## 3 Slip-Based Road Friction Monitoring
### Tyre Self-Alignment Torque
Figure adapted by the authors from [PERSON] et al. [40]. and small and large longitudinal excitation. Update laws for the axle lateral slip and friction potential are provided. Moreover, the observer gains are optimised using Sequential Quadratic Programming (SQP) to guarantee the observer robustness to tyre friction uncertainties. This observer is integrated with the maximum tyre SAT approach described previously, which is used only during high lateral excitation. The authors proposed a rule-based integration scheme to estimate \(\mu_{\max}\) and \(\alpha_{f}\) based on vehicle dynamics measurements such as the lateral acceleration or the yaw rate. The observer was tested experimentally in a mu-jump scenario composed of concrete, ice, packed snow and concrete with ice paths segments. Accurate results were obtained subjecting the test vehicle to sinusoidal-like inputs with \(|a_{y}|<2\) m/s\({}^{2}\). Additional details can be found on the Ph.D. of the author [17]. In [PERSON] and [PERSON] [11] an open loop observer is proposed to estimate the friction potential on the front left and front right wheels using the _Brush_ tyre model. The tyre self-alignment torque is obtained from the axial forces measured by strain-gauge sensors attached to the tie rods. A simplified steering and suspension kinematics model is employed to translate the previous forces from the tie rods to the wheel-ground contact. The observer is validated experimentally under steady-state mu-split and ramp mu-transition manoeuvres. The observer exhibits a good performance and is able to infer the friction potential of high and low mu surfaces (\(\mu_{\max}=0.9\), \(\mu_{\max}=0.4\)) for tyre friction levels ranging from \(\mu=0.6\) to \(\mu=0.4\). The _Brush_ tyre model has been also employed by [PERSON] et al. in [123]. In particular, authors proposed a modified version of the previous model based on test data. The friction potential is obtained using an iterative method that requires the axle lateral slip, and tyre SAT. The former is obtained from a combined auxiliary particle filter and iterated extended kalman filter (APF-IEKF) observer, while the latter is calculated using a linear disturbance observer, as in [17]. The observer is verified under steering inputs below 0.3 G performed at a constant speed on a surface covered by snow (\(\mu_{\max}=0.32\)). Further conclusions regarding grip utilisation levels cannot be extracted. Relevant approaches employing a tyre modelisation different from the _Brush_ model have been also found in the literature [9; 35; 116; 117; 121]. Specifically, in [116] [PERSON] et al. employed two different models for the tyre lateral forces and the tyre SAT respectively: _TMSimple_ and _TMeasy_. The authors grouped the front axle wheel slip and the road friction potential in the term \(x=\alpha_{f}/\mu_{\max}\) with the aim to detect if the steering excitation is large enough to estimate \(\mu_{\max}\) in spite of the current surface grip potential. An input-state-stable (ISS) observer is proposed to estimate \(x\) from lateral acceleration, yaw acceleration, steering wheel angle measurements and an estimate of the total front axle aligning torque obtained from a linear disturbance observer. The road friction potential is obtained from the previous estimate (\(x\)) using recursive total least squares (RTLS). The estimation algorithms are simulated in IPG-CarMaker under sinusoidal and lane change steering inputs on different road  
Figure 18: Estimation structure for medium excitation levels described in [PERSON] et al. [107; 118]. Figure adapted by the authors from [107]. conditions. A threshold on the variable \(x\) is defined (1.3 degrees) in order to update \(\mu_{\text{max}}\) only when enough lateral excitation is present. As this threshold is not directly related to the grip utilisation level extracting further conclusions is complicated. In [117] the same authors proposed a nonlinear adaptive observer and provided a stability analysis of the same.