# A Joint Dual-Frequency GNSS/SINS Deep-Coupled Navigation System for Polar Navigation
## 2 Availability Analysis of GNSS in Polar Areas and Grid SINS Mechanization
### Grid SINS Mechanization and Error Model  
Based on the grid frame described in the previous subsection, the differential equations of SINS attitude, velocity, and position in \(G\) frame are described as follows:  
\[\dot{\mathbf{C}}_{b}^{G}=\mathbf{C}_{b}^{G}(\mathbf{\omega}_{Gb}^{b}\times) \tag{5}\]  
\[\dot{\mathbf{V}}^{G}=\mathbf{C}_{b}^{G}\dot{\mathbf{f}}^{b}-\left(2\mathbf{\omega}_{ie}^{G}+ \mathbf{\omega}_{eG}^{G}\right)\times\mathbf{V}^{G}+\mathbf{g}^{G} \tag{6}\]  
\[\dot{\mathbf{R}}^{e}=\mathbf{C}_{G}^{e}\mathbf{V}^{G}=(\mathbf{C}_{e}^{g})^{\mathrm{T}}(\mathbf{C }_{g}^{G})^{\mathrm{T}}\mathbf{V}^{G} \tag{7}\]  
where \(\mathbf{C}_{b}^{G}\) represents the calculated attitude matrix of SINS from \(b\) to \(G\) frame; \(\mathbf{\omega}_{Gb}^{b}\) is the angular velocity of \(G\) frame relative to \(b\) frame and is expressed in \(b\) frame; \(\mathbf{Vector}\times\) is the antisymmetric matrix of _Vector,_\(\mathbf{V}^{G}\) is the velocity of vehicles expressed in \(G\) frame; \(\dot{\mathbf{f}}^{b}\) is the specific force measured by SINS accelerometers expressed in \(b\) frame; \(\mathbf{\omega}_{ie}^{G}\) is the angular velocity of Earth's rotation expressed in \(G\) frame; \(\mathbf{\omega}_{eG}^{G}\) is the angular velocity of \(e\) frame relative to \(G\) frame and is expressed in \(G\) frame; \(\mathbf{g}^{G}\) is the projection of local gravity acceleration expressed in \(G\) frame; \(\mathbf{R}^{e}\) is the position coordinates expressed in \(e\) frame; and \(\mathbf{C}_{G}^{e}\) is the transformation matrix from \(G\) to \(e\) frame.  
Then, the error models of grid SINS attitude, velocity, and position can be described as follows:  
\[\dot{\mathbf{\phi}}=-\Big{(}\mathbf{\omega}_{iG}^{G}\times\Big{)}\dot{\mathbf{\phi}}^{G}+ \mathbf{C}_{\mathbf{\omega}G\varphi}\delta\mathbf{V}^{G}+(\mathbf{C}_{\mathbf{\omega}eR}+\mathbf{C}_{ \mathbf{\omega}GR})\delta\mathbf{R}^{e}-\mathbf{C}_{b}^{G}\mathbf{e}^{b} \tag{8}\]  
\[\begin{array}{rl}\delta\dot{\mathbf{V}}^{G}&=\mathbf{f}^{G}\times\mathbf{\phi}^{G}+\big{[} \mathbf{V}^{G}\times\mathbf{C}_{\mathbf{\omega}G\varphi}-\big{(}2\mathbf{\omega}_{ig}^{G}+\mathbf{ \omega}_{eG}^{G}\big{)}\times\big{]}\delta\mathbf{V}^{G}\\ &+\mathbf{V}^{G}\times(\mathbf{C}_{\mathbf{\omega}GR}+\mathbf{C}_{\mathbf{\omega}eR})\delta\mathbf{R}^{ e}+\mathbf{C}_{b}^{G}\
abla^{b}\\ &\delta\dot{\mathbf{R}}^{e}=\mathbf{C}_{G}^{e}\delta\mathbf{V}^{G}-\mathbf{C}_{G}^{e}\Big{(} \mathbf{V}^{G}\times\Big{)}\mathbf{C}_{R}\delta\mathbf{R}^{e}\end{array} \tag{9}\]  
where \(\mathbf{\omega}_{iG}^{G}\) is the angular velocity of \(G\) frame relative to \(G\) frame and is expressed in the inertial frame (denoted \(i\)); \(\dot{\mathbf{\phi}}^{G}\) is the attitude error expressed in \(G\) frame; \(\dot{\mathbf{\delta}}\mathbf{V}^{G}\) is the velocity error of vehicles expressed in \(G\) frame; \(\dot{\mathbf{\delta}}\mathbf{R}^{e}\) is the position error expressed in \(e\) frame; \(\dot{\mathbf{\varepsilon}}^{b}\) is gyro noise expressed in \(b\) frame; \(\dot{\mathbf{f}}^{G}\) is the specific force expressed in \(G\) frame; and \(\
abla^{b}\) is accelerometer noise expressed in \(b\) frame. The matrices \(\mathbf{C}_{\mathbf{\omega}G\varphi}\), \(\mathbf{C}_{\mathbf{\omega}eR}\), \(\mathbf{C}_{\mathbf{\omega}GR}\), and \(\mathbf{C}_{R}\) are the interim variables for which the reader can refer to [6].  
The grid SINS mechanization will help the SINS to solve the problem of meridian convergence in polar areas.