## 4 Simulation Results and Analysis
### Results Analysis
Combining Table 2 with the calculated results, we find that, for both the MHE and UKF algorithms, the estimation with wrong initial longitudinal velocity performs slightly worse than that with a correct initial estimate. Even so, MHE still shows more robustness and a faster convergence ability than UKF. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**State**} & \multicolumn{4}{c}{**ME Error**} & \multicolumn{4}{c}{**RMS Error**} \\ \cline{2-7}  & **MHE** & **UKF** & **Improvement** & **MHE** & **UKF** & **Improvement** \\ \hline \(u\) (m/s) & 0.441 & 0.428 & \(-\)3.037\% & 0.127 & 0.335 & 62.090\% \\ \(v\) (m/s) & 0.067 & 0.130 & 48.462\% & 0.024 & 0.029 & 17.241\% \\ \(r\) (rad/s) & 0.017 & 0.022 & 22.727\% & 0.006 & 0.007 & 14.286\% \\ \(F_{s\_fl}\) (N) & 98.429 & 156.903 & 37.268\% & 38.278 & 43.850 & 12.707\% \\ \(F_{s\_fr}\) (N) & 145.710 & 195.451 & 25.449\% & 44.730 & 49.941 & 10.434\% \\ \(F_{s\_rl}\) (N) & 93.551 & 181.288 & 48.396\% & 37.152 & 43.084 & 13.768\% \\ \(F_{s\_rr}\) (N) & 135.059 & 241.798 & 44.144\% & 44.949 & 55.750 & 19.374\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ME and RMS errors comparison of UKF and MHE on the DLC maneuver. ME: maximum estimation; RMS: root mean square. Figure 11: Estimated results and errors of tire lateral forces using UKF and MHE. (**a**) Estimated lateral tire force FL; (**b**) Estimated lateral tire force RL; (**c**) Estimated lateral tire force FR; (**d**) Estimated lateral tire force RR. Figure 12: Estimation performance comparison using UKF and MHE with longitudinal velocity incorrectly initialized. (The red dashed line is estimated by MHE; the black dotted line is estimated by UKF; and the continuous blue line is the reference value). (**a**) Estimated longitudinal speed under initial guess of longitudinal speed 120 km/h; (**b**) Estimated longitudinal speed under initial guess of longitudinal speed 80 km/h; (**c**) Estimated lateral speed under initial guess of longitudinal speed 120 km/h; (**d**) Estimated lateral speed under initial guess of longitudinal speed 80 km/h; (**e**) Estimated yaw rate under initial guess of longitudinal speed 120 km/h; (**f**) Estimated yaw rate under initial guess of longitudinal speed 80 km/h. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Initial Velocity** & \multicolumn{4}{c}{\(\boldsymbol{u=120\;\mathrm{km/h}}\)} & \multicolumn{4}{c}{\(\boldsymbol{u=80\;\mathrm{km/h}}\)} \\ \hline \multirow{2}{*}{**State**} & \multicolumn{2}{c}{**ME Error**} & \multicolumn{2}{c}{**RMS Error**} & \multicolumn{2}{c}{**ME Error**} & \multicolumn{2}{c}{**RMS Error**} \\ \cline{2-9}  & **MHE** & **UKF** & **MHE** & **UKF** & **MHE** & **UKF** & **MHE** & **UKF** \\ \hline \(u\) (m/s) & 5.555 & 5.547 & 0.264 & 1.408 & 5.556 & 5.546 & 0.752 & 1.703 \\ \(v\) (m/s) & 0.205 & 0.279 & 0.050 & 0.101 & 0.084 & 0.287 & 0.032 & 0.090 \\ \(r\) (rad/s) & 0.031 & 0.046 & 0.012 & 0.013 & 0.025 & 0.029 & 0.011 & 0.013 \\ \(F_{s\_fl}\) (N) & 219.867 & 240.493 & 49.126 & 88.098 & 147.926 & 292.768 & 47.576 & 106.835 \\ \(F_{s\_fl}\) (N) & 235.481 & 299.550 & 54.423 & 105.462 & 201.060 & 231.921 & 58.604 & 83.239 \\ \(F_{s\_fl}\) (N) & 263.167 & 275.681 & 63.004 & 89.418 & 214.612 & 356.536 & 50.953 & 111.919 \\ \(F_{s\_tr}\) (N) & 297.239 & 366.145 & 59.385 & 111.808 & 182.882 & 275.438 & 56.724 & 87.101 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: ME and RMS errors comparison of UKF and MHE on the DLC maneuver with an incorrect initial estimate of longitudinal velocity.