# A Combined Dynamic-Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle
## 2 Combined Extended Kalman Filter  
### Dynamic Model  
The dynamic extended Kalman Filter approach is based on a nonlinear single track model (see Figure 1). Only lateral and yaw motion are considered:  
\[\dot{v}_{y} =\frac{1}{m}\left[F_{yf}\cos(\delta)+F_{yr}\right]-v_{x}r \tag{1}\] \[\dot{r} =\frac{1}{I_{zz}}\left[aF_{yf}\cos(\delta)-bF_{yr}\right] \tag{2}\]  
Lateral velocity, \(v_{y}\), and yaw rate, \(r\), are the two state variables, while steering angle, \(\delta\), and longitudinal velocity, \(v_{x}\), are considered to be the input to the system. \[\dot{\mathbf{x}}^{dyn} =f(\mathbf{x}^{dyn},\mathbf{u}^{dyn}) \tag{3}\] \[\mathbf{x}^{dyn} =\left[ \begin{matrix}v_{y}\\ r\end{matrix} \right],\ \mathbf{u}^{dyn}=\left[ \begin{matrix}v_{x}\\ \delta\end{matrix} \right]\]  
\(m\) represents the vehicle mass, \(I_{zz}\) the yaw inertia and \(a\), \(b\) the distance of front and rear axes from vehicle center of gravity (CoG). Sideslip angle is computed from the estimated lateral velocity and from the longitudinal velocity:  
\[\beta=\arctan(\frac{v_{y}}{v_{x}}) \tag{4}\]  
The longitudinal velocity is obtained from the average of the measured longitudinal velocities from the four corners  
\[v_{x,meas}=\frac{1}{4}\sum_{i=1}^{4}v_{cm,i} \tag{5}\]  
where \(v_{cm,i}\) is the longitudinal velocity at the four corners (\(i=1,2,3,4\); \(1=\) front left, \(2=\) front right, \(3=\) rear left and \(4=\) rear right). It is computed from wheel speeds, \(v_{wheel,i}\), considering also the contribution of steering (only at front) and yaw rate:  
\[v_{cm,1} =v_{wheel,1}\cos{(\delta)}+r\frac{t_{f}}{2} \tag{6}\] \[v_{cm,2} =v_{wheel,2}\cos{(\delta)}-r\frac{t_{f}}{2}\] \[v_{cm,3} =v_{wheel,3}+r\frac{t_{r}}{2}\] \[v_{cm,4} =v_{wheel,4}-r\frac{t_{r}}{2}\]  
Figure 1: Single track model. where \(t_{f}\) and \(t_{r}\) are the front and rear tracks, respectively. Longitudinal tyre forces are neglected, while front and rear lateral tyre characteristics are modelled using a customised tyre model, replicating the typical nonlinear behaviour [17]:  
\[F_{y}=\mu F_{z}\tanh\biggl{(}\frac{C_{y}}{\mu F_{z}}\alpha\biggr{)} \tag{7}\]  
where \(\mu\) is the tyre-road friction coefficient, \(F_{z}\) is the vertical load, \(C_{y}\) is the cornering stiffness and \(\alpha\) the tyre slip angle. \(F_{z}\) is considered to be the static load acting on the two axes, namely \(F_{zf}=mg\frac{b}{l}=7984\) N at the front and \(F_{zr}=mg\frac{a}{l}=6976\) N at the rear, where \(l\) is the wheelbase of the vehicle. The experimental results, obtained through the procedure detailed in Section 3.2, are fitted with Equation (7) to compute front and rear cornering stiffness and friction coefficient (reported in Table 1), with the results depicted in Figure 2.  
The system model in (3) is discretised using the Euler approximation to apply the equations of the discrete-time extended Kalman Filter:  
\[\mathbf{x}_{k}^{dyn}=f(\mathbf{x}_{k-1}^{dyn},\mathbf{u}_{k-1}^{dyn})\Delta t +\mathbf{x}_{k-1}^{dyn}=f_{k-1}(\mathbf{x}_{k-1}^{dyn},\mathbf{u}_{k-1}^{dyn}) \tag{8}\] \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Mass & \(m\) & \(1525\,\mathrm{kg}\) \\ Yaw moment of inertia & \(l_{zz}\) & \(2130\,\mathrm{kg}\,\mathrm{m}^{2}\) \\ COG to front axle distance & \(a\) & \(1.315\,\mathrm{m}\) \\ COG to rear axle distance & \(b\) & \(1.505\,\mathrm{m}\) \\ Front track & \(t_{f}\) & \(1.557\,\mathrm{m}\) \\ Rear track & \(t_{r}\) & \(1.625\,\mathrm{m}\) \\ Front cornering stiffness & \(\mathrm{C}_{af}\) & \(115\,\mathrm{kN}/\mathrm{rad}\) \\ Rear cornering stiffness & \(\mathrm{C}_{ar}\) & \(165\,\mathrm{kN}/\mathrm{rad}\) \\ Tyre-road friction & \(\mu\) & \(0.9\) \\ coefficient & \(\mu\) & \(0.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Alfa Romeo Giulia main parameters. Figure 2: Result of the fitting of the lateral characteristic of the front axle (**left**) and of the rear axle (**right**). where \(\Delta t=0.01\) s is the discretisation interval.