## 2 Materials and Methods
### Braking System Model and ABS Model  
When modelling the braking system, the following were taken into account: braking axes (brake cylinders, actuating mechanism), averaged response time on pneumatic lines, trailer brake valve with adjustable predominance, LSV valve (Load Sensing Valve), relay valve, pressure reduction valve and pressure limiting valve. Using the dependencies contained in [44], the braking torque of the wheel depending on the pressure at the entrance to the brake cylinder, we can write:  
\[M_{H}\ =\ k\cdot\Big{(}\big{(}A_{s}\cdot p_{(s)}-B_{s}\big{)}\cdot l-C_{0} \Big{)}\cdot\eta\cdot BF \tag{25}\]  
where: \(A_{s}\), \(B_{s}\)--coefficients describing the equation of force on the brake cylinder piston rod depending on the pressure, \(p_{(s)}\)--pressure at the input to the brake cylinder, \(k\)--number of brake cylinders on the axis, \(l\)--effective length of the brake lever, \(C_{0}\)--minimum brake input torque, \(\eta\)--the efficiency of the brake mechanism, \(BF\)--Brake Factor (the ratio of the braking torque to the torque generated at the end of the actuator lever [44]). The above parameters needed to calculate the braking torque of a wheel can be determined using experimental methods [67; 68] or catalogue data [69; 70]. A correctly designed braking system should ensure optimal distribution of braking forces for an unladen and laden vehicle. It guarantees braking performance (short braking distance) and directional stability during braking, which results in meeting the requirements of ECE regulations on braking systems [29]. If the constant distribution of braking forces does not meet these requirements for both loading conditions, it is necessary to install devices regulating the distribution of braking forces between the different axles of the vehicles set depending on the load condition. The problem of the distribution of braking forces in the set considered in the study will be presented in Section 3.2. The basic formula describing the relationship between the output value (pressure at the actuators of the braking axles) and the input value (the position angle of the main control valve adjuster \(\alpha\) or pressure in the control coupling head \(p_{m}\)) in a pneumatic brake system can be written as:  
\[p_{(s)}\ =\ i_{h1}\cdot(i_{h2}\cdot(\dots-(i_{zh}\cdot(\alpha-\alpha_{0})- \dots)-p_{02})-p_{01})\cdot i_{r} \tag{26}\]  
where: \(i_{zh}\)--slope of the linear characteristic of the main control valve, \(i_{h1}\), \(i_{h2}\), \(\dots\) --slope of the linear characteristic of the other valves (relay valves, reduction valve), \(i_{r}\)--coefficient describing the ratio of the output value to the input value pressure reducer, \(\alpha_{0}\)--minimum angle at which the brakes are actuated ('idle stroke'), \(p_{01}\), \(p_{02}\), \(\dots\) --minimum pressure values needed to actuate the valve (positive value) or pressure values for valve with adjustable predominance (negative value). The above Equation (26) is an example and depends on the model of a specific braking system. This notation makes it easier to add additional valves or not to include them (by replacing \(i_{hr}=1\) and \(p_{0x}=0\)). In addition, it is possible to change the order of the valves without interfering with the mathematical record. The complete model of the pneumatic braking system used for the presented calculations is more complex. It includes many conditional functions, e.g., maximum pressure (brake system maximum working pressure, axle pressure limiters), inability to obtain negative pressures or pressures greater than the supply pressure, which generates a linear mathematical record of pneumatic elements. Also, such functions are needed to describe valve with adjustable predominance, valves with linear-broken characteristics or lifting axle control valve. Additionally, the time delay of pressure \(p_{(s)}\) and \(p_{m}\) on pneumatic lines is simulated by shifting the pressure increase with time.