# On-Board Road Friction Estimation Technique for Autonomous Driving Vehicle-Following Maneuvers
## 4 In-Vehicle Road-Grip Estimation
### From Vehicle Sensors to Tires' State
The estimation of the tangential interaction forces, due an intrinsic non-linearity of each tire system, need an additional convergence algorithm for a correct partition of the global longitudinal force, located at the centre of gravity, into its two contributes based on the front and on the rear axles. Indeed, starting from the vertical loads calculated at each axle the convergence algorithm evaluates the above longitudinal forces, consistent with the vertical loads applied, the kinematics evaluated and the intrinsic dynamic characteristics of a pre-calibrated tire (neglecting the tires' transient behavior at the current stage). * The suspensions and steering system kinematics and compliances have been taking into account by acquiring the invariable KC curves by means of physical bench testor as an output of simulations performed by means of a multibody model. The inputs of the T.R.I.C.K.-based methodology, optimized for the longitudinal vehicle dynamics estimation, comprise the following signals acquired thanks to the sensors acquired and processed directly on-board:  
* Wheels' angular velocity (rad/s). * Longitudinal velocity evaluated at the vehicle's centre of gravity (m/s). * Longitudinal acceleration evaluated at the vehicle's centre of gravity (m/s\({}^{2}\)). * Throttle position (%). * Braking position (%). The model outputs, referring to the axle kinematic and dynamic quantities as well as to the additional, are reported below:  
* Axles' slip ratio (-). * Axles' vertical interaction force (N). * Axles' longitudinal interaction force (N). * Axles' actual friction coefficient (-). Since the double track model, i.e. since the dynamics of the vehicle axle, and analyzing a longitudinal maneuvers, the assumption in [49] related to consider the left and right gear ratio of the steering system almost equal, small steering angles and negligible of lateral load transfer and the body roll effect are accepted. Due to the vehicle body symmetry hypotheses made to develop a single track model, the forces acting on the tire have been considered equal. Therefore, the forces acting on the single tires of a single track model are equal to the forces of the entire axle. The vehicle model and the reference system considered are shown in Figure 2.  
In order to evaluate the vertical forces, the loads acting on axles in a stationary condition (\(v=0\) and \(a=0\)), called \"static loads\", \(W_{f}\) and \(W_{r}\), have to be evaluated. Such values depend on the position of the vehicle body centre of gravity:  
\[W_{f} =\frac{mgl_{r}}{L}, \tag{5}\] \[W_{r} =\frac{mgl_{f}}{L}. \tag{6}\]  
For the longitudinal load equation [49], the load transfer are:  
\[\Delta F_{z}=\frac{mha_{x}}{l}. \tag{7}\]  
The aerodynamic downforces are expressed by following equation:  
\[Fz_{zero_{i}}=\frac{1}{2}\rho A_{v}v^{2}C_{z}, \tag{8}\]  
with \(i=[1,2]\) are defined the axles (respectively front and rear). Therefore, the axles vertical loads result equal to:  
\[F_{z_{i}}=-(W_{i}-\Delta F_{z}+Fz_{zero_{i}}). \tag{9}\]  
The effect due to the inertia resistance of the axles is equal to:  
\[F_{inertia_{i}}=\frac{I_{w}\Omega_{i}}{R_{r_{i}}}. \tag{10}\]  
The longitudinal interaction forces can be estimated starting from the information regarding the velocity estimated at the vehicle centre of gravity, acquirable by means of specific sensors or employing a model-based technique, taking into account the vehicle kinematics and the vertical load estimated at each wheel hub [50]. Therefore, in order to obtain the axle forces, the kinematic and load vehicle state estimator provides the accurate vehicle speed \(v_{x}\), the longitudinal acceleration \(a_{x}\), the wheel speed \(\Omega\), the inclination angle \((IA)\) and the normal load \(F_{z}\). Figure 2: Friction Estimator: Vehicle model and reference system in which the \(\mathbf{z}\) axis is perpendicular to the road equivalent plane \(\mathbf{xy}\).