## 3 Simulation and Analysis
### Straight line  
Position and heading truth values are generated using the train trajectory and speed simulation. Under the given conditions specified in Table 2, the combination navigation process of GNSS/SINS during the train's constant velocity straight-line motion is simulated. The results for the train's position and velocity are shown in Figure 4. Figure 5 demonstrates that, with the assistance of GNSS position and velocity, the navigation results are bounded, with the position converging to approximately 0.1 m.  
During 300 s of navigation time, the heading misalignment angle linearly diverges, confirming the earlier theoretical analysis that the heading is unobservable under a constant velocity straight-line motion. Figure 6 displays the estimated values of gyro and accelerometer biases. The accelerometer bias converges to the simulated setting of 0.1 g. However, under the condition of constant velocity straight-line motion, the gyro bias, which was initially set at 0.05\({}^{\circ}\)/s, diverges to nearly 0.05\({}^{\circ}\)/s within 300 s. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Sensors** & **Contents** & **Parameter Settings** \\ \hline \multirow{4}{*}{IMU} & \multirow{2}{*}{Gyroscope} & Bias \(\epsilon^{b}\) & 0.5\({}^{\circ}\)/s \\  & & White noise & 0.05\({}^{\circ}\)/s \\ \cline{2-4}  & \multirow{2}{*}{Accelerometer} & Bias \(\
abla^{b}\) & 0.1 g \\  & & White noise & 0.001 g \\ \hline \multirow{4}{*}{GNSS} & \multirow{2}{*}{Position error} & 0.1 m (1\(\sigma\)) \\  & & Velocity error & 0.01 m/s \\ \hline \multicolumn{2}{c}{Heading angle} & Heading angle error & 0.25\({}^{\circ}\)(1\(\sigma\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SINS inertial device parameters and GNSS error parameter settings. Figure 4: Diagram of a train’s constant velocity straight-line motion trajectory and velocity. In summary, the accelerometer bias exhibits a strong observability during a constant velocity straight-line train motion, while the heading gyro bias is unobservable. The simulation results are consistent with those of the theoretical analysis. (2) Curved line  
To further validate the observability of the train's heading angle on curved tracks, simulations were conducted on a curved trajectory, with their corresponding trajectory and velocity shown in Figure 7. The minimum curvature radius is approximately 5500 m, as depicted in Figure 8, which complies with the high-speed rail's minimum curvature radius requirements. Figure 5: Velocity and position-assisted navigation of a train’s constant velocity straight-line motion. Figure 6: Gyro and accelerometer bias during a train’s constant velocity straight-line motion. From Figure 9, it can be observed that with GNSS position and velocity assistance, the position convergence is approximately 0.1 m. The heading misalignment angle also converges to around 0\({}^{\circ}\), and as the curvature radius decreases, the convergence performance improves, as is evident in the latter part of the third subplot in Figure 10. As the processing was conducted with simulated data, the accelerometer bias estimates quickly converge to values near their true values. By examining the estimation results of the heading misalignment angle and inertial device biases in Figure 9, it can be observed that they converge to the set values of 0.1 g and 0.5\({}^{\circ}\)/s during curved track segments. This confirms the observability of heading during curved motion. Figure 8: Curvature radius of the curve. Figure 7: Train’s position and velocity on the curve. Finally, based on the theoretical analysis and simulation results presented earlier, it can be concluded that, when the train is moving at a constant speed on a straight track, GNSS velocity and heading updates can be used to avoid heading divergence and the unobservable gyro bias issue.