# On-Board Road Friction Estimation Technique for Autonomous Driving Vehicle-Following Maneuvers
## 4 In-Vehicle Road-Grip Estimation
### On-Board Estimation of Actual and Potential Friction  
It is common knowledge that the tribological characteristics of an asphalt can vary significantly depending on the distributed uniform dry, wet, snow or icy conditions (linked to meteorological aspects), or on the presence of the eventual local singularities as oil spots, puddles, kerbs or potholes (linked to local maintenance conditions of the road surface). In order to guarantee the optimum employment of the advanced functionalities of the autonomous driving logic, besides the information concerning the actual friction condition of the road surface, it is even more important providing the potential friction coefficient and the kinematic conditions, in terms of the tire-road interaction slip ratio quantity, it could be reached applying the external inputs, as throttle or braking pedals. The tire model parameters, employed within the estimation of the potential friction coefficient, depend on the parameters characterized and identified on the road characteristics where the experimental activities took part. Starting from the pre-calibrated set parameters of the tire model, depending, in its turn, on the peculiar dynamic set of equation chosen to describe the tire dynamics, and on the actual grip quantity obtainable from the vehicle state information, the potential friction coefficient achievable by each tire is evaluated. The potential friction quantity is assumed reachable varying only the slip ratio quantity (i.e., relative velocity within the tire-road interface) with all other operating conditions remaining the same (wheel alignment, vertical load and wheel spindle longitudinal speed). There are different approaches to tire modelling in the literature, which can be both physical and empirical. Several authors refer to the tire modelling using the Finite Element Method (FEM), adopted to evaluate static characteristics or to the multi-body tire approaches, as [51; 52; 53], commonly adopted to study dynamic phenomena on uneven surfaces. Although, the above modelling techniques should be evaluated carefully to the purpose of their employment within the embedded on-board control electronics due to their particularly significant computational cost. It becomes, therefore, necessary the adoption of simpler modelling approaches, as semi-empirical and analytical models [54], whose computational cost is compliant with the capabilities of the modern on-board systems. The typical tire characteristics curve is described in Figure 3, where three different regions of tire working range are represented. The ratio between longitudinal and vertical forces gives the instantaneous friction coefficient, i.e., the actual run-time coefficient between road surface and tire, expressed as follows:  
\[\mu_{x_{i,actual}}=\frac{F_{x_{i}}}{F_{x_{i}}}. \tag{17}\]  
The actual friction coefficient \(\mu_{x}\) depends both on the condition of the asphalt and on the peculiar operating conditions the tire is stressed with (i.e., vertical load, wheel alignment, slip ratio, longitudinal speed). Therefore, each tire operating point, describable by the actual friction coefficient \(\mu_{x}\) and the corresponding slip ratio \(\lambda\), can be represented in Figure 4, the point 1. The eventual changes in terms of friction coefficient within the tire-road interface are assumed to be referred only to the road surface, since the tire has been especially pre-calibrated on a reference asphalt surface. Assuming a linear behaviour of the tire in  
Figure 4: Procedure to evaluate potential friction coefficient. Figure 3: Tire characteristics curve and potential friction coefficient. the typical working conditions of the vehicle, a linear proportion between the reference tire-road and the actual tire-road friction coefficients can be assumed.