## 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
The real exemplification of the system proposed will require the use of a few protective elements to limit the values of y\({}_{\rm Oc}\)(t) and \(\psi_{\rm C}\)(t) and, in consequence, of \(\alpha_{\rm k}\)(t) as well as of their first derivatives in respect to time. 4.3 The 1 LCM for the Vehicle with an Active Driver Assistance System, for the Homogeneous Road Surface  
Figure 11 shows results for the single lane change maneuver, where an assistance system was used in comparison with the results with a sinusoidal input (T\({}_{\rm w}\) = 3 s), on a homogeneous road surface. The final lateral displacement and the vehicle speed were pre-set as H = 3.5 m and V = 90 km/h = 25 m/s, respectively. The maneuver began at the instant of t = 1 s. The time histories of the input, i.e., steering wheel angle \(\alpha_{\rm k}\)(t), and of the response, i.e., lateral displacement of vehicle's centre of mass y\({}_{\rm Oc}\)(t) and yaw angle \(\psi_{\rm C}\)(t), have been shown. The curves represent the reference course (Reference), the result obtained in the maneuver controlled by the open system with a sinusoidal input (Sine), and the result of operation of the assistance system described herein (Assist. S.). After performing the maneuver, the car should move parallel to the lane centre line and a sketch of the car should not leave the lane. Knowing vehicle width gives limits for the vehicle's c.g. lateral displacement. These results indicate that the assistance system correctly executes the vehicle motion along the preset trajectory treated as a reference. However, the curve y\({}_{\rm Oc}\)(t) is shifted along the time axis with decreasing tyre-road adhesion coefficient (from dry concrete through wet concrete to ice). This indicates that on the road surfaces with less grip, the vehicle responds with some delay and a bit slower, which results from an increase in the tyre sideslip angles. This is reflected in the graphs that show the time histories of the yaw angle \(\psi_{\rm C}\)(t) and of the controlled quantity, i.e., the steering wheel angle \(\alpha_{\rm k}\)(t). Such a phenomenon could also be observed when comparing the curves in Figure 4 (Section 4.1), which represented the results obtained in the maneuver controlled by the open system with a sinusoidal input applied to the steering wheel, carried out on roads with less and less grip. Moreover, a small overshoot in the y\({}_{\rm Oc}\)(t) curve can be seen for the ice-covered road surface (Figure 11c). Nevertheless, the effect of operation of the assistance system may be considered satisfactory. The system can substitute for the driver in executing the maneuver planned. Figure 10: Schematic diagram of the proposed lane change assistance system for the system being ON and OFF (see the description in the text). Figure 11: Results of the simulation of the vehicle motion (steering wheel angle \(\alpha_{k}\); lateral displacement of vehicle’s centre of mass \(y_{\rm{OC}}\) and yaw angle \(\psi_{\rm{C}}\)) during the 11 CM where an assistance system was used. Vehicle speed: V = 90 km/h. Road surface: **(a)** Dry concrete; **(b)** Wet concrete; **(c)** Ice-covered. 4.4.4. The 1 LCM for the Vehicle with an Active Driver Assistance System, for the Heterogeneous Road Surface  
For the car provided with a lane change assistance system described herein, simulations were carried out that corresponded to those presented in Section 4.2 (Figures 5-8), i.e., for the \(\mu\) split road surfaces in combinations as specified in Table 1, and the effect of the operation of such a system was assessed. For illustration, the most spectacular cases were selected, including those that highlight the possible imperfections of the control type employed (the Dry\(\rightarrow\)Ice and Ice\(\rightarrow\)Dry cases). Figure 12 shows the results obtained for the maneuver with dry concrete surface of the right lane and ice-covered surface of the left lane (the Dry\(\rightarrow\)Ice case) and for the case with ice-covered right lane and dry concrete left lane (the Ice\(\rightarrow\)Dry case).