# On-Board Road Friction Estimation Technique for Autonomous Driving Vehicle-Following Maneuvers
## 5 Design of the Road-Grip Aware Control Modules
### Anti-Lock Braking System  
Once an emergency braking is commanded from the AEB, the ABS has to drive the brake system preventing wheels from locking during the hard braking maneuver. Here we propose a Sliding Mode (SMC) ABS controller that leverages the on-line estimation of the road-grip in order to provide a safe braking automatic maneuver for a vehicle-following process also in the presence of hard rainy or icy pavement. This choice is due to SMC's enhanced stability performances with respect to classical control architectures [58] (e.g., proportional action). In particular it can be shown that matched disturbances (uncertainties entering the system through the same channel as the control) are rejected, at least below the actuation limits, moreover due to the controller nonlinear nature, larger stability margins can be achieved.  
First, let us define a control-oriented model, i.e., the quarter car model, in which we neglect the lateral and yaw motion of the wheel, thus obtaining a model dealing with the wheel rotational dynamics and longitudinal vehicle dynamics. The rotational dynamics of the wheel is described by  
\[I_{w}\dot{\Omega}=-T_{b}-R_{r}F_{x}, \tag{61}\]  
where \(I_{w}\) is the moment of inertia about the wheel axis of rotation, \(\Omega\) is the angular velocity, \(T_{b}\) is the braking torque, \(R_{r}\) is the wheel rolling radius and \(F_{l}\) is the force produced by the friction reaction. The longitudinal vehicle dynamics are simply modeled as  
\[m\dot{v}=-F_{x}. \tag{62}\]  
where \(m\) is the vehicle mass.  
The control goal is to yield \(\lambda\) to a reference value \(\lambda^{*}\) during braking [59]. To this aim we define the following sliding surface  
\[\sigma(t)=\lambda(t)-\lambda^{*}, \tag{63}\]  
where \(\lambda^{*}\) is the optimal slip obtained from the friction estimator (see Section 4) and \(\lambda\) is the longitudinal slip with dynamics as:  
\[\lambda=-\frac{1}{v}(\frac{1-\lambda}{m}+\frac{R_{r}^{2}}{I_{w}})F_{x}+\frac {R_{r}}{vI_{w}}\,T_{b}. \tag{64}\]  
Due to the inertia differences between wheel and vehicle, we can consider the velocity \(v\) as slowly varying, thus reducing Equation (64) to a single-input single-output system, where the control law can be defined as:  
\[u(t)=T_{b}=u_{c}(t)+u_{sw}(t), \tag{65}\]where \(u_{c}\) is the continuous term, or _equivalent control_[58], and \(u_{su}\) the discontinuous term. The equivalent control input is is responsible for keeping the trajectories on \(\sigma\), i.e.,  
\[\dot{\sigma}=0\Rightarrow u_{c}=(\frac{(1-\lambda)I_{w}}{mR_{r}}+R_{r})F_{z}u_{ x_{i,actual}}, \tag{66}\]  
where the force \(F_{x}=F_{z}\mu_{x_{i,actual}}\), where \(F_{z}\) is the tire vertical load, and \(\mu_{x_{i,actual}}\) the instantaneous friction value provided by the estimation module; note that the subscript \(i=[1,2]\) identifies front and rear tire depending from the axle involved (see Equation (17)).  
Closed-loop stability can be easily proven by considering the following Lyapunov function \(V(\lambda)=\frac{1}{2}\sigma^{2}\) and its derivative \(\dot{V}(\lambda)=\sigma\dot{\sigma}\). Substituting Equations (65) and (66) into the expression of \(\dot{V}\), we obtain  
\[\dot{V}(\lambda)=\sigma\dot{\sigma}=\sigma(\frac{R_{r}}{vI_{w}}u_{su}). \tag{67}\]  
Hence, selecting \(u_{su}=-\frac{vI_{w}}{R_{r}}\eta sgn(\sigma)\) it follows that  
\[\dot{V}(\lambda)=-\eta\sigma sgn(\sigma)=-\eta|\sigma|<0, \tag{68}\]  
where \(\eta>0\). In so doing, the surface \(\sigma\) is attractive and the closed-loop is asymptotically stable.  
Note that, in order to avoid the well-known chattering problem of sliding mode controllers, for its practical implementation the sign function in Equation (68) has been substituted by the hyperbolic tangent function. Furthermore, since controllability is lost when the vehicle speed is approaching zero (see Equation (64)), following a common practice for implementing the ABS, the controller is disabled at the very low velocities.