## 2 Materials and Methods
### Extending the Servo-Control Model: A PD Steering Controller with Prediction  
The main common disadvantage of both PD controllers is lagging. It results in an even worse effect on the slippery road. In addition, many real-world physics effects are not taken into account in the controller equations. In practice, these disadvantages lead to the late entrance to the turn, attempts to return to desired trajectory, oversteering and disadvantageous position at the start of second turn, exiting, which could only result in consecutive course deviations, as demonstrated in Figure 6.  
While the second disadvantage was already addressed in previous research [6], we did not manage to find sources regarding the first one.  
The vehicle position could be predicted using current velocity (Figure 7; Figure 8):  
\[\delta=k_{1}e_{predicted}+k_{2}\theta \tag{5}\]  
where \(\theta\) is the angle between the car direction and the road, and \(e_{predicted}\) is the distance between the predicted car center and the road calculated by the following Equations (5)-(9):  
\[e_{predicted}=\;F\big{(}x_{predicted}\;,\;y_{predicted}\big{)} \tag{6}\]  
Figure 6: Tracking of the center of the car on the track during the race with friction 0.1 and 1.05 \(V_{CR}\). In the highest point of the trajectory, a crash was occurring.  
\[x_{predicted}=\ x_{0}+V_{x}t=\ x_{0}+Vt\ \cos\alpha \tag{7}\] \[y_{predicted}= y_{0}+V_{y}t=\ y_{0}+Vt\ \sin\alpha\] (8) \[\alpha\ =\ \theta+A_{road} \tag{9}\]  
Here, \(x_{predicted}\) and \(y_{predicted}\) are predicted coordinates of the position of the car, \(A_{road}\) is the angle of the road at the point that is closest to the car, F is the function calculating the distance between the car and the center of the lane, \(V,\ V_{x}\), is the speed of the car, and its two orthogonal components, respectively, \(t\). is the predicting time interval, and \(\alpha\) is the angular deviation of the car from the center of the lane.  
This controller we called Predictive PD controller (PPD controller).  
It seems natural to also predict the yaw angle. However, it was not performed in our experiments. The reasons for this are not only atomicity of changes that facilitate development and analysis, but the aspiration to achieve human-like behavior of professional and experienced drivers. In contrast to the canonic controller, drivers often use the future position prediction for steering. According to the research [12], having to deal with a lot of information, the human brain experiences cognitive pressure with high speed, which results in a decrease of field of view. In such conditions, it has to operate only the prediction of the position, not the yaw angle. On the other hand, such limitations of human perception do not limit researches in the development of probably more precise models, which predicts both parameters. Nevertheless, such investigations are left for further research and, here, we were curious about how much a change of the value of one of the parameters will affect the quality of driving. We elaborate on this in the discussion section of the paper.  
Figure 8: Real trajectory of the car (**a**) and predicted position of the car (**b**). Time of the prediction—1.5 s, \(\mu=0.1\).  
Figure 7: Predicting the lateral deviation of the car \(\mathrm{e_{pred}}\).