## 2 Observer-Based Estimation
### Vehicle Models  
Two kinds of model can be found in the literature, which are denoted respectively as kinematic and dynamic. On the one hand, the kinematic model is concerned with the vehicle motion with no reference to forces; thus, it does not need complex parameters such as those regarding tyres. Figure 1 depicts a vehicle model showing: the vehicle velocity at the centre of gravity (\(G\)) \(V_{G}\), the longitudinal and lateral components of \(V_{G}\), respectively \(u\) and \(v\), the vehicle yaw rate \(r\), the VSA \(\beta\), the vehicle track \(t\) (here assumed to be the same for the front and the rear), the front and rear semi-wheelbases, respectively \(a_{1}\) and \(a_{2}\), and the vehicle wheelbase \(l\). The vehicle velocity can be written as:  
\[\mathbf{V_{G}}=\mathbf{u}\mathbf{i}+\mathbf{v}\mathbf{j} \tag{1}\]  
where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors that are aligned respectively with the vehicle longitudinal axis \(x\) and the vehicle lateral axis \(y\) (local reference frame). The acceleration of the vehicle at the centre of gravity is, by definition:  
\[\mathbf{a_{G}}=a_{x}\mathbf{i}+a_{y}\mathbf{j}=\frac{\mathrm{d}\mathbf{V_{G}}}{\mathrm{d}t}= \dot{\mathbf{u}}\mathbf{i}+u\frac{\mathrm{d}\mathbf{i}}{\mathrm{d}t}+\dot{\mathbf{v}}\mathbf{j}+v \frac{\mathrm{d}\mathbf{j}}{\mathrm{d}t} \tag{2}\]  
where \(a_{x}\) and \(a_{y}\) are, respectively, the longitudinal and lateral acceleration of the vehicle. Since the local reference frame rotates with the vehicle body, from [PERSON]'s formulae [114]:  
\[ \begin{array}{c}\frac{\mathrm{d}\mathbf{i}}{\mathrm{d}t}=\mathbf{r}\mathbf{j}\\ \frac{\mathrm{d}\mathbf{j}}{\mathrm{d}t}=-\mathbf{r}\mathbf{i}\end{array}  \tag{3}\]  
Therefore, using Equation (3), Equation (2) can be rearranged into [113; 115]:  
Figure 1: Kinematic modelling of a vehicle [113]. \[ \begin{array}{l}a_{x}=\dot{u}-vr\\ a_{y}=\dot{v}+ur\end{array}  \tag{4}\]  
An observer based on a kinematic vehicle model uses Equation (4) to estimate \(u\) and \(v\) (see Section 2.2); therefore, the sideslip angle would be, by definition (see Figure 1):  
\[\beta=\tan^{-1}\Big{(}\frac{v}{u}\Big{)} \tag{5}\]  
Some authors introduce the hypothesis of a small sideslip angle, \(\beta\ll 1\). If so, \(\sin\beta\approx\beta\), \(\cos\beta\approx 1\), and \(v\approx u\beta\); hence, the overall vehicle speed would be \(V=\frac{u}{\cos\beta}\approx u\). Using \(\dot{v}\approx(u\beta)=\dot{u}\beta+\dot{\beta}u\), one can easily rearrange Equation (1) to obtain the following expression, providing the VSA by integration [16]:  
\[\dot{\beta}\approx\frac{a_{y}}{u}-\frac{a_{x}}{u}-r \tag{6}\]  
The main issue of VSA estimation using a kinematic model is that it does not work when the vehicle yaw rate is relatively small or zero, as the system becomes unobservable [116; 117]. The dynamic model, on the other hand, provides a more detailed description of the vehicle dynamics, as it is based on the equilibrium equations. Considering the vehicle as a rigid body, the following is a generic set of equilibrium equations [118; 119]:  
\[ \begin{array}{l}ma_{x}=m\big{(}\dot{u}-vr\big{)}=\sum\limits_{i=1}^{p_{x}}F_ {x,i}\\ ma_{y}=m\big{(}\dot{v}+ur\big{)}=\sum\limits_{i=1}^{p_{y}}F_{y,i}\\ J_{z}\dot{r}=\sum\limits_{i=1}^{p_{z}}M_{i}\end{array}  \tag{7}\]  
where \(m\) is the vehicle mass, \(J_{z}\) is the vehicle moment of inertia with respect to a vertical axis, \(F_{x,i}\) and \(F_{y,i}\) are the generic force contributions, respectively, in the vehicle longitudinal and lateral direction, and \(M_{i}\) is the generic yaw moment contributions. A dynamic model can have different levels of detail/complexity and hypotheses used, with all of them affecting the estimation accuracy. As an example, adopting a standard four-wheel model, as shown in Figure 2, the three equilibrium equations are:  
\[