## Vehicle Dynamics  
### Vehicle Model  
The single-wheel model of the vehicle is shown in Figure 1a. The single-track (bicycle) model is introduced in Figure 1b. Table 1 introduces nomenclature for all variables used in these and other models mentioned in the paper. Single wheel dynamics can be expressed by the following equations:  
\[I_{ui}\cdot\dot{\omega}_{i}=T_{ti}-r_{i}\cdot F_{xi}-T_{bi} \tag{1}\]  
\[F_{xi}=m\cdot\dot{v}_{xi} \tag{2}\]  
\[F_{zi}=m\cdot g. \tag{3}\]  
Brake torque depends on the applied brake pressure:  
\[T_{bi}=r_{i}\cdot k_{b}\cdot p_{bi}, \tag{4}\]  
where \(k_{b}\) is the braking coefficient, which depends on the brake disc friction area, mechanical efficiency of the brake components, and the braking factor is the constant value. In this paper, the ABS controller output variable is the braking pressure for each wheel, \(p_{bi}\). Figure 1: Vehicle model schematic drawing: (**a**) single-wheel model; (**b**) single-track model. The wheel slip at braking is calculated as follows:  
\[\lambda_{i}=\frac{v_{v}-v_{wx,i}}{v_{v}}. \tag{5}\]  
The longitudinal wheel speed be can also simply calculated as:  
\[v_{wx,i}=r_{i}\cdot\omega_{i}. \tag{6}\]  
In reality the tire radius \(r_{i}\) is a dynamic variable. In our case we simplify the equation and consider it as a constant value as the change in radius dimension is negligibly small. The FLC design requires information about the friction-slip curves. Tire-road friction coefficient can be determined as follows:  
\[\mu_{x}(\lambda)=\frac{F_{x}}{F_{z}}. \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Annotation** \\ \hline \(\omega\) & Wheel angular speed 1  & rad/s \\ \(a_{xx}\) & Vehicle longitudinal acceleration 1  & m/s2 \\ \(a_{xy}\) & Vehicle lateral acceleration 1  & m/s2 \\ \(\psi\) & Yaw rate 1  & rad/s \\ \(\delta\) & Steering wheel angle 1  & rad \\ \(p_{b}\) & Braking pressure 1  & bar \\ \(r\) & Wheel radius & m \\ \(m\) & Mass of the vehicle & g \\ \(g\) & Gravitational acceleration & m/s2 \\ \(T_{b}\) & Braking torque & Nm \\ \(T_{t}\) & Traction torque & Nm \\ \(k_{b}\) & Braking coefficient & - \\ \(v_{vx}\) & Vehicle longitudinal velocity & m/s \\ \(v_{wx}\) & Wheel longitudinal velocity & m/s \\ \(\lambda_{i}\) & Wheel slip & \% \\ \(\mu\) & Tire–road friction coefficient (general) & - \\ \(\mu_{x}\) & Tire–road friction coefficient based on vehicle & - \\  & longitudinal acceleration & \\ \(l_{f}\) & Distance from the vehicle Center of Gravity & m \\ \(l_{r}\) & Distance from the vehicle COG to the rear axles & m \\ \(l_{z}\) & Yaw moment of inertia about z-axis & g\(\cdot\)m2 \\ \(F_{x}\) & Longitudinal force & N \\ \(F_{y}\) & Lateral force & N \\ \(F_{z}\) & Vertical force & N \\ \(P_{ABS}\) & Pressure generated for ABS braking & bar \\  & Pressure generated for the yaw rate regulation for & bar \\ \(PESP1\) & the left side wheels of the vehicle & \\  & Pressure generated for the yaw rate regulation for & bar \\  & the right side wheels of the vehicle & m \\ \(s\) & Distance & m \\ \(-a_{average}\) & Average deceleration & m/s2 \\ \(ABS_{lp}\) & ABS operation index of performance & - \\ \(\lambda_{average}\) & Average wheel slip value & \% \\ \(\omega_{p\text{+}p}\) & ABS adaptability peak-to-peak value & \% \\ \(i\) & Subscript for each wheel; \(i\)\(\in\) [FL, FR, RL, RR] 2 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters description. Using Equations (2) and (3), \(\mu_{x}\) can be estimated:  
\[\mu_{x}\approx\frac{\dot{v}_{x}}{g}. \tag{8}\]  
The 3 DOF single-track model, 0b, is required for the formulation of control inputs in the case of vehicle maneuvers with lateral dynamics. The model is described by the following system of equations:  
\[\left\{ \begin{array}{l}m\cdot a_{vx}=F_{xf}\cdot\cos\delta+F_{xr}-F_{yf}\cdot \sin\delta+m\cdot\dot{y}\cdot\psi\\ m\cdot a_{yy}=F_{xf}\cdot\sin\delta+F_{yf}\cdot\cos\delta+F_{yr}-m\cdot\dot{x} \cdot\psi\\ \dot{\psi}\cdot I_{z}=l_{f}\left(F_{xf}\cdot\sin\delta+F_{yf}\cdot\cos\delta \right)-l_{r}\cdot F_{yr}\end{array}