## 2 Materials and Methods  
### The Vehicle Under Test and Its Representative Model  
The vehicle used for the tests was a KIA Ceed SW motor car (Kia Corporation, Zilina, Slovakia). The simulation model representing this vehicle (Figure 1), also shown in [18; 19; 20], consists of nine mass elements, i.e., car body solid, four material points O\({}_{1}\), O\({}_{2}\), O\({}_{3}\), and O\({}_{4}\) (assumed as the points of concentration of car's \"unsprung masses\"), and four rotating solids (wheels). The following coordinate systems have been adopted (see also works [21; 22; 23; 24] with a similar approach to the subject):  
* Oxyz--a system attached to the road (horizontal axes Ox and Oy and vertical axis Oz pointing upwards);
* OCxCyCycZ--a system with axes parallel to the Ox, Oy, and Oz axes (respectively), with its origin situated at the car body centre of mass Oc;
* coordinate systems fixed to model's rigid bodies, i.e., body solid (O\({}_{\mathbb{C}}\)_k_C_NC_C_C_) and four road wheels (O\({}_{1}\xi_{1}\eta_{1}\zeta_{1}\), O\({}_{2}\xi_{2}\eta_{2}\zeta_{2}\), O\({}_{3}\xi_{3}\eta_{3}\zeta_{3}\), O\({}_{4}\xi_{4}\eta_{4}\zeta_{4}\));
* extra systems that define the transformation matrices.  
The translational motion is described using the positions of the centres of mass (O\({}_{\mathbb{C}}\), O\({}_{1}\), O\({}_{2}\), O\({}_{3}\), O\({}_{4}\)) of the said solids. The axes O\({}_{1}\xi_{k}\), O\({}_{1}\eta_{1}\), O\({}_{1}\zeta_{i}\) (i = C, 1, 2, 3, 4) are treated as the principal central axes of inertia of the corresponding rigid bodies.  
To describe the spherical motion of the car body solid around the O\({}_{\mathbb{C}}\) centre, the \"air-craft angles\", also referred to as \"quasi-Euler angles\", have been used (see also works [21; 22; 23; 24] with a similar approach to the subject):  
* yaw angle \(\psi_{\mathbb{C}}\) (rotation around the O\({}_{\mathbb{C}}\)_k_C_ axis);
* pitch angle \(\psi_{\mathbb{C}}\) (rotation around the O\({}_{\mathbb{C}}\)_k_C_ axis);
* roll angle \(\vartheta_{\mathbb{C}}\) (rotation around the O\({}_{\mathbb{C}}\)_k_C_ axis).  
The purpose of introducing the \(\rm{O_{C}xCyCzC}\) system was to show body's c.g. and the reference system that enables the definition of body angular motion coordinates (\"quasi-Euler angles\"): \(\psi_{C}\), \(\psi_{C}\), \(\psi_{C}\).  
The axes of rotations are treated as the principal central axes of inertia. The Lagrange equations of the second kind have been applied to derive equations of motion. Fourteen generalized coordinates were adopted (\(\rm{q_{i}}\); i = 1, 2,...14):  
\(\rm{q_{1}}=x_{OC}\), \(\rm{q_{2}}=y_{OC}\), \(\rm{q_{3}}=z_{OC}\)--the position of the centre of mass of the car body solid (\(\rm{O_{C}}\)) in the inertial reference system Oxyz;  
\(\rm{q_{4}}=\psi_{C}\), \(\rm{q_{5}}=\varphi_{C}\), \(\rm{q_{6}}=\emptyset_{C}\)--the spherical motion of the car body solid around its centre of mass \(\rm{O_{C}}\) (yaw, pitch, and roll angle, respectively);  
\(\rm{q_{7}}=\zeta_{C01}\), \(\rm{q_{8}}=\zeta_{C02}\), \(\rm{q_{9}}=\zeta_{C03}\), \(\rm{q_{10}}=\zeta_{C04}\)--the motion of points \(\rm{O_{1}}\), \(\rm{O_{2}}\), \(\rm{O_{3}}\), \(\rm{O_{4}}\) relative to the car body solid in the direction of axis \(\rm{O_{C}}\zeta_{C}\) of the \(\rm{O_{C}}\xi_{C}\eta_{C}\zeta_{C}\) coordinate system;  
\(\rm{q_{11}}=\varphi_{1}\), \(\rm{q_{12}}=\varphi_{2}\), \(\rm{q_{13}}=\varphi_{3}\), \(\rm{q_{14}}=\varphi_{4}\)--wheel angles of rotation.  
The model describes steering system flexibility and the steered wheels parameters: camber, caster, kingpin, and toe-in. The HSRI-UMTRI tyre-road contact model [25; 26] has been adopted together with the IPG-Tire model of transient states of tyres [23; 27]. The presented model has been successfully experimentally verified for typical vehicle maneuvers: steady-state cornering, step input on the steering wheel, double lane change, single lane change, straight-line braking [28; 29; 30; 31]. So, it also considered the single lane change maneuvers.