## 3 Slip-Based Road Friction Monitoring
### Tyre Self-Alignment Torque
In this case, \(I_{f}\) is estimated rather than \(\mu_{\max}\) with the aim to identify the tyre friction characteristics without requiring a normal force state estimator. Ramp and slalom tests are performed on a dry surface and the discussion of the results is limited to the axle lateral slip estimates accuracy. This estimation structure is also addressed in [122]. In [40] the authors revisit the linearised pneumatic trail model (51). In this work [PERSON] et al. provide a comprehensive explanation regarding the selection of the linearised pneumatic trail model. Specifically, it is argued that while the lateral force remainsinvariant for low lateral slip values the pneumatic trail slope changes drastically with the friction potential, Figure 17.  
Therefore, by monitoring the factor \(I_{f}\) the friction potential can be detected at lower lateral excitation levels. A nonlinear observer based on the tyre lateral force error and similar to [41] is described and a proof of stability is provided. Field tests were executed on dry asphalt and gravel surfaces. Ramp steer and slalom tests are performed in the former surface, and convergence to the true friction values is observed for excitation levels above 50% grip utilisation. Sharp transient turns are executed on gravel and authors remark that the proposed observer provides accurate estimates of the friction potential for grip consumption levels around 50%. Additional details regarding these publications can be found in [120]. The linear pneumatic trail model (51) was also employed by [PERSON] et al. in [39]. In this case, the authors referred to this model as the pneumatic trail stiffness. An open loop observer is presented to estimate the axle lateral slips from lateral acceleration and yaw acceleration measurements, using a simplified single-track planar dynamics model. Moreover, the cornering stiffness is considered constant along the lateral dynamics linear region and RLS is employed to estimate the pneumatic trail stiffness from the axle lateral slip and the measured pneumatic trail. Despite further details regarding the estimation of \(t_{p}\) are not provided, authors highlight that the tyre SAT was directly acquired by wheel force transducers (WFT). Frequency response tests were performed on a high mu road (\(\mu_{\text{max}}\in[0.8\)-\(0.9]\)) and convergence of the friction potential estimates was observed for grip consumption levels between \(\mu=0.2\) and \(\mu=0.3\). In [PERSON] et al. [118] two approaches are proposed to estimate the friction potential. Firstly, the authors introduce a maximum torque method to derive \(\mu_{\text{max}}\) from the peak value of the tyre SAT versus axle lateral slip curve. As this approach can only provide accurate results once the maximum tyre SAT is identified (significant lateral excitation is required), [PERSON] et al. proposed a nonlinear least squares method to estimate the friction potential and the front axle lateral slip from a blended function composed of the lateral force and tyre SAT estimated form readily-available measurements and the same signals obtained from a reference _Brush_ tyre model, Figure 18. Sinusoidal constant speed manoeuvres were simulated in _Carsim_ on different grips (\(\mu_{\text{max}}=1\), \(\mu_{\text{max}}=0.5\) and \(\mu_{\text{max}}=0.2\)). Overall, results evidence a good performance of the nonlinear least squares observer, and a limited accuracy of the maximum tyre SAT approach. Nevertheless, authors indicate that the latter method can be employed to identify the lower bound of the friction potential, using this on the nonlinear least square optimisation. Additional discussion regarding grip utilisation levels is not provided. This work is completed later in [107], where [PERSON] et al. proposed a robust friction potential observer for lateral and longitudinal dynamics. In this case a division is made between medium and high lateral excitation  
Figure 17: (**a**) Normalised tyre lateral forces; (**b**) normalised tyre pneumatic trail for different road friction potential coefficients.