# Polar Region Integrated Navigation Method Based on Covariance Transformation
## 2 The Grid SINS
### Dynamic Model of the Grid SINS  
The mechanization of the grid SINS is accomplished in Section 2.1. Next, the Kalman filter, based on the \(G\)-frame, needs to be designed. In order to design the Kalman filter, the dynamic model of the \(G\)-frame, including three differential equations, is given below, as put forward in [10]. The attitude error is defined as:  
\[\widetilde{\mathbf{C}}_{b}^{G}=\left[I-\mathbf{\phi}^{G}\times\right]\mathbf{C}_{b}^{G} \tag{10}\]  
\[\left[\mathbf{\phi}^{G}\times\right]=-\widetilde{\mathbf{C}}_{b}^{G}\mathbf{C}_{G}^{b} \tag{11}\]  
where \(\widetilde{\mathbf{C}}_{b}^{G}\) is the estimated attitude, expressed in terms of the direction cosine matrix. Differentiating Equation (11) gives:  
\[\left[\dot{\mathbf{\phi}}^{G}\times\right]=-\dot{\widetilde{\mathbf{C}}}_{b}^{G}\mathbf{C} _{G}^{b}-\widetilde{\mathbf{C}}_{b}^{G}\left(\mathbf{C}_{b}^{G}\right)^{T} \tag{12}\]  
Substituting \(\widetilde{\mathbf{C}}_{b}^{G}\) and \(\dot{\mathbf{C}}_{b}^{G}\) from Equation (5) gives:  
\[\left[\dot{\mathbf{\phi}}^{G}\times\right] =-\widetilde{\mathbf{C}}_{b}^{G}\left[\widetilde{\mathbf{\omega}}_{ib}^{B} \times\right]\mathbf{C}_{G}^{b}+\left[\widetilde{\mathbf{\omega}}_{iG}^{G}\times \right]\widetilde{\mathbf{C}}_{b}^{G}\mathbf{C}_{G}^{b}+\widetilde{\mathbf{C}}_{b}^{G} \left[\mathbf{\omega}_{ib}^{B}\times\right]\mathbf{C}_{G}^{b}-\widetilde{\mathbf{C}}_{b}^{G }\mathbf{C}_{G}^{b}\left[\mathbf{\omega}_{iG}^{G}\times\right] \tag{13}\] \[=-\widetilde{\mathbf{C}}_{b}^{G}\left[\delta\mathbf{\omega}_{ib}^{B} \times\right]\mathbf{C}_{G}^{b}+\left[\widetilde{\mathbf{\omega}}_{iG}^{G}\times \right]\widetilde{\mathbf{C}}_{b}^{G}\mathbf{C}_{G}^{b}-\widetilde{\mathbf{C}}_{b}^{G}\mathbf{ C}_{G}^{b}\left[\mathbf{\omega}_{iG}^{G}\times\right]\]  
Substituting \(\widetilde{\mathbf{C}}_{b}^{G}\) from Equation (10) gives:  
\[\left[\dot{\mathbf{\phi}}^{G}\times\right] =-\left[I-\mathbf{\phi}^{G}\times\right]\mathbf{C}_{b}^{G}\left[\delta \mathbf{\omega}_{ib}^{B}\times\right]\mathbf{C}_{G}^{b}+\left[\widetilde{\mathbf{\omega}}_ {iG}^{G}\times\right]\left[I-\mathbf{\phi}^{G}\times\right]-\left[I-\mathbf{\phi}^{G} \times\right]\left[\mathbf{\omega}_{iG}^{G}\times\right] \tag{14}\] \[=-\mathbf{C}_{b}^{G}\left[\delta\mathbf{\omega}_{ib}^{B}\times\right]\mathbf{ C}_{G}^{b}+\left[\delta\mathbf{\omega}_{iG}^{G}\times\right]-\left[\delta\mathbf{ \omega}_{iG}^{G}\times\right]\left[\mathbf{\phi}^{G}\times\right]+\left[\mathbf{\phi} ^{G}\times\right]\left[\delta\mathbf{\omega}_{iG}^{G}\times\right]\]  
According to Equation (12), the attitude error equation is expressed by:  
\[\dot{\mathbf{\phi}}^{G}=-\mathbf{\omega}_{iG}^{G}\times\mathbf{\phi}^{G}+\delta\mathbf{\omega}_ {iG}^{G}-\mathbf{C}_{b}^{G}\delta\mathbf{\omega}_{ib}^{B} \tag{15}\]The velocity error is defined as:  
\[\delta\mathbf{v}^{G}=\widetilde{\mathbf{v}}^{G}-\mathbf{v}^{G} \tag{16}\]  
According to Equation (6), the velocity error equation can be written as:  
\[ \begin{array}{ll}\delta\dot{\mathbf{v}}^{G}&=\widetilde{\mathbf{C}}_{b}^{G}\dot{\mathbf{f }}-\left(2\widetilde{\mathbf{\omega}}_{ie}^{G}+\widetilde{\mathbf{\omega}}_{eG}^{G} \right)\times\widetilde{\mathbf{v}}^{G}+\widetilde{\mathbf{g}}^{G}-C_{b}^{G}\dot{\mathbf{f }}+\left(2\mathbf{\omega}_{ie}^{G}+\mathbf{\omega}_{eG}^{G}\right)\times\mathbf{v}^{G}-\bm {g}^{G}\\ &=\left(\widetilde{\mathbf{C}}_{b}^{G}-\mathbf{C}_{b}^{G}\right)\dot{\mathbf{f}}^{G}+\mathbf{C }_{b}^{G}\dot{\mathbf{g}}\dot{\mathbf{f}}^{G}-\left(2\widetilde{\mathbf{\omega}}_{ie}^{G}+ \widetilde{\mathbf{\omega}}_{eG}^{G}\right)\times\mathbf{v}^{G}-\left(2\delta\mathbf{\omega }_{ie}^{G}+\delta\mathbf{\omega}_{eG}^{G}\right)\times\mathbf{v}^{G}-\delta\mathbf{g}^{G} \\ &=\dot{\mathbf{f}}^{G}\times\mathbf{\phi}^{G}+\mathbf{v}^{G}\times(2\delta\mathbf{\omega}_{ie}^ {G}+\delta\mathbf{\omega}_{eG}^{G})-\left(2\mathbf{\omega}_{ie}^{G}+\mathbf{\omega}_{eG}^{ G}\right)\times\delta\mathbf{v}^{G}+\mathbf{C}_{b}^{G}\dot{\mathbf{g}}\dot{\mathbf{f}}^{G}\\ \end{array}