## 2 Train SINS/GNSS Integrated Navigation System Model
### Measure Equation  
The measured quantities include position, velocity, and heading measurements. In this context, \"position\" can refer to either GNSS position measurements or known reference points; \"velocity\" measurements can be the GNSS velocity or the odometer velocity, among others; and \"heading\" measurements can be from GNSS dual antennas, track centerline electronic map heading angles, or dual-wheel speedometer heading measurements.  
For position measurements, we use the following equation:  
\[\left\{\begin{array}{l}p_{mN}=p_{N}+n_{pN}\\ p_{mE}=p_{E}+n_{pE}\end{array}\right. \tag{14}\]  
where \(p\) represents position.  
For velocity measurements, we use the following equation:  
\[\left\{\begin{array}{l}v_{mN}=v_{N}+n_{vN}\\ v_{mE}=v_{E}+n_{vE}\end{array}\right. \tag{15}\]  
where \(v\) represents position.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Motion Mode** & **Straight/Curve** & **Observability** \\ \hline Constant Speed & Straight & Unobservable \\ Acceleration/Deceleration & Straight & Observable \\ Constant Speed & Curve & Observable \\ Acceleration/Deceleration & Curve & Observable \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of heading observability in different motion modes and track curvatures.  
For heading measurements, we use the following equation:  
\[\psi_{m}=\psi+n_{\psi} \tag{16}\]  
where \(\psi\) represents heading angle.  
The system equations decouple the heading obtained from the other states, and when heading measurements are available, both the heading misalignment angle and gyroscope bias will converge as follows:  
\[\dot{\psi}=\varepsilon_{\mathrm{z}} \tag{17}\]  
\[\mathrm{y}=\psi+v \tag{18}\]  
Equations (17) and (18) are transformed into matrix forms and provided as Equations (19) and (20), respectively.  
\[\begin{bmatrix}\dot{\psi}\\ \dot{\varepsilon}_{\mathrm{z}}\end{bmatrix}=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\begin{bmatrix}\psi\\ \dot{\varepsilon}_{\mathrm{z}}\end{bmatrix} \tag{19}\]  
\[\mathrm{y}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}\psi\\ \dot{\varepsilon}_{\mathrm{z}}\end{bmatrix}+v \tag{20}\]  
\[v\,\sim(0,\ R) \tag{21}\]  
Then, Equations (14)-(21) are transformed into the standard Riccati differential equation as shown in Equation (22).  
\[\left\{\begin{array}{ll}\dot{P}=AP+PA^{T}-PC^{T}R^{-1}CP+Q\\ \left[\begin{array}{cc}P_{11}&P_{12}\\ \dot{P}_{12}&P_{22}\end{array}\right]=\begin{bmatrix}2P_{12}-P_{11}^{2}/R&P_{2 2}-P_{11}P_{12}/R\\ P_{22}-P_{11}P_{12}/R&-P_{12}^{2}/R\end{array}\right] \tag{22}\]  
In Equation (22), \(P\) represents the covariance matrix, \(A\) is the system matrix that can be expressed as \(A=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\), \(C\) is the measurement matrix that can be expressed as \(C=\begin{bmatrix}1&0\end{bmatrix}\), \(R\) is the measurement noise, and \(Q\) is the system noise based on matrix calculations. The analytical solution for the matrix can be obtained as follows:  
\[\Delta=12R^{2}+P_{11}(0)P_{22}(0)t^{4}+12P_{11}(0)tR+4P_{22}(0)t^{3}R \tag{23}\]  
\[P_{11}(t)=\frac{1}{\Delta}4R\Big{[}P_{11}(0)P_{22}(0)t^{3}+3 RP_{11}(0)+3t^{2} P_{22}(0)R\Big{]} \tag{24}\]  
\[P_{12}(t)=\frac{1}{\Delta}6 RP_{22}(0)t[P_{11}(0)t+2R] \tag{25}\]  
\[P_{22}(t)=\frac{1}{\Delta}12 RP_{22}(0)[P_{11}(0)t+R] \tag{26}\]  
The \(\Delta\)'s magnitude is of the order of \(t^{4}\) in terms of time \(t\); thus, the \(P\) will eventually converge to zero. Therefore, according to the definition of random observability, when accurate velocity and heading observations are available, the heading will converge.