# Polar Region Integrated Navigation Method Based on Covariance Transformation
## 1 Introduction  
Considering that the distance of a great circle flight route is shorter, using trans-arctic routes can accomplish great savings in flying time when aircraft make transcontinental flights. Due to the demands of flight safety, each aircraft usually uses an INS/GNSS integrated navigation system to provide high-precision navigation information. The INS/GNSS integrated navigation system has broad development prospects. Previous literature [1] proposed an integrated navigation scheme based on INS and GNSS single-frequency precision point positioning, which is expected to be an advantage for low-cost precise land vehicle navigation applications. Several researchers [2, 3] have discussed the application of GNSS/INS on railways. Traditional INS/GNSS-integrated navigation algorithms are based on a north-oriented geographic frame. However, as the latitude increases, the traditional algorithms lose their efficacy in the polar region because of the meridian convergence. To solve this problem, when the aircraft is in the polar region, pilots usually plan their route based on polar-adaptable coordinate frames, such as the Earth-centered Earth-fixed frame (e-frame) [4], transversal Earth frame (t-frame) [5, 6], pseudo-Earth frame [7], wander frame [8] and grid frame (G-frame) [9, 10]. Although these coordinate frames are adaptable to polar regions, they cannot accomplish successful global navigation individually because some of them have specific mathematical singularities, such as the t-frame, pseudo-Earth frame, wander frame, and G-frame. These coordinate frames are usually adopted only in the polar region, and the local geographic frame (n-frame) is used as the reference navigation frame in non-polar regions. The e-frame can be used for continuous worldwide navigation. However, because the e-frame adopts Cartesian coordinates, the height channel is coupled with three rectangular coordinates but this causes position errors to diverge rapidly and brings difficulties to damping filtering. In addition, the e-frame does not have an explicit azimuth, which isinconvenient for flight route planning. Usually, the INS/GNSS integrated navigation system takes the local geographic frame as the navigation frame at low and middle latitudes and turns instead to grid frames at high latitudes. When the navigation frame is switched between different coordinate frames, such as the G-frame and n-frame, the structure of the filter changes. In this case, as another study [11] points out, if the consistency of the error state estimation cannot be guaranteed, this will cause the integrated navigation filter to overshoot and trigger error discontinuity. However, the current research [12; 13; 14] on polar region navigation mainly focuses on the design of an integrated navigation algorithm within the polar region or on looking for a navigation frame to achieve worldwide navigation independently and to avoid the problem caused by switching between navigation frames. One study [15] proposed the virtual sphere n-vector algorithm and derived detailed mechanization and dynamic equations. Their virtual sphere n-vector algorithm used the surface normal vector of the ellipsoid model to represent the aircraft's position, and did not have specific mathematical singularities. Essentially, the virtual sphere n-vector algorithm is the same as the e-frame algorithm and its azimuth definition is indistinct. The researchers of [11] and [16] proposed a hybrid polar navigation method, which accomplishes the inertial navigation mechanization in the e-frame, whereas it outputs the navigation parameters in the G-frame or t-frame. In addition, the studies of [11; 16] introduce a position matrix to decouple the height channel and three rectangular coordinates, which can solve the problem of position error divergence. In this way, the continuity of global navigation is guaranteed. However, it completely changes the navigation frame of the current airborne inertial navigation system, which is not conducive to system upgrades.