# A Novel SINS/SRS/CNS Multi-Information Fusion Global Autonomous Navigation Method
## 3 The Multi-Information Fusion Autonomous Navigation System Based on SINS/SRS/CNS
### The Principle of Spectral Redshift Navigation
#### 3.3.1 State Equations  
In the high latitudes, the \(G\) frame was selected as the navigation coordinate frame. The coordinates in the \(e\) frame are used to represent the position of the UAV. Therefore, the error equations are as follows [9]:  
\[\begin{cases}\dot{\mathbf{\Phi}}^{G}=\mathbf{\Phi}^{G}\times\left(\mathbf{\omega}_{ie}^{G} +\mathbf{\omega}_{eG}^{G}\right)+\mathbf{C}_{\circ}\delta\mathbf{v}^{G}-\mathbf{C}_{b}^{G}\mathbf{e }^{b}\\ \delta\dot{\mathbf{v}}^{G}=\mathbf{f}^{G}\times\mathbf{\Phi}^{G}+\left(\mathbf{v}^{G}\times\mathbf{ C}_{v}-\left(2\mathbf{\omega}_{ie}^{G}+\mathbf{\omega}_{eG}^{G}\right)\times\right)\delta \mathbf{v}^{G}+\mathbf{C}_{b}^{G}\
abla^{b}\\ \delta\dot{\mathbf{p}}^{G}=\mathbf{C}_{G}^{c}\delta\mathbf{v}^{G}-\mathbf{C}_{c}^{c}\left(\mathbf{v }^{G}\times\right)\mathbf{C}_{P}\delta\mathbf{p}^{c}\end{cases} \tag{18}\]  
where \(\mathbf{\Phi}^{G}\) is the mathematical platform error angle in the \(G\) frame; \(\mathbf{v}^{G}\) is the velocity in the \(G\) frame, and \(\delta\mathbf{v}^{G}\) is the error; \(\delta\mathbf{P}^{c}\) is the position error in the \(e\) frame; \(\mathbf{\omega}_{ie}^{G}\) is the projection of the Earth's rotation angular velocity, \(\mathbf{\omega}_{ie}\), in the \(G\) frame; \(\mathbf{\omega}_{eG}^{G}\) is the angular velocity of the \(G\) frame to the \(e\) frame, projected in the \(G\) frame; \(\mathbf{f}^{G}\) represents the specific force in the \(G\) frame; \(\mathbf{C}_{b}^{G}\) is the direction cosine matrix from the \(b\) frame to the \(G\) frame; \(\mathbf{C}_{G}^{c}\) is the direction cosine matrix from the \(G\) frame to the \(e\) frame;  
\[\mathbf{C}_{v}=\frac{1}{\sqrt{(R_{M}+h)(R_{N}+h)}}\begin{bmatrix}0&-1&0\\ 1&0&0\\ 0&-\cot L\sin\sigma&0\end{bmatrix};\sigma\text{ is the angle between the grid north and the geographic north; }\mathbf{C}_{P}=\frac{\xi}{\sqrt{(R_{M}+h)(R_{N}+h)}}\begin{bmatrix}\sin L&0&- \cos\lambda\cos L\\ \sin\lambda\sin L\cos^{2}L&1/\xi^{2}&-\sin\lambda\sin L\cos L\\ \cos L\sin L\sin\lambda&0&-\cos^{2}L\sin\lambda\cos\lambda\end{bmatrix},\]  
and \(\xi=\frac{1}{\sqrt{1-\cos^{2}L\sin^{2}\lambda}}\).  
The gyroscope and accelerator drift error equations are the same as Equation (12).  
According to Equations (18) and (12), select \(\mathbf{\Phi}^{G}\), \(\delta\mathbf{v}^{G}\), \(\delta\mathbf{P}^{c}\), \(\varepsilon^{b}\), and \(\
abla^{b}\) as the state vector:  
\[\mathbf{X}^{G}(t)=\begin{bmatrix}\mathbf{\Phi}^{G}&\delta\mathbf{v}^{G}&\delta\mathbf{p}^{c}& \varepsilon^{b}&\
abla^{b}\end{bmatrix}^{\mathrm{T}} \tag{19}\]  
Then, the state equation of the SINS/SRS/CNS multi-information fusion autonomous navigation system in the high latitudes can be established:  
\[\dot{\mathbf{X}}^{G}(t)=\mathbf{A}^{G}(t)\mathbf{X}^{G}(t)+\mathbf{G}^{G}(t)\mathbf{W}^{G}(t) \tag{20}\]  
where \(\mathbf{X}^{G}(t)\) is the state vector; \(\mathbf{A}^{G}(t)\) is the state transition matrix; \(\mathbf{G}^{G}(t)\) is the system noise coefficient matrix; and \(\mathbf{W}^{G}(t)\) is the system noise vector.