## 3 Slip-Based Road Friction Monitoring
### Longitudinal Dynamics
Specifically, it is highlighted that a grip utilisation between 70% and 85% is required to estimate \(\mu_{\max}\) with an accuracy within the 10% band. The _Brush_ model was also employed by [PERSON] et al. [13]. In this case, the tyre longitudinal stiffness is assigned a fixed value (under the assumption of invariability during a short period of time) and only the \(\mu_{\max}\) coefficient is estimated. The nonlinear _Brush_ model is linearised to apply an RLS identification routine and several braking manoeuvres are simulated under the actuation of a threshold-based ABS and a Sliding Mode control (SMC) based ABS. Moreover, the estimator was verified experimentally under the actuation of the former ABS. In this case, the analysis is reduced to limit braking manoeuvres, and further detailsregarding the estimation accuracy on lower consumption thresholds are not provided. [PERSON] et al. [19] proposed a third-order force model of the form,  
\[\mu_{x}=\frac{a_{1}}{3}(\lambda-a_{2})^{3}+\frac{a_{1}a_{2}^{3}}{3} \tag{45}\]  
where \(a_{1}\) and \(a_{2}\) are model parameters estimated using linearised recursive least squares (LRLS) and the friction potential is defined as \(\mu_{\max}=\frac{a_{1}a_{2}^{3}}{3}\). Several simulations and experimental tests are conducted and authors state that this estimation approach requires around 60-70% of the total friction consumption to achieve accurate estimates. In [PERSON] [94] a tyre model designated as BW and developed by [PERSON] at _BorgWarner TorqTransfer Systems AB_ was employed. Instead of assuming a constant \(C_{\lambda}\), the author used the linear slip slope relationship [44] described in the previous paragraphs. The \(A\) and \(C\) parameters were obtained from field tests performed in dry asphalt and ice/snow surfaces. Once again, the nonlinear tyre model was linearised in order to apply the RLS routine and a predefined friction consumption threshold was set to carry out the estimation only during dynamic excitation exceeding this value. A combined linear-nonlinear identification routine was employed in [PERSON] [18]. Specifically, the author employed the least squares method in a linearised version of the _Brush_ model during low excitation levels and a _Gauss-Newton_ optimisation routine during high longitudinal excitation levels. Experimental tests (acceleration ramp) were executed with winter tyres on asphalt, snow and ice surfaces. The maximum friction consumption level during the execution of these tests was maintained below the 0.4 band in snow and asphalt surfaces, and below the 0.1 band in icy conditions. [PERSON] [18] remarked that good results were obtained in snow conditions (converging the friction estimate to the true value at a 50% grip consumption level) and ice conditions (at a full tyre saturation level), but accurate results were not achieved on asphalt due to the reduced friction potential utilisation. [PERSON] et al. [21] proposed an algorithm to classify the slipperiness of the road surface. A rule-based classifier is proposed to identify the slipperiness grade (ranging from 1-asphal to 3-oil or icy patches) based on the slip estimated at each wheel and the vehicle longitudinal acceleration. Moreover, the algorithm includes a _Magic Formula_ curve fitting step executed when enough data points spread along the \(a_{x}\)-\(\lambda\) curve are available. Experimental tests at reduced acceleration (\(a_{x}<0.2\) G) are performed on a path composed of sand and gravel segments. Despite accurate results are obtained on the sandy segment (detected as a grade 3 segment), the algorithm misclassified the gravel terrain. The authors remark that a classifier relying on the wheel slip might not be accurate enough to distinguish rough road segments from rigid ones, and propose a second classifier based on the wheel speed fluctuations. Other works employing different friction models have been found in the literature. A dynamic _Lugre_ model was employed in [PERSON] et al. [93].