## 2 Mathematical Model of the Experimental ABS Laboratory Setup
### Mathematical Model of the Experimental ABS Laboratory Setup  
The mathematical model of the ABS experimental setup was developed under the assumptions of imperceptible lateral and vertical dynamics and negligible rolling resistance force with respect to braking. The dynamic equations of the experimental setup were as follows [24; 27; 28]:  
\[ \begin{split}\dot{\omega}_{1}&=\frac{r_{1}}{J_{1}} F_{I}-\frac{1}{J_{1}}\left(d_{1}\omega_{1}+T_{b}\right)\\ \dot{\omega}_{2}&=-\frac{r_{2}}{J_{2}}F_{I}-\frac{1 }{J_{2}}d_{2}\omega_{2}\end{split}  \tag{1}\]  
where \(\omega_{1}\) and \(\omega_{2}\) represented the angular velocities of the upper and lower wheels, respectively; \(J_{1}\) and \(J_{2}\) were inertia moments; and \(r_{1}\) and \(r_{2}\) were the radii. Moreover, \(d_{1}\) and \(d_{2}\) were the coefficients of the viscous friction of the upper and lower wheel, respectively (the nominal parameters are given in Table 1). During the acceleration and braking processes, a longitudinal tire force \(F_{I}\) is achieved during contact between the upper and lower wheels (see Figure 1). This traction force is proportional to the usual load of the vehicle via the tire-road friction coefficient \(\mu\in[0,1]\) and is a nonlinear function of the longitudinal wheel-slip ratio, i.e., the relative velocity difference:  
\[\lambda=\frac{v_{x}-v_{w}}{v_{x}}=\frac{r_{2}\omega_{2}-r_{1}\omega_{1}}{r_{2} \omega_{2}}=1-\frac{r_{1}}{r_{2}}\frac{\omega_{1}}{\omega_{2}} \tag{2}\]  
Figure 1: The Inteco experimental setup and its scheme. where the quantities \(v_{x}=r_{2}\omega_{2}\) and \(v_{w}=r_{1}\omega_{1}\) represent the vehicle longitudinal velocity and the vehicle wheel velocity, respectively. According to [24], \(T_{b}\) is the braking torque and is designed by using a first-order equation:  
\[\dot{T}_{b}=c\Big{(}-T_{b}+b(u)\Big{)} \tag{3}\]  
where \(c>0\) is a constant and \(b(u)\) is the control input that expresses the relationship between \(u\) and the voltage applied to the coupled DC motor (which drives the action of the brake pads). This relation can be calculated using an equation comparable with the brake-pedal model in a vehicle [25; 34; 38]:  
\[b(u)=\left\{ \begin{array}{ll}b_{1}u-b_{0}&\text{if }u\geq u_{0}\\ 0&\text{if }u<u_{0}\end{array} \right.\]  
where \(u_{0}\) is the operating threshold of the brake-driving system and \(b_{1}\) and \(b_{0}\) are constants. In this case, the control signal \(u\) is the voltage applied to the small DC motor. This motor commands brake pads located in the upper wheel. Considering the constant steady-state value \(T_{b}\) in (3), the control signal \(u\) can be obtained by solving for \(u=(T_{b}+b_{0})/b_{1}\). Additionally, to design the traction force \(F_{t}\) in the linear and nonlinear region, in this paper, the widely used [PERSON]'s magic formula [39] model was selected. This method approaches the response curve of the braking process based on experimental data. Then, the traction force is defined as follows:  
\[F_{t}=\mu D\sin\Big{(}C\arctan(B\lambda)\Big{)} \tag{4}\]  
where \(\mu\) is the friction coefficient between the two wheels and simulates road conditions and the coefficients \(B\), \(C\), and \(D\) are determined experimentally. Hence, considering (3) and the definition (4), the dynamic equations of the ABS laboratory setup (1) can be rewritten in the following form:  
\[\dot{\omega}_{1} =\frac{r_{1}}{f_{1}}F_{t}-\frac{1}{f_{1}}\big{(}d_{1}\omega_{1}+T _{b}\big{)} \tag{5}\] \[\dot{\omega}_{2} =-\frac{r_{2}}{f_{2}}F_{t}-\frac{1}{f_{2}}d_{2}\omega_{2}\] \[\dot{T}_{b} =c(-T_{b}+b(u)).\]