# Driving Safety Improved with Control of Magnetorheological Dampers in Vehicle Suspension
## 1 Introduction
### Modelling of the Vehicle  
Numerous studies that are presented in the literature are devoted to simulation research that precede work on prototyping and commercial applications of MR dampers. They usually represent some kind of compromise between the accuracy of modelling and the requirement of simplicity of solutions, because, in the next stage, the developed control algorithm should be implemented in the microcontroller. Moreover, the use of very complex vehicle models requires the design of appropriate experiments in order to identify a large number of parameters, especially when it is necessary to take the measurement uncertainty into account [16]. Models of suspension are commonly grouped into a quarter-car, half-car, or full-car models. The quarter-car model is recommended when an initial analysis of the mechanical system is performed. Furthermore, it allows for a simple demonstration of control algorithm, like e.g., Skyhook control [17]. Half-car model is recommended when the pitch or roll dynamics of vehicle body have to be analysed, e.g., minimisation of quantities that are related to both ride comfort or driving safety while using H-infinity optimal control was presented in [18]. The full-car suspension model, which usually exhibits seven degrees of freedom, takes into account the vertical motion of the wheels, as well as heave, pitch, and roll motion of the body [19]. The next problem that should be taken into account is how to model a wheel. Usually, for the purpose of vehicle dynamics simulation the wheel is decomposed into the rim and tire. Depending on the model complexity vertical motion only or additionally lateral and rotational motion of the rim can be included in the wheel model. The dynamics of the rim depends on several factors, i.e., force and torque that are exerted by the road surface, as well as torques that are generated by the drive and braking system. The modelling of tire behaviour during the ride is complex due to the susceptibility of rubber to large deformations. The complex tire structure is a second reason, where several elements of tire can be distinguished, e.g., tire treads, belt, sidewalls, or bead bundle [2]. Different phenomena are observed for the driving tire, such as longitudinal and lateral stiffness and damping, rolling resistance, or significantly nonlinear tire friction on the road. In [20], it was stated that four categories of possible approaches for developing a tire model can be distinguished, i.e., based on experimental data only, using similarity methods, based on a simple or more complex physical model. The vertical tire model (i.e., the tire dynamics is considered vertically) that is presented in [21] is the simplest and most widely used in research. Further, the model can be extended by longitudinal or lateral forces [2]. Also, a group of radially and circumstantially located dampers and springs can be used. The most complex tire models are described using final element methods (FEM), as it was shown in [22; 23], where modal analysis for steady or rolling tire was performed or circumferential waves at wheel's critical speed were considered [2]. Other types of models allow for precise analysis of interactions between tire and road, as in the case of FEM models [24] or based on evaluation of contact mechanics between tread blocks and road surface [23]. Moreover, an alternative analytical model of the tire was applied in [25] in the case of uneven roads. Rolling resistance is observed for rolling objects, while it is especially noticeable in the case of elastic tire where the energy for the expanding tire on the rear of the wheel is not fully restored, but dissipated. It results in a non-uniform distribution of ground normal stress, more generally, the resultant ground reaction force is shifted forward with respect to tire contact patch, as shown in [26; 27] or [2]. It results in reaction force, which not only acts on the wheel center of gravity, but also generates a braking torque for the tire.