Control and stability of the lateral dynamics of the vehicle to avoid road accidents

The paper presents elements related to the control and stability of the lateral dynamics of the vehicle. The mathematical model of the ESC system is given. The study of stability is presented, including by using the phase portrait. The control strategies and algorithms used in ESC are presented. The results obtained using the technical data of the Volkswagen Touareg and Chevrolet Camaro cars are presented.


Introduction
Current vehicles are equipped with active stability control systems, which ensure the prevention of leaving the desired trajectories of movement, horizontally and vertically; for this purpose, the control of the yaw and rolling movements is ensured, separately or through an integrated solution [1,2,3]. Current solutions also ensure the control of longitudinal dynamics, for example with the help of the adaptive cruise control system (ACC: Adaptive Cruise Control) and the antilock braking system (ABS: Anti-lock Brake System). The purpose of this paper is to study the control of the lateral dynamics of the vehicle using the electronic stability control system (ESC: Electronic Stability Control; ESP: Electronic Stability Program).

Mathematical model
In the literature, the most used mathematical model of the lateral dynamics of the vehicle is the one with two degrees of freedom [4]. The two degrees of freedom of the model are the angular speed of yaw movement z (yaw rate) and the angle of lateral sliding .  The lateral sliding angle of the vehicle is determined by the relation, with vy and vx the axial components of the velocity v: fig.1 are shown the steering angle of the front wheels , as well as the distances a and b from the center of gravity G to the axles.
The mathematical description of the mathematical model in fig.1

Stability of lateral movement
Due to the presence of trigonometric functions in expressions (2), the mathematical model is nonlinear, which is a difficulty in controlling stability using ESP. For this reason, the model is usually linearized taking into account that in practice the angle values  are small at each time/calculation step and consequently the system (2) becomes: In the frequently adopted hypothesis of a linear model of the tire, the relations result: (4) in which f C  and s C  represents the transverse stiffness of the front and rear axle tires, respectively; in addition, f and s represent the lateral sliding angle of the front wheel and the rear wheel, respectively, established with the relations (the steering is only at the front axle): The system of differential equations is finally obtained: Applying the Laplace transform to the relations (6) we obtain the main determinant: from which results the characteristic equation of the form: Where: From systems theory it is known that a system is stable if the roots of the characteristic equation A(s) = 0 are real and negative, or complex with the real negative part. For a second order system (as in this case), the necessary and sufficient condition for the system to be stable is that all the coefficients of the characteristic equation are strictly positive, in which case the roots satisfy the mentioned conditions. From expressions (9) it is found that a2> 0 and a1> 0. From the condition that a0> 0 results:  As can be seen from fig. 2, all curves tend towards a finite value, which is the stationary value, or the static equilibrium value; since the stationary value is finite, it follows that the stability of the stationary regimes is ensured and therefore the condition is fulfilled (10). It is therefore necessary to analyze the stability of stationary regimes for different values of longitudinal speed, wheel steering angle and tire side stiffness. For this purpose, the expressions (6) are known as:

Results and discussion
where: ; ; ; ; In steady state,  = const. and z = const. and as a result in the left limb of equations (11) The matrix is thus obtained: The type of critical points (also called singular points, or equilibrium points) depends on the eigenvalues 1 and 2 of the matrix A; at the same time, the eigenvalues also show whether a critical point is stable or unstable [5]:  In this way other aspects can be addressed. For example, fig. 7a shows the determination of the maximum longitudinal speed that ensures the stability of the lateral dynamics for certain values of tire stiffness. As can be seen from fig.7a, the maximum longitudinal speed for stability reasons is less than 151.4 km/h, value over which one of the poles p of the transfer function becomes positive; the poles of the transfer function are the roots of the characteristic equation (8). This aspect is confirmed in fig.7b for Vx = 180 km/h, at which the two quantities do not tend towards a finite value but towards infinity cumulative with time.
Similarly, the minimum values of the lateral stiffness of the tires can be set at a certain longitudinal speed so as to ensure the stability of the lateral dynamics of the car.

Lateral dynamics control
One of the important problems is the electronic control of the lateral dynamics of the car, the main role of the ESC system. There are three control strategies frequently applied to lateral dynamics control systems [6,7,8]:  differential braking, which uses the ABS braking system for the different application of braking to the car's wheels in order to control the torque;  active steering control, which ensures the correction of the steering angle given by the driver, so that the vehicle returns to the desired/imposed trajectory;  active distribution of torque to the wheels, which uses active differential and traction on all wheels, to control the traction torque.
In order for the ESC to fulfill its role, the vehicle is composed of transducers for measuring longitudinal acceleration, lateral acceleration, angular speed, wheel speed, steering angle and braking pressure.
For example, Fig. 8 shows the movement on a slalom route of a Chevrolet Camaro car that moved with ESC out of order and ESC functional, with the detail in fig. 9. As it turns out, in case of decommissioning of the ESC system, the car leaves the required route, has a pronounced turning motion, which results in loss of control by the driver, thus increasing the risk of a car accident.  As can be seen from the upper graphs in fig.10, in the case of ESC malfunction, the rotation movement has amplified over time. For example, the rotation angle increased to 99.8 degrees clockwise (it has a negative value), which can be seen in fig.9.
The lower graphs in fig.10 show the pronounced increase of the steering angles of the wheels in case of ESC malfunction.The graphs in fig.11 show the application of the differential braking algorithm of the wheels to keep the car on the required route in case of ESC operation.

Figure 11
Wheel braking forces with ESC in operation, movement on wet asphalt Differential wheel braking satisfies the control algorithm of the ESC system, which aims to obtain the smallest possible values of two quantities: the distance between the center of gravity of the car and the required trajectory, as well as the difference between the required turning angle and the actual turning angle.
ESC systems used on current vehicles use various control algorithms in order to obtain required performance, the main ones being: feedback loop control; PID control; adaptive control; MPC -Model Predictive Control; SMC -Sliding Mode Control; intelligent control (based on artificial intelligence algorithms: genetic algorithms, fuzzy logic, neural networks, neuro-fuzzy algorithms); robust control (H2 control and H∞ control); FTC -Fault Tolerant Control.

Conclusion
Electronic control of the lateral dynamics is of great importance, because an impermissible deviation of the trajectory from the one desired by the driver can lead to road accidents with serious consequences for people and significant damage to the vehicles involved. For this reason, the control and stability of the lateral dynamics concern aspects related to the vehicle, the road and the driver.
For the study of lateral motion stability, algorithms from systems theory are applied, the phase portrait being widely used, which allows the establishment and visualization of critical points.
The presented study highlights the need to establish the speed and lateral stiffness of tires that lose the stability of lateral movement.
For a more in-depth study of lateral dynamics, other mathematical models can be used, with an increased number of degrees of freedom [9,10,11].