## 1 Introduction
This value is related to both parameters mentioned above, and the driver's actions in accident hazard situations (including response times) are functions of this value (Equation (1)). \(TTC\) (in seconds) is the ratio of the distance from an obstacle (m) to the vehicle speed (m/s)--Figure 1[66; 67]. \[TTC=\frac{S}{V} \tag{1}\]  
In the authors' earlier works, this time calculation was called \"Risk Time\" [68]. In the driver model, a segment called \"distance reserve\" \(y_{lat}\)--\(y_{v}\) was introduced into the equations. The relationship formulated in this way caused the dependence of the amount of deceleration \(b\) of the vehicle (during braking) on the change of its lateral position in relation to the lateral position of the obstacle. The increase in the deceleration was adopted as a variable; therefore, on the left side of the equation, a derivative of the deceleration \(\dot{b}\) was introduced. Finally, the braking model (Equation (2)) was supplemented with a segment related to the driver's reaction time for the braking manoeuvre \(BRT\)[69]. \[b(t)+W_{1}\!\cdot\!\dot{b}(t)=\left\{ \begin{matrix}0\\ W_{2}(y_{lat}(t)-y_{v}(t))+W_{3}\Big{(}\frac{1}{TTC}\Big{)}\end{matrix} \right. \text{ for }\left\{\begin{array}{c}t<BRT\\ t\geq BRT\end{matrix}\right. \tag{2}\]  
where  
\(W_{1}\), \(W_{2}\), \(W_{3}\)--braking model coefficients;  
\(BRT\)--braking reaction time (s);  
\(TTC\)--time to collision (s);  
\(y_{lat}\)--rate of penetration of an obstacle into the vehicle motion area (m);  
\(y_{v}\)--lateral position of the centre of the vehicle mass (m);  
\(b\)--deceleration (m/s\({}^{2}\)). The final variant of the driver model, in terms of steering manoeuvre, is described by Equation (3). The steering wheel angle \(\delta\) depended on the difference between the distance of the obstacle and the change in the lateral position of the vehicle \(y_{lat}\)--\(y_{v}\). The greater the angle, the \"calmer\" the driver's reaction. At the same time, a segment related to the  
Figure 1: Scheme of a road situation. derivative of the steering wheel angle \(\dot{\delta}\) and to the driver's reaction time for the steering manoeuvre, _SRT_[69], was introduced in the notation. \[\delta(t)+W_{4}\dot{\delta}(t)=\left\{ \begin{array}{l}0\\ W_{5}(y_{lat}(t)-y_{s}(t))\end{array} \right.\text{for }\left\{ \begin{array}{l}t< SRT\\ t\geq SRT\end{array} \right. \tag{3}\]  
where  
\(W_{4}\), \(W_{5}\)--steering model coefficients;  
_SRT_--steering reaction time (s);  
\(\delta\)--steering wheel angle (rad). This paper presented a proposition of the driver models related to the driver's actions in critical situations. The main purpose of this model is to reproduce the driver's actions in pre-accident situations. It can be used as an additional element in programs used to reconstruct road accidents. Currently, these programs simulate the way the vehicle moves, and in most of them, the impact of the driver's behaviour is not taken into account at all. The method of describing the mutual position of obstacles and the vehicle makes it rather easy to apply the proposed algorithm to them. This does not exclude the possibility of applying this model to solutions related to assistant systems in the future, but further work is still necessary for such situations, taking into account their complexity. The model parameters used must be determined in future studies for different runs. Such studies are planned for the future. The second chapter presents the most important assumptions of the model, while the third chapter presents the results of computer simulations of the avoiding manoeuvre for an exemplary road situation. It describes the driver's action algorithm based on the location of the vehicle and other objects (including an obstacle) around the vehicle. The model introduces a different safety zone--an area that should be intact during such situations. As mentioned earlier, the use of this model can be rather broad.