# Polar Region Integrated Navigation Method Based on Covariance Transformation
## 2 The Grid SINS  
### Grid Frame and Grid SINS Mechanization  
The definition of the grid reference frame is shown in Figure 1. The grid plane is parallel to the Greenwich meridian, and its intersection with the tangent plane at the position of the aircraft is the grid's north. The angle between geographic north and grid north provides the grid angle, and its clockwise direction is the positive direction. The up direction of the grid frame is the same as that of the local geographic frame and forms an orthogonal right-handed frame with the orientations at grid east and grid north.  
The grid angle \(\sigma\) is expressed as found in [9]:  
\[\sin\sigma=\frac{\sin L\sin\lambda}{\sqrt{1-\cos^{2}L\sin^{2}\lambda}}\] \[\cos\sigma=\frac{\cos\lambda}{\sqrt{1-\cos^{2}L\sin^{2}\lambda}} \tag{1}\]  
The direction cosine matrix \(\mathbf{C}_{e}^{G}\) between the G-frame and the e-frame (earth frame) is as found in [9]:  
\[\mathbf{C}_{e}^{G}=\mathbf{C}_{n}^{G}\mathbf{C}_{e}^{n} \tag{2}\]  
where \(n\) refers to the local horizontal geographic frame. \(\mathbf{C}_{e}^{n}\) and \(\mathbf{C}_{n}^{G}\) are expressed as:  
\[\mathbf{C}_{e}^{n}=\left[\begin{array}{ccc}-\sin\lambda&\cos\lambda&0\\ -\sin L\cos\lambda&-\sin L\sin\lambda&\cos L\\ \cos L\cos\lambda&\cos L\sin\lambda&\sin L\end{array}\right] \tag{3}\]  
\[\mathbf{C}_{n}^{G}=\left[\begin{array}{ccc}\cos\sigma&-\sin\sigma&0\\ \sin\sigma&\cos\sigma&0\\ 0&0&1\end{array}\right] \tag{4}\]  
The updated equations of the attitude, the velocity, and the position in the grid frame are expressed as:  
\[\mathbf{\dot{C}}_{b}^{G}=\mathbf{C}_{b}^{G}\Big{[}\mathbf{\omega}_{ib}^{b}\times\Big{]}- \Big{[}\mathbf{\omega}_{iG}^{G}\times\Big{]}\mathbf{C}_{b}^{G} \tag{5}\]  
\[\mathbf{\dot{v}}^{G}=\mathbf{C}_{b}^{G}\mathbf{\dot{p}}^{b}-\left(2\mathbf{\omega}_{ie}^{G}+\bm {\omega}_{eG}^{G}\right)\times v^{G}+\mathbf{g}^{G} \tag{6}\]  
\[\mathbf{\dot{R}}^{G}=\mathbf{C}_{e}^{G}\mathbf{v}^{G} \tag{7}\]  
Figure 1: The definition of the grid reference frame. The blue arrows represent the three coordinate axes of the local geographic frame. The orange arrows represent the three coordinate axes of the grid frame.  
where \(\mathbf{\omega}_{iG}^{G}\) is the turn rate of the G-frame with respect to the i-frame.  
\[\mathbf{\omega}_{iG}^{G}=\mathbf{\omega}_{ie}^{G}+\mathbf{\omega}_{eG}^{G}=\mathbf{C}_{e}^{G}\bm {\omega}_{ie}^{e}+\mathbf{\omega}_{eG}^{G}\]  
\[\mathbf{\omega}_{ie}^{G}=\left[\begin{array}{c}-\omega_{ie}\sin\sigma\cos L\\ \omega_{ie}\cos\sigma\cos L\\ \omega_{ie}\sin L\end{array}\right],\mathbf{\omega}_{eG}^{G}=\left[\begin{array}[ ]{cc}\frac{1}{\gamma_{f}}&-\frac{1}{R_{y}}\\ \frac{1}{R_{x}}&-\frac{1}{\gamma_{f}}\\ \frac{\kappa}{\gamma_{f}}&-\frac{\kappa}{\gamma_{y}}\end{array}\right]\left[ \begin{array}{c}v_{E}^{G}\\ v_{N}^{G}\end{array}\right] \tag{8}\]  
where \(R_{x}\) is the radius of curvature of the grid east, \(R_{y}\) is the radius of curvature of the grid north, and \(\gamma_{f}\) is the distorted radius.  
Since the meridian converges rapidly in the polar region, the position of the aircraft in the polar region is usually expressed in the ECEF frame. The relationship between the coordinates \(x,y,z\) and the latitude \(L\) and the longitude \(\lambda\) is given by:  
\[\left\{\begin{array}{c}x=(R_{N}+h)\cos L\cos\lambda\\ y=(R_{N}+h)\cos L\sin\lambda\\ z=\left[R_{N}(1-f)^{2}+h\right]\sin L\end{array}\right. \tag{9}\]