# Sensor Fusion of GNSS and IMU Data for Robust Localization via Smoothed Error State Kalman Filter
## 4 Results and Discussions
### Oval Track Simulation Analysis
The experimental results shown in Table 5 once again demonstrated the feasibility and robustness of the error state Kalman filtering algorithm, indicating that the algorithm can achieve stable and accurate integrated navigation under various operating conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Position Error RMS (m)** & **Lateral** & **Longitudinal** & **Vertical** \\ \hline GNSS & 3.370 & 3.308 & 8.988 \\ EKF & 1.231 & 1.735 & 1.453 \\ ESKF & 0.635 & 0.890 & 0.957 \\ ESKF\(-\)RTS & 0.368 & 0.422 & 1.456 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMS values of localization errors for different algorithms (Serpentine 2). Figure 11: Lateral, Longitudinal and Vertical positions of ESKF (Serpentine 2). Figure 12: Lateral, Longitudinal and Vertical positions of ESKF\(-\)RTS (Serpentine 2). Figure 14: Lateral, Longitudinal and Vertical positions of EKF (Polygonal). Figure 13: Motion trajectory of polygon. The above data show the results of testing the navigation system using EKF, ESKF, and ESKF\(-\)RTS algorithms. By analyzing these results, we can draw the following conclusions. The above data demonstrate the performance of three different filtering algorithms in GNSS navigation. First, we can see that the performance of the EKF and ESKF algorithms is relatively similar, with position errors around 1.3 m. This is because both algorithms use Kalman filtering, but the ESKF algorithm introduces error states while considering errors, which can more accurately estimate errors and improve accuracy. At the same time, the ESKF algorithm can also reduce the impact of errors in future time by predicting error states, thus improving the stability of the algorithm. Figure 16: Lateral, Longitudinal and Vertical positions of ESKF\(-\)RTS (Polygonal). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Position Error RMS (m)** & **Lateral** & **Longitudinal** & **Vertical** \\ \hline GNSS & 2.601 & 2.677 & 8.516 \\ EKF & 1.243 & 1.289 & 4.328 \\ ESKF & 1.253 & 1.320 & 1.237 \\ ESKF\(-\)RTS & 0.900 & 0.896 & 1.999 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: RMS values of localization errors for different algorithms (Polygonal). Figure 15: Lateral, Longitudinal and Vertical positions of ESKF (Polygonal). On the other hand, the ESKF\(-\)RTS algorithm performs even better, with position errors even lower than 1 m. This is because the ESKF\(-\)RTS algorithm uses segmental smoothing to optimize the filtering results, which can more accurately estimate and correct errors. In the ESKF\(-\)RTS algorithm, RTS stands for \"Recursive Least Squares Smoothing\" which can combine the prediction results of the ESKF algorithm with the observation results to obtain more accurate state estimation results. Overall, the above data indicate that the ESKF\(-\)RTS algorithm performs well in GNSS navigation. The advantage of this algorithm is that it not only considers error states, but also further optimizes the filtering results through smoothing algorithms. Therefore, in practical applications, selecting the ESKF\(-\)RTS algorithm for navigation filtering can achieve more accurate and stable results, improving the reliability and accuracy of the navigation system.