## 4 Results and Discussion
### The 1 LCM for the Vehicle with No Driver Assistance System, for the Heterogeneous Road Surface  
Lane change (1 LCM) simulation results without a driver assistance system on a heterogeneous road surface have been presented in an analogous form in Figures 5-7. They are time histories of steering wheel angle \(\alpha_{\rm k}\), lateral displacement of vehicle's centre of mass (y\({}_{\rm Oc}\)), and time derivative of the displacement (y\({}_{\rm Oc}\)Prim \(\equiv\) dy\({}_{\rm Oc}\)/dt), obtained for three vehicle speeds (50, 90, and 140 km/h). In this case, as previously mentioned, the steering wheel input applied to carry out the maneuver was adopted as identical to that used for the homogeneous road surface and kept unchanged as chosen for the surface of the lane where the maneuver was started.  
Figure 5: Lane change (1 LCM) simulation results. Change in the road surface condition: (**a**) the Dry\(\rightarrow\)Wet type, i.e., from dry concrete (right lane) to wet concrete (left lane); (**b**) the Wet\(\rightarrow\)Dry type, i.e., from wet concrete (right lane) to dry concrete (left lane).  
The change in the road surface condition encountered when the vehicle changed the right lane to the left one significantly affected the time histories of the quantities represented in the graphs, i.e., lateral displacement and lateral velocity of vehicle's centre of mass (\(y_{\text{oc}}\) and \(\text{dy}_{\text{oc}}\)/dt, respectively, the latter being denoted in the drawings as \(y_{\text{oc}}\)Prim). Due to the use of the same form of the steering wheel input as it was in the case of the homogeneous road surface (based only on the right lane), the intended lateral displacement of the vehicle of 3.5 m could not be achieved. For the differences to be better illustrated, the trajectories of vehicle's centre of mass plotted for 6 variants of the maneuver carried out on heterogeneous road surfaces have been compared in Figure 8 with the trajectories obtained for the homogeneous road surfaces.  
Figure 6: Lane change (1 LCM) simulation results. 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).  
Figure 7: _Cont._  
The differences between the lateral vehicle's position relative to the line y = 3.5 m (i.e., the centreline of the left lane) reached (at the end of the maneuver) c.a. +2 m and -3 m, which indicates a great hazard, including even the risk of lane departure. Having noticed such a situation (excessive deviation from the desired trajectory), the driver should correct the input (after the necessary reaction time) in order to implement the lane change as assumed. Due to the typical reaction times of average drivers (1 s or even more [10; 40; 41; 42; 43]), it may be too late for any driver's defensive action to be effective. However, we can imagine the use of an assistance system that could support or replace the driver in such situations. This will be the subject of the subsequent part of this article.  
Figure 8: Comparison of the trajectories of vehicle’s centre of mass observed for the 1 LCM carried out on homogeneous and \(\mu\)-split road surfaces: (**a**) Dry and Wet; (**b**) Dry and Ice; (**c**) Wet and Ice.  
Figure 7: Lane change (1 LCM) simulation results. Change in the road surface condition: (**a**) the Wet\(\rightarrow\)Ice type, i.e., from wet concrete (right lane) to ice (left lane); (**b**) the Ice\(\rightarrow\)Wet type, i.e., from ice (right lane) to wet concrete (left lane).