## 3 Slip-Based Road Friction Monitoring
### Longitudinal Dynamics

Table 3: Longitudinal stiffness \(C_{\lambda}\) values obtained in [17]. A linear regression was performed to obtain these values setting the maximum longitudinal slip to 2%. [PERSON] presents the \"equivalent tyre-road stiffness\" theory [42; 53] to explain the differences seen on \(C_{\lambda}\). Specifically, this theory attributes the variations in slip slope to the presence of soft material (e.g., gravel, snow or wet ice) in the road-rubber interface. Under these conditions, the surface can no longer be considered infinitely rigid with respect to the tyre carcass, as states the accepted tyre modelling theory [67]. In [PERSON] et al. [44] the slip slope \(\kappa\) is estimated using RLS from the linear relationship  
\[\frac{F_{x}}{F_{z}}=\kappa\lambda \tag{44}\]  
In addition, the authors proposed a linear expression to relate the slip slope to the road friction potential, \(\mu_{\max}=A\kappa+C\), where \(A\) and \(C\) are constant parameters determined empirically. In particular, the authors found the previous coefficients from the slip slope estimates obtained in dry asphalt and gravel surfaces. In this case, the \"equivalent tyre-road stiffness\" theory holds, and the abrupt reduction of \(\kappa\) in gravel conditions correlates well with the experimental tyre model parameterisations on gravel performed by other authors [71]. Unfortunately, the previous expression has not been reproduced experimentally in a wider range of rigid road surfaces. The same expression was adopted by [PERSON] et al. [91]. The authors mention that the coefficients \(A\) and \(C\) were obtained experimentally, but additional details are not provided. In [PERSON] et al. [92] more details are provided regarding the linear \(\mu_{\max}\)-\(\kappa\) relationship. Specifically, the slip slope is estimated from gentle braking inputs performed with a winter maintenance vehicle on dry concrete and concrete with light loose snow covering, and the slip slope values 9.8 and 7.0 are obtained for each test case. An attempt to clarify whether the slip slope can be related to the road friction potential has been performed by [PERSON] et al. [45]. The authors performed field trials on five different rigid pavements of different micro and macro-textures (with friction potential coefficients ranging from 0.1 to 1.2). In addition, manoeuvres were repeated with summer and winter tyres, two tyre manufacturers, new and worn tyres, and a wet road surface with a water film thickness of 0.4 mm. The combination of the above conditions provided 40 cases for each one where braking at different levels of deceleration and slip were conducted. [PERSON] et al. found that the longitudinal stiffness \(C_{\lambda}\) of summer tyres is greater than that of winter tyres and the longitudinal stiffness of worn tyres is greater than that of new tyres. They also concluded that the longitudinal stiffness \(C_{\lambda}\) does not allow discrimination between the pavements, while their maximum friction coefficients varied almost three times and presented very different textures. In conclusion, they stated that a clear relationship \(C_{\lambda}\)-\(\mu_{\max}\) was not verified, and therefore it may be impossible to estimate the maximum available grip \(\mu_{\max}\) from the longitudinal stiffness \(C_{\lambda}\). Due to the fuzziness associated to slip slope methods, other authors have opted for using more sophisticated friction modelisations (e.g., _Dugoff model_) to estimate the road grip potential [13; 16; 18; 19; 21; 94]. These are designated here as model-based approaches. In brief, in these works linear and nonlinear regression algorithms are employed to infer the parameter \(\mu_{\max}\) from a friction model updated with suitable force and slip estimates. In [PERSON] [16] nonlinear least squares (NLLS) are applied to the analytical _Brush_ type model. The authors provide different minimum grip consumption thresholds based on the results obtained in high and low mu conditions (\(\mu_{\max}\approx 1\) and \(\mu_{\max}\approx 0.2\)).