## 3 Slip-Based Road Friction Monitoring
### Longitudinal Dynamics  
A large number of works have been presented to estimate the friction potential from the vehicle longitudinal dynamics (e.g., braking manoeuvres). These can be classified into longitudinal stiffness or slip slope-based approaches [6; 7; 17; 44; 91; 92], friction model-based approaches [12; 13; 16; 18; 19; 93; 94; 95] and active force excitation approaches [71; 96]. Despite slip slope approaches belong to the friction model-based group (i.e., linear longitudinal stiffness tyre model), a distinction has been made to treat them in greater detail. Furthermore, a new promising approach recently developed by [PERSON] et al. (active force excitation) has been added to the discussion. Slip slope approaches are based on the assumption that the tyre longitudinal stiffness depends on the maximum road friction coefficient [6]. Therefore, provided that a suitable function \(C_{\lambda}=f(\mu_{\text{max}})\) is found, the friction estimation problem can be solved by monitoring the tyre longitudinal stiffness. In [PERSON] [6] a linearised friction model is presented to estimate the slip slope \(\kappa=C_{\lambda}/F_{z}\) and the friction bias \(\rho\) from the tyre longitudinal slip \(\lambda\) and the normalised longitudinal force (\(\mu=\frac{F_{z}}{F_{z}}\)),  
\[\lambda=\frac{\mu}{\kappa}+\rho \tag{43}\]  
where the force \(F_{x}\) is obtained from the measured engine torque and \(\lambda\) is calculated using the difference between the driven and non-driven wheel speeds. Different roads (asphal, wet asphalt, gravel, snow and ice) and tyres (MXT, MXV2 (almost worn out), NCT2 and Eurofrost (winter tyres)) were tested, and a rule-based classifier was proposed based on the empirical results. Slip slope coefficients varied significantly depending on the tyre considered, and well-clustered surfaces were obtained only for the Eurofrost tyre. In addition, authors remarked that the values obtained in these tests were very sensitive to variations in tyre parameters such as the tyre pressure or the tyre tread depth (tyre wear). In [PERSON] et al. [7] several braking manoeuvres were performed on dry (\(\mu_{\text{max}}\approx 1.1\)) and soapy roads (\(\mu_{\text{max}}\approx 0.7\)). The tyre forces were measured from a strain-based torque sensor and the braking action was performed only at the front wheels. The slip slope was calculated using recursive least squares (RLS) on the model given by expression (43) and significant differences between the tested surfaces were obtained for grip consumption levels of \(\mu=0.4\). These tests were repeated employing a cost-effective experimental setup and the following slip slope ranges were found, Table 2:  
Once again, the high variability of the slip slope coefficient was highlighted by the authors. Moreover, [PERSON] et al. indicated that the \(\kappa-\mu_{\text{max}}\) relation that works at the present time might not work during the next month, thus remarking the strong dependence of this coefficient on tyre parameters such as inflation pressure or tyre wear. In Ahn [17] longitudinal stiffness values of a Pirelli 255/50R-17 installed in a Jaguar S-type are provided, Table 3. Additional details regarding the temperature at which the tests on ice were performed are not given. As will be seen in Section 4.1, some authors argue that the ice temperature can have a significant impact on the tyre longitudinal stiffness or slip slope factor. \begin{table}
\begin{tabular}{c c c c} \hline
**Road** & \(\kappa_{\text{min}}\) & \(\kappa_{\text{max}}\) & \(\mu_{\text{max}}\) \\ \hline Dry asphalt & 23 & 40 & [0.85,1.15] \\ Soapy asphalt & 17 & 28.2 & [0.45,0.75] \\ \hline \end{tabular}
\end{table} 
Table 2: Slip slope \(\kappa\) and \(\mu_{\text{max}}\) ranges obtained in [7]. \begin{table}
\begin{tabular}{c c c c} \hline
**Type Parameter** & **Concrete** & **Snow** & **Ice** \\ \hline \(\mu_{\text{max}}\) & [0.85–1] & [0.35–0.4] & [0.15–0.2] \\ C\({}_{\lambda}\) & \(16\times 10^{4}\) & \(6.6\times 10^{4}\) & \(1.8\times 10^{4}\) \\ \hline \end{tabular}
\end{table}