# Sensor Fusion of GNSS and IMU Data for Robust Localization via Smoothed Error State Kalman Filter
## 4 Results and Discussions
### Oval Track Simulation Analysis  
Figure 2 illustrates the simulated oval shape trajectory, which comprises a long straight line with small angle turns in two dimensions. The localization results for the various filtering algorithms are shown in Figures 3-5 and Table 2. It can be seen that the ESKF algorithm exhibits better target tracking accuracy compared to the EKF algorithm, as evidenced by its RMS values of 1.633 m, 1.782 m, and 1.476 m for Lateral, Longitudinal, and Vertical directions, respectively. By reducing the error state parameter, the accuracy of the three poses is improved by 2.8%, 2.1%, and 52.0%, respectively. After performing the backward smoothing filtering process, the RMS values of the three attitudes are further optimized to 1.463 m, 1.588 m, and 1.393 m, respectively. Moreover, the target tracking accuracy of ESKF-RTS is superior to ESKF, with the attitude accuracy in the three directions improving by 10.4%, 10.9%, and 5.6%, respectively. The trajectory also reveals that the curve of ESKF-RTS is smoother, further highlighting the advantages of ESKF-RTS over the other two algorithms in addressing the problem of smooth estimation of nonlinear states. Figure 2: Oval shape motion trajectory. Figure 4: Lateral, Longitudinal and Vertical positions of ESKF (Oval). Figure 3: Lateral, Longitudinal and Vertical positions of EKF (Oval). Figure 8: Lateral, Longitudinal and Vertical positions of ESKF (Serpentine 1). Figure 7: Lateral, Longitudinal and Vertical positions of EKF (Serpentine 1). Figure 9: Lateral, Longitudinal and Vertical positions of ESKF\(-\)RTS (Serpentine 1). Table 3 illustrates the improved performance of the ESKF and ESKF\(-\)RTS algorithms compared to the EKF algorithm in the serpentine condition. The RMS of Lateral, Longitudinal, and Vertical in EKF are 0.955 m, 1.585 m, and 5.823 m, respectively. After reducing the state value, the RMS of the three positions for the ESKF algorithm are 0.464 m, 0.641 m, and 1.700 m, which are 51.4%, 59.6%, and 70.8% better than the EKF algorithm. The ESKF\(-\)RTS algorithm further improves the position accuracy, with RMS values of 0.206 m, 0.243 m, and 0.912 m for the three directions, respectively, which are 55.6%, 62.1%, and 46.4% higher than those of the ESKF algorithm. Figures 7-9 show the comparison of the root mean square errors, which further demonstrates the superior performance and stability of the ESKF\(-\)RTS algorithm. However, since the simulation was conducted with a relatively good GNSS signal, the robustness of the algorithm could not be well evaluated. Subsequently, the GNSS data for the serpentine condition are partitioned into eight segments, and the error magnitude is amplified by a factor of two at intervals of 600 s in order to evaluate the algorithm's robustness. The serpentine working condition with further exaggerated GNSS signal error is analyzed, as shown in Figures 10-12 and Table 4. The RMS values for the Lateral, Longitudinal, and Vertical directions are 1.231 m, 1.735 m, and 1.453 m, respectively, for the EKF algorithm. Meanwhile, the ESKF algorithm improves these values by 48.4%, 48.7%, and 34.1%, respectively, resulting in the RMS values of 0.635 m, 0.890 m, and 0.957 m. Moreover, the ESKF\(-\)RTS algorithm improves the position accuracy in three directions by 42.1%, 52.6%, and 52.1%, respectively, compared to the ESKF algorithm, with RMS values of 0.368 m, 0.422 m, and 1.456 m. Even though the accuracy of ESKF\(-\)RTS decreases on the Vertical axis, the higher degree of smoothing in Figure 12 suggests that the ESKF\(-\)RTS smoothing algorithm can significantly enhance robustness and achieve precise localization, even with lower accuracy of GNSS sensors. Next, we continued the simulation of circular operating conditions and divided the entire simulation process into four sections to validate the feasibility of the algorithm by doubling the GNSS signal error, as shown in Figures 13-16.