## 2 Materials and Methods
### Motion and Dynamics Model of a Set of Vehicles  
Considered (Figure 2) is the translational motion of rigid bodies illustrating set elements: vehicle A (tractor) and vehicle B (semitrailer) in the vertical plane (O\(\chi\)z coordinate system related to the road, horizontal \(x\)-axis, vertical \(z\)-axis), on a horizontal, even road surface. Alternatively, the vehicles set can be a truck with a trailer). The model assumes a rectilinear motion of the set along the \(x\)-axis. The movement of the masses of the set in the \(z\) direction is omitted, so the deflections of the suspension and tyres are not considered. The bodies of the set are modelled as a system of concentrated masses connected with weightless beams. They are subject to a flat system of forces. The motion resistances of the air drag force of the towing and towed vehicle (omitting wind speed) and rolling resistance forces of vehicle wheels are considered. The coupling model assumes the interaction of forces in two directions (\(x,z\)). The relative movements of vehicles are omitted (rigid articulated connection). Multiple (complex) axles are described with a single equivalent axle where the reactions acting on a single axle are  
Figure 1: Functional diagram of the proposed research method. replaced by a sum of forces concentrated in the centre of the equivalent axle. The equivalent axle's position corresponds to the axles set geometric centre. The quasistatic equilibrium equations for vehicle A have the following form:  
\[\sum F_{x}\ =\ -T_{A1}-T_{A2}+G_{A}\cdot\gamma_{h}+F_{sA}-F_{PA}-F_{TA1}-F_{ TA2}\ =\ 0 \tag{1}\]  
\[\sum F_{z}\ =\ R_{A1}+R_{A2}-G_{A}-R_{sA}\ =\ 0 \tag{2}\]  
\[\sum M_{1}\ =\ G_{A}(L_{A}-b_{A})-G_{A}\cdot\gamma_{h}\cdot h_{A}-R_{A2}\cdot L_{A} +R_{sA}(L_{A}-c_{A})-F_{sA}\cdot h_{s}+F_{PA}\cdot h_{PA}\ =\ 0 \tag{3}\]  
and for vehicle B:  
\[\sum F_{x}\ =\ -T_{B2}+G_{B}\cdot\gamma_{h}-F_{sB}-F_{PB}-F_{TB2}\ =\ 0 \tag{4}\]  
\[\sum F_{z}\ =\ R_{B2}-G_{B}+R_{sB}\ =\ 0 \tag{5}\]  
\[\sum M_{2}\ =\ -G_{B}\cdot\gamma_{h}\cdot h_{B}-G_{B}\cdot b_{B}+R_{sB}\cdot c _{B}+F_{sB}\cdot h_{s}+F_{PB}\cdot h_{PB}\ =\ 0 \tag{6}\]  
The symbols used in Figure 2 and the above formulas mean: _SM_--vehicle centre of mass, _L_--wheelbase, _h_--height of the vehicle's centre of mass above the ground, _b_--distance of the vehicle's centre of mass from the vertical plane passing through the rear axle, _h_s--height of the coupling (fifth-wheel) position above the ground, _c_--distance of the coupling position from the vertical plane passing through the rear axle, _h_p--height above the ground of the resultant force of air drag. Symbols \(T\) denote braking forces, _G_--vehicle weight, _R_--normal ground reactions, and _F_--forces of resistance to motion: _F_--air drag force, _F_--rolling resistance forces. Index \(A\) indicates the values for the towing vehicle (_A_), and index _B_--for the towed vehicle (_B_). Indexes 1, 2--consecutive axles of the vehicle (front and rear, respectively). The quantity \(\gamma_{h}\) appearing in the dependencies is the braking intensity (braking ratio) ([26, 27, 28, 29, 51]):  
\[\gamma_{h}\ =\ \frac{-(m_{A}+m_{B})\cdot\ddot{x}_{s}}{G_{A}+G_{B}}\ =\ \frac{-\ddot{x}_{s}}{g}\ =\ \frac{-\dot{v}_{s}}{g} \tag{7}\]  
where: \(m_{A/B}\)--mass of vehicle _A_/_B_, _v_--vehicle velocity, _g_--acceleration of gravity, \(\ddot{x}_{s}\)--vehicle acceleration. Figure 2: Physical model: (**a**) Forces acting on vehicle A (tractor); (**b**) Forces acting on vehicle B (semitrailer). The equilibrium conditions for the coupling of \(A\) and \(B\) vehicles can be written as follows:  
\[F_{sA}\ =\ F_{sB}\ =\ G_{B}\cdot\gamma_{h}-T_{B2}-F_{PB}-F_{TB2} \tag{8}\]  
\[\sum F_{x}\ =\ -T_{A1}-T_{A2}+G_{A}\cdot\gamma_{h}+F_{sA}-F_{PA}-F_{TA1}-F_{ TA2}\ =\ 0 \tag{1}\]0  
Dependencies (8) and (9) combine Equations (1)-(3), (5) and (6), i.e., the tractor and semitrailer model.