# A Novel FECAM-iTransformer Algorithm for Assisting INS/GNSS Navigation System during GNSS Outages
## 2 Theory of INS/GNSS Integrated Navigation System  
The INS and GNSS loosely integrated navigation system combines the high-frequency, short-term accuracy of the INS and the long-term stability of the GNSS to achieve long-term, high-precision, and stable navigation. Initially, both the INS and GNSS independently measure position, velocity, and attitude. A Kalman filter algorithm is then employed to estimate the position, velocity, and attitude errors of the INS relative to the GNSS. These estimated errors are used to correct the INS, thereby reducing the navigation bias caused by the accumulation of errors in the INS system. In this subsection, we first derive the error propagation model of the inertial guidance system and then introduce the combined navigation model based on the extended Kalman filter. The INS/GNSS navigation system used in this paper is shown in Figure 1.  
The INS is the backbone of the INS/GNSS integrated navigation system. In the system, the positioning information provided by GNSS and AI is primarily used to correct the positioning errors of the INS. The sensor used by the INS is IMU, and the raw data from the IMU serve as the input feature for the AI algorithm [35]. The INS outputs information on the attitude, position, and velocity of the vehicle. There are errors in this information before applying the extended Kalman filter, which is then corrected by the positioning information from the GNSS receiver The selection of reference frames is crucial for the navigation system. In this paper, the navigation frame is set as the East-North-Up (ENU) geographic coordinate, while the body frame is set as the Right-Front-Up (RFU) frame. The navigation coordinate system is called the \(n\)-frame, and the earth coordinate system is called the e-frame. Based on the established reference frames, the INS error model can be derived, incorporating elements such as attitude, velocity, and position errors. These elements are also included in state estimation as components of the state transition matrix of the EKF. The attitude error equation is shown in Equation (1) [33]:  
\[\dot{\theta}=\theta\times\omega_{in}^{n}+\triangle\omega_{in}^{n}-\varepsilon ^{n} \tag{1}\]  
where \(\theta\) is the attitude error vector. \(\omega_{in}^{n}\) and \(\triangle\omega_{in}^{n}\) are the angular velocity of the navigation frame to the inertial frame and its error. \(\varepsilon^{n}\) is the drift vector of the gyroscopes. The velocity error equation is shown in Equation (2) [33]:  
\[\Delta\dot{V}^{n}=f^{n}\times\theta^{n}+V^{n}\times(\Delta\omega_{en}^{n}+2 \Delta\omega_{ie}^{n})+\Delta V^{n}\times(\omega_{en}^{n}+2\omega_{ie}^{n})+ \
abla^{n} \tag{2}\]  
where \(V^{n}\) represents the kinematic velocity vector of the vehicle, while \(\Delta V^{n}\) denotes the associated errors. \(f^{n}\) stands for the specific force vector. \(\omega_{en}^{n}\) and \(\Delta\omega_{en}^{n}\) represent the angular velocity and its errors relative to the \(n\)-frame with respect to the e-frame. \(\omega_{ie}^{n}\) and \(\Delta\omega_{ie}^{n}\) represent the angular velocity and errors associated with the earth's rotation. Finally, \(\
abla^{n}\) denotes the accelerometer bias vector. The position error equation is shown in Equations (3)-(5) [33]:  
\[\Delta\dot{L}=-\frac{V_{N}}{\left(R_{M}+h\right)^{2}}\Delta h+\frac{1}{R_{M}+ h}\Delta V_{N} \tag{3}\]  
\[\Delta\dot{\lambda}=\frac{V_{E}\tan L\sec L}{R_{N}+h}\Delta L+\frac{\sec L}{R_ {N}+h}\Delta V_{E}-\frac{V_{E}\sec L}{\left(R_{N}+h\right)^{2}}\Delta h \tag{4}\]  
\[\Delta\dot{h}=\Delta V_{U} \tag{5}\]  
where \(L\), \(\lambda\), and \(h\) represent the latitude, longitude, and height of the vehicle in the \(n\)-frame, respectively, while \(\Delta L\), \(\Delta\lambda\), and \(\Delta h\) denote their corresponding errors. \(V_{N}\), \(V_{E}\), and \(V_{U}\) indicate the velocities in the north, east, and upward directions within the \(n\)-frame, with \(\Delta V_{N}\), \(\Delta V_{E}\), and \(\Delta V_{U}\) representing their respective errors.