# A Combined Dynamic-Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle
## 4 Results
Moreover, the kinematic part does not lead to diverging results, thanks to \(\phi_{y}^{dyn}\) entering as a measure. Table 2 reports a comparison of the root mean square error (RMSE) of the \(\beta\) estimations, highlighting the higher performance of the DK-EKF, with respect to both D-EKF and K-KF. The cumulative distribution function of the estimation error of \(\beta\), \(F_{\beta}(x)\), is also shown in the bottom-right plot of Figure 8. \(F_{\beta}(x)\) expresses the probability that the estimation  
error of \(\beta\) remains below a certain value \(x\), namely \(F_{\beta}(x)=\mathrm{P}(\beta\leqslant x)\). This metric clearly shows that DK-EKF outperforms both D-EKF and K-KF, with the latter returning the least reliable estimation. The results of the slalom manoeuvre are reported in Figure 9. Even in this case, the \(\beta\) estimation quality is enhanced by the combined approach, while less significantly. This is confirmed by the difference between the \(\beta\) RMSEs (see Table 2). When the D-EKF prediction returns a relevant error, the DK-EKF corrects such behaviour, in general (see, for instance, the predicted sideslip angle between 10 s and 12 s, in the bottom-right plot). The K-KF performs significantly better for the DLC case, due to the high lateral dynamic content of the slalom manoeuvre, as may be seen by looking at the yaw rate signals in the top-right of Figure 9. This is also suggested by the cumulative distribution function of the \(\beta\) estimation error, which is very similar to the one of the D-EKF (bottom-right plot of Figure 9). The \begin{table}
\begin{tabular}{l c c} \hline \hline
**Strategy** & **DLC** & **Salam** \\ \hline D-EKF & 0.40 deg & 0.33 deg \\ K-KF & 0.60 deg & 0.33 deg \\ DK-EKF & 0.27 deg & 0.22 deg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of RMSE of sideslip angle estimation strategies. Figure 8: DLC at 50 km/h: longitudinal velocity (**top-left**), yaw rate (**top-right**), sideslip angle (**bottom-left**) and cumulative distribution function of \(\beta\) estimation error (**bottom-right**). difference between the RMSE of the DLC and the one of the slalom obtained with the K-KF underlines the weaknesses of this method, due to a drifting behaviour of the estimation when the vehicle lateral dynamics are not sufficiently excited. The RMSEs obtained with the combined approach are instead consistent in the two manoeuvres, highlighting the capability of compensating for the observability issues of the kinematic part of the DK-EKF. Longitudinal velocity signals are again similar and characterised by small errors. Both test results highlight that the proposed combined approach can give more precise sideslip angle estimation, without displaying drifting of the estimation and effectively compensating for model mismatch. Moreover, the DK-EKF may also be exploited further, by defining variable \(Q^{kin}\) and \(R^{kin}\) matrices, based on driver inputs. This modification would allow to define how much the estimation relies on the dynamic or on the kinematic part, depending on the current driving condition.