# Extended Particle-Aided Unscented Kalman Filter Based on Self-Driving Car Localization
## 4 Simulation Results  
The performance of the previous basic PAUKF and the extended PAUKF were compared based on the root mean square error (RMSE). The performance of filters can be realized by the numerical number of the RMSE of the estimation error. The calculation method of the RMSE of the estimation error is shown as Equation (8). \[\left[ \begin{array}{c}\mathrm{RMSE}_{\mathrm{est}}\\ \mathrm{RMSE}_{\mathrm{noise}}\end{array} \right]=\left[ \begin{array}{c} \sqrt{\left[\sum_{i=1}^{N}\left(\mathrm{Position}_{\mathrm{esti}}-\mathrm{ Position}_{\mathrm{mean}_{\mathrm{eosti}}}\right)^{2}\right]/N}\\ \sqrt{\left[\sum_{i=1}^{N}\left(\mathrm{Position}_{\mathrm{noise}}-\mathrm{ Position}_{\mathrm{mean}_{\mathrm{eosti}}}\right)^{2}\right]/N}\end{array} \right]. \tag{8}\]  
Figure 4a,b show the previous basic PAUKF and the extended PAUKF at 60 km/h, respectively. In both figures, the blue line with the circles is the ground truth trajectory of the vehicle, the red line with the triangles is the noisy vehicle position, the yellow line with the square is the particle filter's estimation result, and the black line with the circle is the final estimated vehicle position. The estimated trajectory of the previous basic PAUKF showed a significant error in relation to the ground truth trajectory, as shown in Figure 4a, since the previous basic PAUKF did not consider the geometry  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Noise Name** & **Generate Method** \\ \hline vehicle x-axis error (m) & \(\sim 15\sin(\mathrm{N}(0,\ 1))+\mathrm{N}(9.65,\ 12.2)+5)\) \\ vehicle y-axis error (m) & \(\sim 15\sin(\mathrm{N}(0,\ 1))+\mathrm{N}(8.34,\ 12.33)+5)\) \\ vehicle z-axis error (m) & \(\sim\mathrm{N}(0,\ 0.09)\) \\ Velocity error (m/s) & \(\sim\mathrm{sin}(\mathrm{N}(0,\ 0.09))\) \\ Yaw error (degree) & \(\sim\mathrm{sin}(\mathrm{N}(0,\ 0.09\ ))\) \\ Yaw rate error (degree/s\({}^{2}\)) & \(\sim\mathrm{sin}(\mathrm{N}(0,\ 0.09))\) \\ Feature x, y, z error (m) & \(\sim\mathrm{N}(0,\ 0.09)\) \\ Feature z position (m) & Random(0,10) \\ Range sensor bearing error (degree) & \(\sim\mathrm{N}(0,\ 0.09)\) \\ Range sensor elevation error (degree) & \(\sim\mathrm{N}(0,\ 0.09)\) \\ \(\sigma_{\  
effect of perception. Furthermore, the previous basic PAUKF did not consider the effect of noise in the vertical direction. As a result, it cannot generate and select appropriately weighted particles. The extended PAUKF by contrast achieved better performance than the previous basic PAUKF, by not only improving the calculation of the geometry, but also by considering the vertical noise effect on the weight generation and the selection scheme. The trajectory of the extended PAUKF was very close to the ground truth data, as shown in Figure 4b. Figure 5 shows the visualization result of the probability distribution of the features' position, the ground truth position, the position from the GPS, and the estimated position from the extended PAUKF at 250 th sample time. In order to make the figure easy to understand, the origin coordinate of probability distributions translates into the position of features, GPS, and vehicle. One thousand random numbers were used for visualizing the probability of each parameter. In the figure, the red square value is the position from the GPS, the black color with start marker is the position that is estimated from the extended PAUKF, and the blue data with the square is the ground truth data of the vehicle. The figure shows that all the features have different vertical values. The position from the GPS has a large error in relation to the ground truth data. The position from the extended PAUKF estimation is closest to the ground truth data. The extended PAUKF fused the surrounding feature position information to localize the vehicle itself in the map coordinates. This shows the effectiveness of the extended PAUKF. Figure 4: (**a**) Position estimation result of the basic particle-aided unscented Kalman filter (PAUKF) at 60 km/h; (**b**) position estimation result of the extended PAUKF at 60 km/h.