\mathbf{C}_{\text{G}}^{G}\mathbf{C}_{b}^{G} \tag{30}\]  
where \(\mathbf{\[PERSON]}_{S\text{G}}^{G}= \begin{bmatrix}0&0&-\delta\sigma\end{bmatrix} ^{ \text{T}}\); \(\delta\sigma=\frac{\sin\sigma\cos\sigma\cos L}{\sin L}\delta L+\frac{1-\cos^{ 2}\sigma\cos^{2}L}{\sin L}\delta\lambda\). By comparing Equations (29) and (30), we obtain:  
\[(\mathbf{\[PERSON]}^{G}\times)=\mathbf{C}_{S}^{G}(\mathbf{\[PERSON]}^{g}\times)\mathbf{C}_{\text{G}}^{ \mathcal{G}}+\Big{(}\mathbf{\[PERSON]}_{S\text{G}}^{G}\times\Big{)} \tag{31}\]  
This can be reorganized as:  
\[\mathbf{\[PERSON]}^{\text{G}}=\mathbf{C}_{g}^{G}\mathbf{\[PERSON]}^{g}+\mathbf{\[PERSON]}_{S\text {G}}^{G}=\mathbf{C}_{g}^{G}\mathbf{\[PERSON]}^{g}+ \begin{bmatrix}0\\ 0\\ -\delta\sigma\end{bmatrix} =\mathbf{C}_{g}^{G}\mathbf{\[PERSON]}^{g}- \begin{bmatrix}0\\ 0\\ \frac{\sin\sigma\cos\sigma\cos L}{\sin L}\delta L+\frac{1-\cos^{2}\sigma\cos^{ 2}L}{\sin L}\delta\lambda\end{bmatrix}  \tag{32}\] \[=\mathbf{C}_{g}^{G}\mathbf{\Phi}^{g}- \begin{bmatrix}0&0&0\\ 0&0\\ \frac{\sin\sigma\cos\sigma\cos L}{\sin L}&\frac{1-\cos^{2}\sigma\cos^{2}L}{ \sin L}&0\end{bmatrix} \delta\mathbf{p}^{g}\] \[\triangleq\mathbf{C}_{g}^{G}\mathbf{\Phi}^{g}-\mathbf{T}_{1}\delta\mathbf{p}^{g}\]  
The conversion relationship between \(\delta\mathbf{v}^{g}\) and \(\delta\mathbf{v}^{G}\) is as follows:  
\[\delta\mathbf{v}^{G}=\mathbf{C}_{g}^{G}\delta\mathbf{v}^{g}+\delta\mathbf{C}_{g}^{G}\mathbf{v}^{g }=\mathbf{C}_{g}^{G}\delta\mathbf{v}^{g}-\Big{(}\mathbf{\Phi}_{S\text{G}}^{G}\times\Big{)} \mathbf{C}_{g}^{G}\mathbf{v}^{g}=\mathbf{C}_{g}^{G}\delta\mathbf{v}^{g}-\Bigg{[} \begin{array}[ ]{c}0\\ -\delta\sigma\end{array} \times\Bigg{]}\mathbf{C}_{g}^{G}\mathbf{v}^{g} \tag{33}\] \[=\mathbf{C}_{g}^{G}\delta\mathbf{v}^{g}-\Bigg{[} \begin{array}{ccc}0& \delta\sigma&0\\ -\delta\sigma&0&0\\ 0&0&0\end{array} \Bigg{]} \begin{bmatrix}\cos\sigma&-\sin\sigma&0\\ \sin\sigma&\cos\sigma&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}v_{E}\\ v_{N}\\ v_{U}\end{bmatrix} =\mathbf{C}_{g}^{G}\delta\mathbf{v}^{g}+