## 2 Description of the Assumption of the Driver Model
It will be different when the vehicle is moving on winding, hilly roads, at high traffic density, in dynamically changing traffic situations. Then, the distance to the target point is shortened. While driving, the observation point can \"change\" its position as the moving vehicle changes its position. It can therefore be assumed that when driving at a constant speed along a straight road section, the distance to the target point remains constant and the influence of the attraction field is also constant. When the road situation becomes dynamic (there are some disturbances, e.g., sudden obstacles), the observation distance \(L\) can also  
Figure 2: Scheme of interactions in the potential field. be subject to dynamic changes. Important elements affecting the driver included in the model are obstacles near the vehicle. The repelling potential \(U_{O}\), acting on the vehicle from an obstacle with coordinates (\(x_{o}\),\(y_{o}\)), depends on the scaling coefficient \(k_{O}\) and is inversely proportional to the distance of the obstacle from the vehicle \(p_{O}\) (Equation (6)):  
\[U_{O}(q)=\frac{1}{2}k_{o}\frac{1}{p_{O}^{n}(q)} \tag{6}\]  
where  
\(k_{O}\)--scaling coefficient for the potential repelling from obstacles;  
\(p_{O}(q)\)--a distance of the obstacle from point \(q\);  
\(n\)--exponent \(n=2\dots\) 4. Another important factor affecting the driver is the position of the vehicle (distance) in relation to the edges of the roadway \(p_{BL}\) and \(p_{BL}\). In this respect, the behaviour of the driver may also be influenced by the type of road edge and the type of roadside; hence, the \(k_{Bj}\) coefficient may be different. For a driver travelling on a roadway separated from a hard shoulder with a painted line, the influence of the edge will be lesser. In this case, the occurrence of a slight steering error by the driver, resulting in driving out of the road, does not cause any dangerous consequences. However, a lower roadside, barriers, a high kerb, or trees near the road will affect the driver otherwise. In such situations, driving a vehicle out of an assumed lane may result in serious damage to the vehicle or an accident. In this case, the scaling coefficient should therefore be higher. The value of the potential field \(U_{B}\), repelling from the edge of the roadway, acting on the point \(q\) of the vehicle, will be:  
\[U_{B}(q)=\frac{1}{2}k_{BL}\frac{1}{p_{BL}^{2}(q)}+\frac{1}{2}k_{BR}\frac{1}{p_ {BR}^{2}(q)} \tag{7}\]  
where  
\(k_{BL}\)--scaling coefficient for the repelling potential from the left edge of the road;  
\(k_{BR}\)--scaling coefficient for the repelling potential from the right edge of the road;  
\(p_{BL}(q)\)--a lateral distance of point q from the left edge of the road;  
\(p_{BR}(q)\)--a lateral distance of point q from the right edge of the road. A vehicle moving symmetrically in the middle of the right lane will be affected by potential hazards from the left and right edges of the road, and, despite the different distances from the edges, the values of these potentials will be identical in value but will differ in turn. The effect of these potentials, when moving in the middle of the right lane, should not generate the need to change the lateral position of the vehicle (vehicle movement should be stable). The values of the coefficients taking into account the potential values from the left and right edges of the road will also depend on the width of the road \(L\), because \(p_{BL}+p_{BR}=L\). The following relationship should be fulfilled:  
\[k_{BL}\frac{1}{p_{BL}^{2}(q)}=-k_{BR}\frac{1}{p_{BR}^{2}(q)} \tag{8}\]  
Therefore, if the value of the scaling coefficient \(k_{BR}\) is selected (identified), then the \(k_{BL}\) value must be dependent on the current distance of the car from the right edge of the roadway, according to the following Formula (9).