The anti-lock braking system is a braking control that prevents the wheels from locking up during braking to avoid uncontrolled skidding, and controls the slip of the wheel precisely around 0.2 during deceleration to achieve maximum braking force. This is done by receiving data from speed sensor in each wheel and adjusts the braking by sending signal to the hydraulic unit. The ABS control module constantly monitors the rotational speed of each wheel, if it detects a wheel rotating significantly slower than the others, it actuates the hydraulic valves to reduce the brake pressure to that wheel, to stop the locking.
Figure 7.3.2(1) Bosch ABS9 control unit [15] Figure 7.3.2(2) Bosch ESP9 control unit [15]
In the overall simulation (see section 10), the simulation is done with an assumption of ‘skillful driver threshold braking’, that is the driver can control the brake pedal extremely well to avoid any slipping, and achieve maximum braking force at the same time. That assumption is achieved by setting a maximum limit to the braking force to avoid slipping. Without the limitation, slipping occurs as the wheel speed goes to zero, as shown in the figure (3), the blue colour represent the vehicle speed and the red colour represent the wheel speed. And the figure (4) shows the slip versus time plot during the deceleration.
* Note that the slip equation is:
Slip=1− Wheel angular velocity
Vehicle Speed divided by wheel radius
Figure 7.3.2(3) Speed versus time plot without ABS Figure 7.3.2(4) Wheel slip versus time plot without ABS
So slip is 1 means the wheel speed is zero, which means the wheel is locked and the skidding happens. In the reality, the ABS rapidly applies or releases brake pressure to maintain the braking force at the threshold braking point, to achieve maximum braking force without slipping. In the simulation, this is done by controlling the rate of change of brake pressure, as shown in the figure (5) below.
Figure 7.3.2(5) ABS action simulation
The slip error is the difference between the current wheel slip and the desired slip, and the desired slip is set to be 0.2 according to the empirical function between slip and friction coefficient, to maximize the adhesion between the tire and road and minimize the stopping distance with available friction. Then the error is passed into a Bang-bang controller, which outputs a 1 for a positive input, and a -1 for a negative input. This tells the ABS whether the wheel requires more brake force or less brake force. This on/off rate passes through a first-order lag that represents the delay associated with the hydraulic lines of the braking system, the 5*10^5 is an estimated amount of brake pressure that the brake can apply or release each time. Then the model integrate the
filtered rate to yield the actual brake pressure, the upper limit of the integrator is set to be equal to the maximum brake pressure of our brake system, which is 5.4*10^6. The resulting brake pressure, multiplied by the piston area and radius with respect to the wheel, is the brake torque applied to the wheel. Finally the wheel speed is imported into another system to calculate the current slip. [16]
Figure 7.3.2(6) Overall ABS simulation
The figure (6) shows the overall ABS simulation. The tire torque is the product of friction and tire radius, and the friction is product of friction coefficient and weight. The experiment shows that the friction coefficient between tire and road surface is an empirical function of slip, known as mu-slip curve. During braking, there is no thrust force coming from the engine, so the only force acts on the car is the friction, which includes air drag and friction force between wheel and road. So the deceleration of the car can be calculated by friction over mass, and then integrated to get wheel speed. Then the current slip can be calculated by the wheel speed and car speed, a loop system is created.
It can be seen from the figure (7) and (8) that the slip varies around 0.2 as expected, the wheel is not locked during braking and both the time for the deceleration and the stopping distance are reduced. The 0.2 slip control is the threshold braking technique that all the professional driver are practicing. The simulation result has proven that the ABS can achieve that precisely, so that the car equipped with an ABS system can save the driver a lot of trouble during the deceleration and
ensure the safety. What’s more, the driver normally practices the threshold braking skills on a normal road condition, so in a different situation such as wet condition or soft road condition, the coefficients of friction are different in these situations causes the threshold braking skills which largely based on driver’s experience no longer hold. The ABS, however, can still work precisely in these conditions since the control is achieved bases on the real-time data and calculations rather than experience.
Figure 7.3.2(7) speed versus time plot with ABS Figure 7.3.2(8) wheel slip versus time plot with ABS
It is interesting that, there is a chance that the skilful driver can beat the ABS during deceleration.
As shown in the figure (9), the bump before the activation of ABS indicates the area where a human can beat the ABS, if the driver is properly threshold braking, properly modulating the brake pedal force to stop just short of activating the ABS.
Figure 7.3.2(9) Driver versus ABS [17]