# A Combined Dynamic-Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle
## 2 Combined Extended Kalman Filter
### Combined Dynamic-Kinematic EKF  
Kinematic estimators return good predictions of the state variables, but they are affected by observability issues [18]. From the kinematic state-space model in (14) the observability matrix \(\mathcal{O}\) may be computed, resulting in that the rank of \(\mathcal{O}\) is lower than the dimension of the state vector in (12) when \(r=0\). Consequently, the estimation tends to drift away when the vehicle is driving straight. The observability of the linear system can also be studied by applying the Popov-Belevitch-Hautus (PBH) test [19]. Considering a continuous-time state-space model  
\[\dot{\mathbf{x}} =\mathrm{F}\mathbf{x}+\mathrm{G}\mathbf{u} \tag{15}\] \[\mathbf{y} =\mathrm{H}\mathbf{x}\]\[\mathbf{x}\in\mathbb{R}^{n}\qquad\mathbf{u}\in\mathbb{R}^{m}\qquad\mathbf{y}\in \mathbb{R}^{p}\qquad\mathrm{F}\in\mathbb{R}^{n\times n}\qquad\mathrm{G}\in \mathbb{R}^{n\times m}\qquad\mathrm{H}\in\mathbb{R}^{p\times n}\]  
the PBH test states that the system is observable over \(\mathbb{R}\) if, and only if  
\[\mathrm{rank}\begin{bmatrix}\mathrm{F}-\lambda\mathrm{I}\\ \mathrm{H}\end{bmatrix}=n,\;\forall\lambda\in\mathbb{C} \tag{16}\]  
If the PBH test is performed for the kinematic model (14) it is clear that, when \(r=0\), the lateral velocity is unobservable:  
\[\mathrm{rank}\begin{bmatrix}-\lambda&r\\ -r&-\lambda\\ 1&0\end{bmatrix}=\mathrm{rank}\begin{bmatrix}-\lambda&0\\ 0&-\lambda\\ 1&0\end{bmatrix}<2,\;\mathrm{for}\;\lambda=0 \tag{17}\]  
On the other hand, dynamic estimators do not suffer from observability issues, but, unlike kinematic ones, they are strongly problem-dependent, meaning that a good characterisation of vehicle parameters is needed [20].  
To exploit the potential and reduce the impact of weaknesses of both methods, a combined dynamic-kinematic EKF approach is proposed. Figure 3 shows a schematic of the combined algorithm. Estimated quantities are distinguished from measured ones by the use of circumflex (\(\hat{\ }\)) and the subscript _meas_ is dropped for the sake of clarity. Both dynamic and kinematic parts have a prediction block--fed with measurements at step \(k-1\)--and an update block--fed with measurements at step \(k\). The lateral velocity estimate resulting from the dynamic EKF is used as an additional measurement for the update of the kinematic estimation part. Consequently, the measurement equation of the kinematic model becomes the following:  
\[\mathbf{y}_{k}^{kin}=\begin{bmatrix}v_{x}\\ \phi_{y}^{dyn}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\hat{\mathbf{x}}_{k}^{kin} \tag{18}\]  
This modification improves the observability of the kinematic model, avoiding the estimate to drift during straights. Applying the PBH observability test to the kinematic part of the combined EKF, it results that the model is observable \(\forall\lambda\), also when \(r=0\):  
\[\mathrm{rank}\begin{bmatrix}-\lambda&0\\ 0&-\lambda\\ 1&0\\ 0&1\end{bmatrix}=2 \tag{19}\]  
The output of the kinematic model is directly fed into the kinematic part for the subsequent estimation step. Moreover, the resulting longitudinal velocity estimation is used as input to the dynamic part, while \(\phi_{y}^{kin}\) and \(\hat{r}^{dyn}\) form the state for next prediction in the dynamic part.  
The proposed configuration aims at increasing the robustness of kinematic estimators and improving the dynamic estimation, by correcting it with a part that is independent of vehicle parameters. Consequently, by tuning the covariance matrix of measurement noises of the kinematic part, \(\mathrm{R}^{kin}\), it is possible to define the level of importance of one part with respect to the other.