## 5 Conclusions  
A change in the condition of the road surface on which the vehicle is moving during the lane change maneuver (from the right lane to the left one) significantly affects the time histories of the lateral displacement of vehicle's centre of mass and of the yaw angle as well as of their time derivatives, i.e., velocities. If the input applied to the steering wheel is in the same form as it would be in the case of the homogeneous road surface (i.e., where the left lane surface is identical to that of the right one), then it will be impossible to achieve the intended lateral displacement and the yaw angle of the vehicle. Having noticed such a situation, the driver should correct the input (after the necessary reaction time) in order to implement the lane change as assumed. Unfortunately, in many cases of the vehicle motion under analysis, it is too late (due to the driver reaction time of 1 s or more) to eliminate the threat that has emerged in the form of large differences in vehicle's position and yaw angle from those expected (intended). This leads to a very high risk of an accident resulting from vehicle's departure from the occupied lane or even the road. The proposed assistance system based on a PID controller correctly implements the motion along the assumed trajectory, treated as a reference (a pre-set one) on a homogeneous surface. It has been observed, however, that the course of the vehicle trajectory (transformed to the time domain) 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 showing the yaw angle and the time history of the controlled quantity, i.e., the steering wheel angle. Moreover, a slight inaccuracy in the course of vehicle's lateral position can be seen for the ice-covered road surface. 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. The results presented indicate that the assistance system definitely improves vehicle's behaviour in comparison with that observed during the maneuver carried out on a heterogeneous road surface and controlled by the open system with a sinusoidal input applied to the steering wheel. The vehicle remains within the second lane. In this work, the components of the control system have been indicated that have an impact on the error of lateral vehicle's displacement. The said error can be reduced, but the reduction may be accompanied by greater variability of the course of the steering wheel angle and oscillations of vehicle's yaw angle. Such functioning of the assistance system may cause some anxiety for the driver. This possible negative aspect should be tested when checking prototype devices or during driving simulator tests. It is worth emphasizing that the assistance system proposed in this article fulfils its tasks even for the motion with a very high speed (140 km/h) on a homogeneous road surface (even if ice-covered) as well as for the case where the tyre-road adhesion is changing during a lane change maneuver. Thanks to this, it is possible to avoid the risk of an accident, which (as shown) is real when the vehicle is driven in a routine way. The presented results show that traditional, well-known (and tested in many real-world applications) PID controllers are enough effective compared with newer methods (e.g., MPS--Model Predictive Control or SMC--Sliding Mode Control). The authors claim that for reproducibility of the presented work, the most important are the described method, simulation model, block diagram of the proposed lane change assistant system (Figure 10), and description of the way how to determine the controller parameters (see Section 4.4.2).