## 3 Slip-Based Road Friction Monitoring
### Lateral Dynamics
Such discussion is also missing in the solution proposed by [PERSON] et al. [105], where an observer to estimate simultaneously the vehicle longitudinal and lateral velocities and the road friction potential from wheel speed and chassis acceleration measurements is proposed. Despite an observer convergence analysis is provided, details regarding the lateral acceleration levels at which the observer converges are not provided. To continue with this model-based discussion, [PERSON] et al. [100] proposed a feedforward approach to estimate the friction potential during pure lateral dynamics conditions from a _Duggoff_ tyre model. Using this model the friction potential was expressed as a function of the lateral force, vertical force, cornering stiffness and tyre lateral slip angles. The cornering stiffness was assumed a known constant parameter, the tyre forces were obtained from the measurements provided by a 6-dof inertial motion unit, and the vehicle body slip necessary to compute the tyre lateral slips from a Luenberger yaw-sideslip observer. Simulation results were presented and accurate friction estimates were obtained for \(\mu_{\max}=0.5\) and \(\mu_{\max}=0.3\) at the expenses of normalised lateral force levels of \(\mu_{y}\approx 0.30\)-0.4 and \(\mu_{y}\approx 0.20\)-0.25 respectively. [PERSON] et al. [99] proposed an estimation structure to calculate the road friction potential and the tyre longitudinal and lateral stiffnesses (\(C_{\lambda},C_{a}\)), Figure 16. In this structure, the tyre forces and lateral and longitudinal slips are estimated in separated blocks, and fed  
Figure 15: Estimation structure proposed in [PERSON] et al. [103]. Figure adapted by the authors from [103]. through a linearised recursive least squares block (LRLS) based on a linearised _Brush_ tyre model. An adaptive multiple forgetting factor strategy is adopted to compensate the variation rate of the tyre properties (\(C_{\lambda},C_{\alpha}\), which vary slowly) and the friction potential (\(\mu_{\max}\), which can vary quickly during mu-jump situations). Moreover, the structure is enhanced to estimate left and right friction potentials in case of transversal mu-split conditions. Simulation results are presented in Carsim, and an accurate tracking of the friction potential and tyre parameters is achieved in a road formed by \(\mu_{\max}=0.9,\mu_{\max}=0.3,\mu_{\max}=0.6\) under a continuous lateral excitation level of approximately 4 m/s\({}^{2}\). The _Brush_ tyre model has been also employed by [PERSON] et al. [34; 43]. In this case, the authors proposed an adaptive law to estimate the tyre cornering stiffness and friction potential using a differential Global Positioning System (DGPS) setup. Several experiments were performed on high \(\mu_{\max}=0.9\) and low mu \(\mu_{\max}=0.4\) roads and accurate results were obtained by the proposed algorithms. The author remarked that the results obtained evidenced a good performance even under small lateral slip values, but the lack of lateral acceleration or grip utilisation results complicates extracting further conclusions. [PERSON] [111] presented a Bayesian Hypothesis selection process to infer the friction potential from a parameterised tyre model. The same approach is also reported in [PERSON] et al. [43]. In brief, this method provides a most likely friction potential value \(\mu_{\max}\) for a given set of tyre forces and tyre slips. Following a similar approach than the works presented previously, the tyre forces and tyre slips are estimated in an external Extended Kalman-Bucy filter (EKBF). The estimation routine is validated experimentally under J-turn and braking manoeuvres performed on a dry asphalt (\(\mu_{\max}\approx 0.9\)) segment. The authors remark that for low longitudinal or lateral slip levels (e.g., \(|a_{x}|<0.3\) G, \(|a_{y}|<0.4\) G) the algorithm does not perform well due to the proximity of the tyre force curves in the low slip regions, Figure 11a.