## 2 System Structures
### IMU Factor and Bias Factor Modeling  
The IMU pre-integration method proposed in [37] is used to construct the main structure of the factor navigation system, and other sensors are introduced to correct it. In addition, the bias function is applied to correct the errors of IMU and to obtain the accurate navigation state in the whole system. The acceleration and angular velocity information is obtained from the IMU sensor, and the IMU model in the body system at time \(k\) can be defined as follows:  
\[\left\{ \begin{aligned} \hat{\mathbf{a}}_{k}^{B}=\mathbf{R}_{W}^{B_{k }}(\mathbf{a}_{k}^{W}-\mathbf{g}_{k}^{W})+\mathbf{b}\mathbf{a}_{k}+\mathbf{n }_{a}\\ \hat{\mathbf{\omega}}_{k}^{B}=\mathbf{\omega}_{k}^{B}+\mathbf{b}\mathbf{g}_ {k}+\mathbf{n}_{g}\end{aligned} \right. \tag{4}\]  
where \(\hat{\mathbf{a}}_{k}^{B}\) represents the raw acceleration measurements in the IMU body frame and \(\hat{\mathbf{\omega}}_{k}^{B}\) represents the raw gyroscope measurements. \(\mathbf{a}_{k}^{W}\) represents the theoretical values of acceleration in the world frame and \(\mathbf{g}_{k}^{W}\) represents the gravity in the world frame. \(\mathbf{\omega}_{k}^{B}\) represents theoretical values of the gyroscope. \(\mathbf{R}_{W}^{B_{k}}\in\text{SO}_{3}\) is the transfer matrix from the world frame to the body frame, \(\mathbf{b}\mathbf{a}_{k}\) represents the slowly changing acceleration bias, and \(\mathbf{b}\mathbf{g}_{k}\) represents the slowly changing bias of the gyroscope. \(\mathbf{n}_{a}\sim N(0,\sigma_{a}^{2})\) and \(\mathbf{n}_{g}\sim N(0,\sigma_{g}^{2})\) are the random noise functions which follow the Gaussian distribution. The pre-integration of the rotation, velocity, and position between the two consecutive states \(\mathbf{X}_{k}\) and \(\mathbf{X}_{k+1}\) is defined as follows:  
\[ \begin{aligned} \mathbf{p}_{B_{k+1}}^{W}& =\mathbf{p}_{B_{k}}^{W}+\mathbf{v}_{B_{k}}^{W}\Delta t+\underset{t \in[k,k+1]}{\iint}(\mathbf{R}_{B_{k}}^{W}(\hat{\mathbf{a}}_{k}^{B}-\mathbf{b }\mathbf{a}_{k}-\mathbf{n}_{a})-\mathbf{g}_{k}^{W})dt^{2}\\ \mathbf{v}_{B_{k+1}}^{W}&=\mathbf{v}_{B_{k}}^{W}+ \underset{t\in[k,k+1]}{\int}(\mathbf{R}_{B_{k}}^{W}(\hat{\mathbf{a}}_{k}^{B}- \mathbf{b}\mathbf{a}_{k}-\mathbf{n}_{a})-\mathbf{g}_{k}^{W})dt\\ \mathbf{q}_{B_{k+1}}^{W}&=\mathbf{q}_{B_{k}}^{W} \underset{t\in[k,k+1]}{\int}\frac{1}{2}\mathbf{R}_{B_{k}}^{W}\otimes(\hat{ \mathbf{\omega}}_{k}^{B}-\mathbf{b}\mathbf{g}_{k}-\mathbf{n}_{g})dt\end{aligned}  \tag{5}\]  
where \(\Delta t\) denotes the time interval between \(B_{t}\) and \(B_{t+1}\), and \(\mathbf{R}_{B_{k}}^{W}\) is the transfer matrix from the body frame to the world frame at time \(k\). The pre-integration of IMU assumes that the integration processes last a short time and the bias value at each state is a variable. According to the IMU factor node, the reference frame is under the IMU coordinate and the formula of the pre-integration between time \(k\) and \(k+1\) can be transformed into the following:  
\[ \begin{aligned} \mathbf{R}_{W}^{B_{k}}\mathbf{p}_{B_{k+1}}^{W}= \mathbf{R}_{W}^{B_{k}}(\mathbf{p}_{B_{k}}^{W}+\mathbf{v}_{B_{k}}^{W}\Delta t- \frac{1}{2}\mathbf{g}_{k}^{W}\Delta t^{2})+\mathbf{\omega}_{B_{k+1}}^{B_{k}}\\ \mathbf{R}_{W}^{B_{k}}\mathbf{v}_{B_{k+1}}^{W}=\mathbf{R}_{W}^{B_{k}}( \mathbf{v}_{B_{k}}^{W}-\mathbf{g}_{k}^{W}\Delta t)+\mathbf{\beta}_{B_{k+1}}^{B_{k}} \\ \mathbf{q}_{W}^{B_{k}}\otimes\mathbf{q}_{B_{k+1}}^{W}=\mathbf{ \gamma}_{B_{k+1}}^{B_{k}}\end{aligned}  \tag{6}\]where \(\mathbf{\alpha}_{B_{k+1}}^{B_{k}}\), \(\mathbf{\beta}_{B_{k+1}}^{B_{k}}\), and \(\mathbf{\gamma}_{B_{k+1}}^{B_{k}}\) represent the true value of the navigation state of pre-integration and can be defined as follows:  
\[