## 3 Slip-Based Road Friction Monitoring
### Tyre Self-Alignment Torque  
The discussion on slip-based methods is completed with the revision of road friction potential estimation approaches which rely on the tyre self-alignment torque (SAT) signal. Among the papers consulted, the vast majority employ an analytical friction modelisation to estimate the coefficient \(\mu_{\max}\)[9; 11; 17; 35; 37; 39; 40; 87; 88; 107; 116; 117; 118; 119; 120; 121; 122; 123]. In particular, the _Brush_ tyre model has received greater attention on the works reviewed. On the other hand, a reduced number of approaches avoiding the use of any particular friction model have been found in this survey [10; 68; 124]. Both, tyre model-less and tyre model-based approaches are treated briefly in the following. Regarding the tyre model-based works, an extensive research has been performed by different authors with the aim to extract the road friction potential from the variations of the tyre pneumatic trail. As was mentioned at the beginning of this section, the tyre SAT peaks at lower lateral slip levels than the tyre forces. As the tyre SAT \(\tau_{a}\) is directly related to the tyre lateral force \(F_{y}\) by the pneumatic trail \(t_{p}\),  
\[\tau_{a}=F_{y}t_{p} \tag{49}\]  
some authors argue that by monitoring the evolution of the tyre pneumatic trail, changes on the friction potential could be identified at significantly lower excitation levels compared to the longitudinal and lateral dynamics-based works described previously. In [PERSON] et al. [88] the road friction potential and the front axle cornering stiffness \(C_{\lambda}\) are estimated from body slip, yaw rate, longitudinal velocity, steering wheel angle, and steer-by-wire motor current measurements. The total steering torque is obtained from the steer-by-wire current using a second-order steering system model. A nonlinear least square (NLS) optimisation routine is run continuously to estimate the cornering stiffness assuming a constant \(\mu_{\max}=\mu_{\max,0}\). In parallel, the least squares (LS) algorithm is run and the residuals are compared to those obtained with the NLS. When a large difference is obtained, enough information of the nonlinear tyre characteristics is available and \(\mu_{\max}\) is estimated and updated in the _Brush_ model. A ramp experimental test is performed on dry asphalt and steady-state road potential estimates are obtained when the lateral acceleration exceeds 0.5 G.  
The previous work was continued in [87]. In this case, nonlinear adaptation laws are presented to estimate the friction potential and the front axle lateral slip (\(\alpha_{f}\)) from vehicle speed, yaw rate, steering wheel angle, lateral acceleration and the total aligning torque seen at the steering system. Regarding the latter, a linear disturbance observer is employed and the steering system is modelled as a second-order system. The nonlinear update laws are based on the error between the measured lateral tyre forces and front total aligning torque, and the estimates of these signals, which are obtained from a _Brush_ tyre model. The additional torque components derived from the vertical loads are compensated using suspension and steering kinematic models. The friction estimation algorithm is tested experimentally in ramp and slalom tests performed on a dry road. In order to avoid issues during moments of low lateral excitation, the algorithm only provides \(\mu_{\max}\) estimates once the lateral acceleration is above the 0.5 G threshold. In [PERSON] et al. [41] the authors employed a _Brush_ tyre model for the lateral forces and a linearised model for the pneumatic trail,  
\[t_{p}=c_{0}+c_{1}|\tan\alpha_{f}|I_{f} \tag{50}\]  
where \(I_{f}=\frac{1}{\mu_{\max}F_{z}}\) and it is assumed that the tyre is not fully sliding. An adaptation law is proposed to estimate the front axle lateral slip based on the error between the measured and estimated tyre lateral force.