## 2 Background
### Road-Rubber Friction
#### 2.2.3 Tyre Models as a Function of the Friction Coefficient  
Relevant models have been presented in the literature to infer the maximum road-rubber friction characteristics employing the surface profile and the rubber characteristics [60; 66]. A brief introduction was provided in the previous paragraphs. In practice, such an analysis is not adopted in vehicle dynamics applications. Instead, the road-rubber friction is expressed as a function of the lateral (4) and (5) and longitudinal (6) slip quantities, which can be easily computed from standard vehicle on-board measurements. Thus, analytical (e.g., _Brush model_[67]), empirical (e.g., _Magic Formula_[67]) and data-based (e.g., _Neural Networks_[3; 68]) friction models are constructed to calculate the friction forces derived from a certain combination of \(\alpha\) and \(\lambda\) values, expression (15). \[\mu=f(\alpha,\lambda)\big{|}_{\gamma_{0},P_{0}} \tag{15}\]  
The evolution of the road friction \(\mu\) with the tyre slips is often portrayed (see Figure 7) assuming a fixed wheel inclination angle (\(\gamma_{0}\)), and tyre pressure (\(P_{0}\)). The lateral or longitudinal slip value at which the maximum friction \(\mu_{\text{max}}\) is located varies significantly with the road characteristics and the tyre tread. As a reference, low values are seen on competition tyres (high longitudinal and lateral tyre stiffness), and high values are obtained on loose surfaces (e.g., deep snow, gravel) [2; 31; 69; 70; 71]. The most extended friction models are the _Magic Formula_[67] presented by [PERSON], the _[PERSON]_[14; 72] tyre model, and the _Brush_ analytical formulation [16; 17; 67]. The _Dugoff_ and _Brush_ tyre models are often given preference in road friction recognition problems due to their implementation easiness and cost-effectiveness [53]. Other friction modelisations have been proposed in the literature (e.g., _Burckhardt_ friction model or _[PERSON]_ and _[PERSON]_ dynamic tyre models) [73; 74], but these are omitted in this section due to their reduced use in the reviewed works and their added complexity. \begin{table}
\begin{tabular}{c c} \hline \hline
**Road Category** & **Friction Coefficient Range \(\mu_{\text{max}}\)** \\ \hline Macro rough/micro rough surface (draining mixes, bituminous concretes) & 0.5–0.9 \\ \hline Macro smooth/micro rough surface (fine mixes) & 0.4–0.8 \\ \hline Macro rough/micro smooth surface (rolled aggregates) & 0.2–0.3 \\ \hline Macro smooth/micro smooth surface (flushing asphalt) & 0.1–0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Friction coefficient for four different broad categories of wet road surfaces [2]. Figure 7: Friction versus longitudinal slip curves representative of (**a**) high mu asphalt and (**b**) gravel surface. Curves generated with the tyre parameters presented in [31]. In the following section the _Magic Formula_, the _Brush_ formulation, and the _Dugoff_ model are introduced. For additional friction models and a more detailed discussion [53] can be consulted. * Magic Formula The Magic Formula [67] consists of a nonlinear formulation based on arctan functions. The coefficients \(D_{j},C_{j},B_{j},E_{j}\), with \(j\in\{x,y\}\), are determined empirically, based on experimental data. A simplified Magic formula formulation is presented here [8; 31]. More sophisticacp and complex formulations can be consulted in [PERSON] [67]. Firstly, the one-directional normalised tyre forces (\(\mu_{x0},\mu_{y0}\)) are computed from expressions (16) and (17).