## 2 Train SINS/GNSS Integrated Navigation System Model
### SINS State Equation  
We use the expanded Kalman filter (EKF) as our data fusion architecture and loosely coupled loop integration architecture for the data of GNSS and INS. The state quantity is chosen as the error state, and the whole process is shown in Figure 3.  
In order to emphasize the role of observability analysis in train positioning and to simplify the theoretical derivation process, train motion is simplified to a motion in a two-dimensional plane. This makes it easier to highlight the non-observability of heading when the train is moving in a uniform straight line. Thus, only two horizontal accelerometers and one celestial gyroscope are used in the article instead of the usual inertial measurement  
Figure 3: Schematic diagram of the integration scheme of the GNSS/INS system. Figure 2: Train’s curved motion and sideslip angle schematic. (black line: rail; red line: bogie axle; blue dot: blue dot: center point of train carriage). component including three gyroscopes and three accelerometers. The error equation for inertial navigation is as shown in Equation (1):  
\[\left\{ \begin{aligned} &\delta^{r^{n}}=-\omega_{en}^{n}\times\delta r^{n}+ \delta\theta\times v^{n}+\delta v^{n}\\ &\delta v^{n}=C_{b}^{n}\delta f^{b}+f^{n}\times\phi-\left(2\omega _{in}^{n}+\omega_{en}^{n}\right)\times\delta v^{n}+v^{n}\times\left(2\delta \omega_{ie}^{n}+\delta\omega_{en}^{n}\right)+\delta g_{p}^{n}\\ &\dot{\phi}=-\omega_{in}^{n}\times\phi+\delta\omega_{in}^{n}- \delta\omega_{ib}^{n}\end{aligned} \right. \tag{1}\]  
where, \(r\), \(v\), \(\phi\) represents position, speed, and attitude. \(\omega_{en}^{n}\) represents the projection of the angular velocity of the n-system relative to the e-system under the n-system, \(\omega_{in}^{n}\) represents the projection of the angular velocity of the n-system relative to the inertial coordinate system under the n-system, \(\omega_{ie}^{n}\) represents the projection of the angular velocity of the e-system relative to the inertial coordinate system under the n-system, The superscript n denotes the projection into the n-coordinate system and the prefix \(\delta\) denotes the error of the corresponding variable, \(C_{b}^{n}\) represents the rotation matrix of the b system relative to the n system. The error of the specific force vector resolved in the n-frame is referred to as \(\delta f^{b}\), while the error of gyro output is denoted as \(\delta\omega_{ib}^{n}\). To use the error-state Kalman filter as the combination framework for integrated navigation computation, the continuous-time system equation used is as follows:  
\[\dot{X}(t)=F(t)X(t)+G(t)w(t) \tag{2}\]  
where \(X\) is the system state, \(F\) is the system matrix, \(G\) is the noise covariance matrix, and \(w\) is the system noise. The system error state is as follows:  
\[X=\left[ \begin{array}{cccccccc}\phi_{z}&\delta p_{E}^{n}&\delta p_{N}^{n}& \delta v_{E}^{n}&\delta v_{N}^{n}&\
abla_{x}&\
abla_{y}&\varepsilon_{z}\end{array}  \right]^{T} \tag{3}\]  
where \(\phi_{z}\) represents the misalignment angle in the heading axis between the computed navigation frame and the relative navigation frame; \(\delta p_{E}^{n}\), \(\delta p_{N}^{n}\) are the eastward and northward position errors; \(\delta v_{E}^{n}\), \(\delta v_{N}^{n}\) are the eastward and northward velocity errors; \(\
abla_{x}\)\(\
abla_{y}\) are the eastward and northward accelerometer biases; and \(\varepsilon_{z}\) represents vertical gyroscope drift. Under the conditions of plane navigation for trains, due to the gyroscope drift, there is a misalignment angle between the integrated true heading angle and the computed heading angle \(\phi_{z}\). \[\left\{ \begin{aligned} &\phi_{z}=\int_{0}^{t}\varepsilon_{x}+n_{g}d \tau\\ &\dot{\phi}_{z}=\varepsilon_{x}+n_{g}\end{aligned} \right. \tag{4}\]  
where \(n_{g}\) is noise of gyroscope. Position error differential equation is as follows:  
\[\left\{ \begin{aligned} &\delta p_{E}^{n}=\delta v_{E}^{n}\\ &\delta p_{N}^{n}=\delta v_{N}^{n}\end{aligned} \right.