## 1 Introduction
This curve speed model could be applied in a curve speed warning system. A similar subject concerning driver modelling was described in the paper [46]. In this paper, the authors calculated a driver behaviour factor, defined for different driving styles as the ratio of drivers' actual selected speed to the theoretical curve speed. In the paper [47], a model for predictions of driver behaviour for a turning manoeuvre was presented. The authors described that the motion parameters from a human-driven vehicle could be used to predict the human driver's decision to stop before executing a left turn. A very important group of driver models are models that reproduce the driver's behaviour in near-accident situations. Such a model should usually take into account the defensive manoeuvres undertaken by the driver, namely steering and braking. Models, among which it is possible to search for such solutions, may also include:  
* Speed control models [48; 49] designed for driving control systems in the ICC (intelligent cruise control) column [50; 51; 52], in autonomous vehicles [53; 54], or Internet of Vehicles [55; 56];
* Longitudinal [57; 58] and lateral [59] motion control models [60];
* Models for traffic flow simulation, used, e.g., to analyse the fluidity of traffic on motorways and road networks, including the assessment of the probability of pile-ups, etc. ;
* Models for software used in accident reconstruction, to reproduce the behaviour of drivers in emergencies. The traffic stream model presented in the paper [50] contains a description of a car following its predecessor (car-following model). The assumptions used to create this model can be considered typical for this group of models. The driver can use one of four modes of driving: free driving, approach, stabilisation of driving in a column, and braking. In another paper [61], a model which characterises macroscopic traffic was described. This model is based on analogies with the ideal gas law, known in thermodynamics. A traffic constant, that uses both physiological and psychological driver factors, is proposed. Physiological driver behaviour includes observing and processing local traffic conditions. A traffic constant encompassing the perception, awareness, attitude, and reaction of a driver was described. In critical situations, one of the basic defensive manoeuvres is braking. The construction of an emergency braking model proposed in the paper [62] was preceded by studies of the behaviour of a driver driving a vehicle at a constant speed, who, at some point, encountered an obstacle in his field of vision. This is what forced the start of the braking manoeuvre to avoid a collision. An interesting review of driver models, used in critical situations, was presented by [PERSON] and [PERSON] et al. in their work [63]. Many authors state in their works that the actions of drivers in critical situations depend on the distance from an obstacle and the speed of driving. All mathematical dependencies that make up the driver model are then the functions of these two variables. In the previous work of the authors of this article [64], a model covering two main defensive models of the driver's actions, observations made during driving in real traffic, as well as during tests in simulated conditions on a track, were used. During the avoidance manoeuvre, as well as during braking, the drivers reacted to an obstacle penetrating the roadway area. The different behaviour patterns of the driver were described, namely speed reduction (braking) and obstacle avoidance. The main parameters determining the possibility of undertaking a given defensive manoeuvre (and its intensity) was the lateral position of the obstacle concerning the lateral position of the vehicle. The vehicle speed and distance from the obstacle were equally important. In their work, e.g., [36; 65], the authors showed that it is much more advantageous to use in the driver model the distance from an obstacle, but measured in time--called \(TTC\) (time to collision).