## 3 Factor Graph Integrated Navigation Framework  
### Factor Graph Information Fusion Theory  
Considering the problems of occlusion and signal interference in the urban road environment, this paper constructs a GNSS/INS integrated navigation information fusion framework based on the factor graph to meet the needs of in-vehicle navigation systems for positioning accuracy and robustness. The integrated navigation information fusion factor graph framework is shown in Figure 2. In the figure, the IMU pre-integration factor node \(f^{imu}\) is connected to the navigation state variable node \(x\) and the system error variable node \(\alpha\) in the neighboring moments, and the GNSS factor node is connected to the navigation state variable node in the corresponding moments. At the same time, the system error factor node \(f^{bias}\) is connected to the system error variable node of the neighboring moment, and the factor graph model expands with the access of measurement information [27]. A factor graph is a graph structure consisting of factor nodes and variable nodes used to represent a probabilistic model, which is able to visually describe the dependencies and conditional probabilities between random variables. In the factor graph framework, the system state variables \(x=\left[\varphi,v,p\right]^{T}\), where \(\varphi=\left[\psi,\theta,\gamma\right]\) is the 3D attitude angle vector, \(v=\left[v_{N},v_{E},v_{D}\right]\) is the velocity vector, and \(p=\left[L,\lambda,h\right]\) is the position vector. The system deviation variables \(\alpha=\left[b_{\alpha},b_{\alpha}\right]^{T}\), \(b_{\omega}\) and \(b_{\alpha}\) are the three-axis deviations of the gyroscope and accelerometer, respectively. The set of system state variables and system error variables  
Figure 1: Programmatic framework for integrated navigation systems. Figure 2: GNSS/INS integrated navigation information fusion factor graph framework. from the initial moment \(t_{0}\) to the current moment \(t_{k}\) is defined as \(X_{k}=\{x,\alpha\}_{i=0}^{k}\). The set of measurement information of the system from the initial moment \(t_{0}\) to the current moment \(t_{k}\) is defined \(Z_{k}=\left\{z_{i}^{min},z_{i}^{gnss}\right\}_{i=0}^{k}\), where \(z_{i}^{min},z_{i}^{gnss}\) are the IMU and GNSS measurement values, respectively. The optimal navigation solution corresponding to the navigation system is the maximum a posteriori estimate of the system state. According to Bayesian theory, the a posteriori probability \(P(X_{k}|Z_{k})\) based on the navigation system state variables \(X\) can be factorized as:  
\[ \begin{split} P(X_{k}|Z_{k})&=\frac{P(Z_{k}|X_{k}) P(X_{k}|X_{k-1})}{P(Z_{k})}P(X_{k-1}|Z_{k-1})\\ &=\prod\limits_{i=1}^{k}\frac{P(z_{i}|x_{i})P(x_{i}|x_{i-1})}{P( z_{i})}P(X_{0})\end{split}  \tag{1}\]  
where \(P(X_{0})\) denotes the a priori information of the initial state, \(P(Z_{k})\) is the probability density of the sensor measurements, \(P(Z_{k}|X_{k})\) is the a priori probability density function of the navigation system's state quantity and sensor quantity measurements. The Bayesian estimation process consists of two parts: prediction and updating. In the prediction process of the factor graph, the probability density \(P(X_{k}|X_{k-1})\) is utilized to pass the state backward, providing a priori information about the state quantities at each moment in time. The updating process, on the other hand, corrects the prior probability density of the state quantities by using the most recent observations to obtain the posterior probability density \(P(X_{k}|Z_{k})\). Based on the maximum a posteriori probability estimation, the optimal estimation of the navigational state variables can be expressed as follows:  
\[X^{map}=\text{argmax}P(X_{k}|Z_{k}) \tag{2}\]  
It is functionally decomposed according to the principle of factor graph, and each probability density from the decomposition corresponds to the factor node in it, respectively.