# A Combined Dynamic-Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle
## 4 Results  
The performance of the combined DK-EKF is compared to an extended Kalman filter composed by the dynamic part alone (D-EKF), and to a kinematic Kalman filter (K-KF). As is well-known, the performance of Kalman filter estimators depends on the choice of process noise and measurement noise covariance matrices, Q and R, respectively [25]. The R matrix contains the variances of sensor noise, while Q is tuned during the estimator implementation [26]. Since the combined DK-EKF is composed by one dynamic EKF and  
Figure 7: Schematic representation of the position of IMU and S-Motion with respect to vehicle COG. one kinematic KF, it is necessary to define two pairs of Q, R. The following covariance matrices are selected for the dynamic part:  
\[ \begin{split}\mathrm{Q}_{DK}^{dyn}&=\begin{bmatrix}q_{v }^{dyn}&0\\ 0&q_{v}^{dyn}\end{bmatrix}=\begin{bmatrix}3\cdot 10^{-6}\,(\mathrm{m/s})^{2}&0\\ 0&10^{-6}\,(\mathrm{rad/s})^{2}\end{bmatrix}\\ \mathrm{R}_{DK}^{dyn}&=\begin{bmatrix}r_{r}^{dyn}&0\\ 0&r_{a_{y}}^{dyn}\end{bmatrix}=\begin{bmatrix}4\cdot 10^{-7}\,(\mathrm{rad/s})^{2}&0 \\ 0&6\cdot 10^{-2}\,(\mathrm{m/s})^{2}\end{bmatrix}\end{split}  \tag{28}\]  
and, for the kinematic part  
\[ \begin{split}\mathrm{Q}_{DK}^{kin}&=\begin{bmatrix}q_{v_{x}}^{ kin}&0\\ 0&q_{v_{y}}^{kin}\end{bmatrix}=\begin{bmatrix}5\cdot 10^{-3}\,(\mathrm{m/s})^{2}&0\\ 0&p_{v_{y}k}^{dyn}\cdot 10^{-1}\,(\mathrm{m/s})^{2}\end{bmatrix}\\ \mathrm{R}_{DK}^{kin}&=\begin{bmatrix}r_{v_{x}}^{kin}&0\\ 0&r_{v_{y}}^{kin}\end{bmatrix}=\begin{bmatrix}2\cdot 10^{-3}\,(\mathrm{m/s})^{2}&0 \\ 0&p_{v_{y}k}^{dyn}\,(\mathrm{m/s})^{2}\end{bmatrix}\end{split}  \tag{29}\]  
where \(p_{v_{y}k}^{dyn}\) is the component related to \(v_{y}\) of the covariance of the estimation error computed by the dynamic part of the filter at time \(k\), \(P_{k}\), that is estimated according to the extended Kalman filter formalism [25]. Since \(\phi_{y}^{dyn}\) is used as a measurement in the kinematic part, but it comes from an estimator rather than an actual sensor, \(p_{v_{y}k}^{dyn}\) expresses the estimated variance of the signal. Moreover, \(q_{v_{y}}^{kin}\) is set equal to \(p_{v_{y}k}^{dyn}\cdot 10^{-1}\) to have a constant ratio \(q_{v_{y}}^{kin}/r_{v_{y}}^{kin}\). The D-EKF covariance matrices are set equal to the ones in (28), \(Q_{D}=Q_{DK}^{dyn}\), \(R_{D}=R_{DK}^{dyn}\). The covariance matrices of the K-KF are instead different from the ones in (29), since the structure of the filter is also different. \[ \begin{split}\mathrm{Q}_{K}&=\begin{bmatrix}5\cdot 1 0^{-3}\,(\mathrm{m/s})^{2}&0\\ 0&3\cdot 10^{-3}\,(\mathrm{m/s})^{2}\end{bmatrix}\\ \mathrm{R}_{K}&=2\cdot 10^{-3}\,(\mathrm{m/s})^{2}\end{split}  \tag{30}\]  
Moreover, the estimated lateral velocity is set to zero when \(|r|<0.01\) rad/s, to reduce drift during straights. Figure 8 shows the results of a DLC at 50 km/h. The estimated longitudinal velocity is shown in the top-left plot. For the D-EKF, \(v_{x}\) is not an output of the extended Kalman filter, but it is computed from Equation (5). The three estimations are very similar and close to the reference value (the maximum error is lower than 0.15 m/s), even if all three tend to slightly over-estimate \(v_{x}\). This is due to the averaging of the longitudinal velocities from the four corners, that do not account for longitudinal slips. Figure 8 also shows the estimation of the yaw rate computed by D-EKF and DK-EKF, compared to the signal of the measured one. Both Kalman filters significantly rely on the measured yaw rate, since the variance of the sensor noise is rather low. The K-KF is not displayed, since it directly uses the measured signal, shown as reference yaw rate. The sideslip angle estimate is presented in the bottom-left plot. The D-EKF is able to predict the general behaviour of \(\beta\), but it is not precise. The K-KF estimation is affected by a drifting behavior, but it is better at capturing the high-frequency features of the reference sideslip angle. The DK-EKF corrects the dynamic part and is closer to the reference.