## 4 Results and Discussion
### The 1 LCM for the Vehicle with a Driver Assistance System
#### 4.4.2 The Steering Wheel Angle Control System Adopted for the Lane Change Maneuver
All the other test parameters were as before, i.e., maneuver duration time \(T_{w}=3\) s, anticipated final lateral vehicle displacement H = 3.5 m, the maneuver began at the instant of t = 1 s, and the vehicle speed was V = 90 km/h = 25 m/s. The curves in the graphs represent steering wheel angle \(\alpha_{k}\)(t), lateral displacement of vehicle's centre of mass y\(\mathrm{{}_{OC}}\)(t), and yaw angle \(\psi_{C}\)(t) and show the reference course (Reference), the result obtained in the maneuver controlled by the open system with a sinusoidal input and carried out on a homogeneous dry concrete road surface (Sine Dry), and the result of operation of the assistance system under consideration on the heterogeneous road surface (Assist. S.). Figure 12: Results of the simulation of the vehicle motion (steering wheel angle \(\alpha_{k}\); lateral displacement of vehicle’s centre of mass y\(\mathrm{{}_{OC}}\) and yaw angle \(\psi_{C}\)) during the 1 LCM where an assistance system was used for a change in the road surface condition: (**a**) the Dry\(\rightarrow\)Ice type, i.e., from dry concrete (right lane) to ice (left lane); (**b**) the Ice\(\rightarrow\)Dry type, i.e., from ice (right lane) to dry concrete (left lane). Vehicle speed: V = 90 km/h. It can be seen that the vehicle response is significantly delayed when the vehicle is on the ice-covered surface at the start of the maneuver (similar vehicle behaviour can be observed when the vehicle is moving without the assistance system, especially at high speeds--see Figures 4c, 6b and 7b). The results presented indicate that the assistance system definitely improves vehicles' behaviour compared to that observed during the maneuver carried out on a \(\upmu\)-split road surface and controlled by the open system with a sinusoidal input applied to the steering wheel. The vehicle remains within the target lane, although the actual final lateral displacement values differ from that having been preset (the difference is about 0.4 m, both positive and negative), and the vehicle moves parallel to the road direction (Ox). The error of the lateral vehicle's position can be reduced by lowering the proportional control gain \(\text{K}_{\text{p}\text{p}}\) (for the yaw angle). However, this would raise the variability of the steering wheel angle and oscillations of the yaw angle \(\text{\psi}_{\text{C}}\)(t). The oscillations, in turn, can be reduced by extending the lane change time. For vehicle speeds of V = 90 km/h and higher, the lane change time (the \(\text{T}_{\text{w}}\) period) was extended to a value of 4 s (without changing the values of \(\text{K}_{\text{py}}\) and \(\text{K}_{\text{p}\text{u}}\)). Such an action improves the conditions of operation of the assistance system and lowers the intensity of turning the steering wheel and the values of lateral acceleration and yaw velocity. The effects of these actions have been presented in Figure 13, which corresponds to Figure 12. Thus, the functioning of the assistance system is markedly improved, and the error of the lateral vehicle's position is decreased. Figure 13: Results of the simulation of the vehicle motion (steering wheel angle \(\alpha_{k}\); lateral displacement of vehicle’s centre of mass \(\text{y}_{\text{Oc}}\) and yaw angle \(\text{\psi}_{\text{C}}\)) during the 1 LCM where an assistance system was used for a change in the road surface condition: (**a**) the Dry\(\rightarrow\)Ice type, i.e., from dry concrete (right lane) to ice (left lane); (**b**) the Ice\(\rightarrow\)Dry type, i.e., from ice (right lane) to dry concrete (left lane). Vehicle speed: V = 90 km/h. Maneuver duration time \(\text{T}_{\text{w}}=4\) s.  
It was decided to check whether the assistant system proposed would work correctly for the most difficult case of those considered (the lane change at a speed of V = 140 km on the \"u-split\" road surface, Dry\(\rightarrow\)Ice and Ice\(\rightarrow\)Dry lane change combinations).