# On-Board Road Friction Estimation Technique for Autonomous Driving Vehicle-Following Maneuvers
## 5 Design of the Road-Grip Aware Control Modules
### Predictive ACC Design
\tag{28}\]  
However, in order to synthesize the MPC controller, Equation (26) are discretized with a fixed sample time \(T_{s}\) leveraging the zero-order-hold method, thus yielding:  
\[x(k+1) =Ax(k)+Bu(k)+Ww(k), \tag{29}\] \[y(k) =Cx(k)-Z, \tag{30}\]  
where, with an abuse of notation, the discrete-time system matrices have been labelled as the ones of the continuous-time model. Moreover, by augmenting the state vector as \([x(k)\ u(k)]^{T}\), we can resort to the following off-set free formulation as:  
\[x(k+1) = \begin{bmatrix}A&B\\ 0&1\end{bmatrix} x(k)+ \begin{bmatrix}B\\ 1\end{bmatrix} \Delta u(k)+ \begin{bmatrix}W\\ 0\end{bmatrix} w(k), \tag{31}\] \[y(k) = \begin{bmatrix}C&0_{4\times 1}\end{bmatrix} x(k)-Z. \tag{32}\]  
Note that the above formulation is really beneficial since the increment \((\Delta u(k)/T_{s})\) is the chassis jerk in discrete time, which is the crucial index for the driving comfort. The ACC problem in Equation (1) is solved when system in Equation (31) is regulated to the origin while fulfilling at the same time additional tracking, comfort, consumption and safety constraints for all times \(k\). In this perspective, the cost function embeds three different indexes, namely tracking capability, energy consumption and driving comfort. Tracking capability measures the performances in terms of distance and velocity errors, driving comfort is evaluated in terms of acceleration and jerk, finally the acceleration command is chosen as the performance index for energy consumption, yielding the following cost:  
\[J_{1}(y(k),\Delta u(k))=q_{1}|d(k)-d_{des}(k,\widehat{u})|^{2}+q_{2}|\Delta v (k)|^{2},+q_{3}|a(k)|^{2}+q_{4}|u(k)|^{2}+r|\Delta u(k)|^{2} \tag{33}\]  
which can be recasted in matrix form as:  
\[J(y(k),\Delta\mathcal{U})=\sum_{i=0}^{H_{p}-1}y_{k+i|k}^{T}Qy_{k+i|k}+\Delta u _{k+i}^{T}R\Delta u_{k+i}, \tag{34}\]  
where \(Q=\text{diag}\{q_{1},q_{2},q_{3},q_{4}\}\) is a positive definite diagonal matrix, \(R\in\mathbb{R}^{+}\), \(H_{p}\) and \(H_{c}\) are the prediction and control horizon, respectively, \(\Delta\mathcal{U}\ \overset{\Delta}{=}\ [\Delta u_{k},\dots,\Delta u_{k+H_{c}-1}]\) is the control sequence during the horizon \(H_{c}\), while \(y_{k+i|k}\) is the predicted output vector at time \(k+i\), obtained by applying the input sequence starting from the state \(x(k)\). With respect to safety, road-grip constraints are introduced for the desired and actual acceleration, as:  
\[u_{min}(\widehat{\mu})\leq u(k)\leq u_{max}(\widehat{\mu}), \tag{35}\] \[a_{min}(\widehat{\mu})\leq a(k)\leq a_{max}(\widehat{\mu}), \tag{36}\]  
with \(a_{max}(\widehat{\mu})=u_{max}(\widehat{\mu})=\min(2,\widehat{\mu}g)\), \(a_{min}(\widehat{\mu})=u_{min}(\widehat{\mu})=\max(-4,-\widehat{\mu}g)\), where \(\widehat{\mu}\) is the estimated maximum available grip and \(g\) the acceleration of gravity. The saturation values ( i.e., 2 and \(-4\)) are chosen as upper and lower limit, related to the ideal value for the grip set as 1 [28]. The constraints on the spacing and the maximum velocity are given as:  
\[d_{min}\leq d(k) \leq d_{max}, \tag{37}\] \[v(k) \leq v_{max}, \tag{38}\]  
where \(d_{min}\) is set to the standstill value \(d_{0}\) (see Equation (3)) and \(v_{max}\) is the maximum admissible speed depending on the legal requirements on the specific traveled road (urban, extra-urban, etc.). Note that this information can be acquired from a map-based on board service leveraging the GPS (Global Positioning System).