# A Novel FECAM-iTransformer Algorithm for Assisting INS/GNSS Navigation System during GNSS Outages
## 2 Theory of INS/GNSS Integrated Navigation System
The terms \(R_{M}\) and \(R_{N}\) are the radius of the curvature of the meridian and the prime vertical, respectively. In an INS/GNSS integrated navigation system, a 15-dimensional state vector is commonly used for state estimation, which is defined as follows [17]:  
Figure 1: INS/GNSS integrated navigation system. \(f_{ib}^{b}\) and \(\omega_{ib}^{b}\) represent the specific acceleration and angular velocity vectors measured by the IMU, respectively. The variables \(a_{INS}\), \(v_{INS}\), and \(p_{INS}\) are the attitude, velocity, and position of the vehicle calculated by INS. \(p_{GNSS}\) is the positions obtained with the GNSS. \(\Delta p_{GNSS}\) is the position correction value obtained from the EKF calculations. The remaining parts of the EKF calculation are also used to correct errors in the INS. \[X=[\theta_{R},\theta_{P},\theta_{H},\Delta V_{E},\Delta V_{N},\Delta V_{U},\Delta L,\Delta\lambda,\Delta h,\
abla_{x},\
abla_{y},\
abla_{z},\varepsilon_{x}, \varepsilon_{y},\varepsilon_{z}] \tag{6}\]  
where \(\theta_{R}\), \(\theta_{P}\), and \(\theta_{H}\) are the roll, pitch, and yaw angle errors, respectively. \(\varepsilon_{x}\), \(\varepsilon_{y}\) and \(\varepsilon_{z}\) are the gyroscopes' drifts in the three directions in the body frame. \(\
abla_{x}\), \(\
abla_{y}\), and \(\
abla_{z}\) are the accelerometers' biases in the three directions in the body frame. The discrete expressions of the state and measurement models for the INS/GNSS loosely coupled navigation are defined as follows [17]:  
\[\left\{ \begin{array}{l}X_{k}=F_{k,k-1}X_{k-1}+G_{k-1}W_{k-1}\\ Z_{k}=H_{k}X_{k}+V_{k}\end{array} \right. \tag{7}\]  
where vectors \(X_{k}\) and \(X_{k-1}\) represent the states at moments \(k\) and \(k-1\), \(F_{k,k-1}\) is the state transition matrix, \(H_{k}\) is the measurement matrix, \(G_{k-1}\) is the noise input matrix, \(W_{k-1}\) is the system process noise sequence, \(V_{k}\) is the system measurement noise sequence, and \(Z_{k}\) is the measurement vector. The filtering algorithm is the core of the INS/GNSS integrated navigation system, providing more accurate positioning results through state estimation. In this paper, we use the extended Kalman filter, a nonlinear version of the Kalman filter, which is better suited for working in real-world nonlinear environments. The computation is shown in Algorithm 1[34]. ```
```
0: Nonlinear model: \(x(t)=f(x(t-1),u(t-1))+w(t-1)\) \(z(t)=h(x(t))+v(t)\) Initial value: \(\xi_{0},P_{0},Q,R\) for i = 1,2,3,..,N do  Prediction phase: \(x^{-}(t)=f(\tilde{x}(t-1),u(t-1),w(t-1))\) \(\Phi(t-1)=\frac{\tilde{x}}{\tilde{x}}(f(x^{-}(t-1),u(t-1)))\) \(P^{-}(t)=\Phi(t-1)P(t-1)\Phi(t-1)^{T}+Q\) Update phase: \(y(t)=z(t)-h(x^{-}(t))\) \(H(t)=\frac{\tilde{x}}{\tilde{x}}h(x^{-}(t))\) \(K(t)=P^{-}(t)H(t)^{T}\Big{[}H(t)P^{-}(t)H(t)^{T}+R\Big{]}^{-1}\) \(\tilde{x}(t)=x^{-}(t)+K(t)y(t)\) \(P(t)=(l-K(t)H(t))P^{-}(t)\) endfor
```

**Algorithm 1**Extended Kalman Filter. ## 3 Design of Integrated Navigation System Based on FECAM-iTransformer

### Features of IMU and GNSS Signals

As shown in Figure 1(a), when the vehicle's motion state changes, the IMU measurement data displaying the specific acceleration and angular velocity values correspond to the vehicle's state changes. Meanwhile, GNSS receiver outputs the location of the vehicle, and the change between successive locations also indicates the vehicle's state changes. This suggests a potential correlation between IMU data and GNSS position changes. Ideally, the vehicle's position could be directly calculated by integrating the IMU measurements, but noise in the IMU signals introduces significant errors [14]. Thus, we propose utilizing neural networks to learn this potential correlation and the AI will be used to output the GNSS position increment \(\delta p_{GNSS}\). Furthermore, the multi-channel measurements from the IMU collectively reflect the vehicle's state changes, and they exhibit kinematic interdependencies [36].