Figure 1: Antilock brake system components.

1 Antilock Braking System Modeling

The main components of an antilock brake system (ABS) are: a control unit, a brake force actuator, and wheel speed sensors. The controller unit monitors the wheel speed, performs calculations based on the wheel speed, and sends the control signal to the brake force actuator. A diagram of an ABS unit is shown in Figure 1. The ABS system is considered a safety feature in that the main purpose of an ABS is to minimize stopping distance under emergency braking conditions and yet maintain vehicle stability. Antilock brake systems minimize the stopping distance by preventing the wheels from locking up during braking. The vehicle brake force is proportional to the normal force on the tires. The proportionality factor that relates the vehicle brake force to the tire’s normal force is the road coefficient of adhesion. The road coefficient of adhesion is a function of wheel slip as can be seen in Figure 2. The wheel slip is defined as the ratio of the slip velocity in the contact patch (forward velocity minus tire circumferential speed) to forward velocity. Let λ denote the wheel slip, then

λ = \frac{v − ωR_{w}}{v} ,  (1)

where

v = vehicle forward velocity
ω = tire angular velocity

Figure 2: Road coefficient of adhesion versus wheel slip curves for different road surfaces.

R_{w} = tire rolling radius.

The above parameters are shown in Figure 3. Note that 0 ≤ λ ≤ 1. Following the convention, we specify the value of λ using %. Brake lock-up corresponds to 100% wheel slip, and Figure 2 shows that at 100% slip the road coefficient of adhesion is not maximized. The maximum road coefficient of adhesion is attained when the wheels are slipping. Therefore, the maximum braking force acting on the vehicle occurs when the wheels are slipping. Hence, the vehicle stopping distance is minimized by controlling the wheels to slip in the range where the road coefficient of adhesion is maximal. The maximum coefficient of adhesion is in the 5% to 30% slip range. Thus, by allowing the wheels to slip in the 5% to 30% range, we maximize brake force. The operation of an ABS systems is shown in Figure 4 that was adapted from Gillespie [1, page 68]. The application region is the range of slip (point A) where the driver applies the brakes and where the slip level is below impending wheel lock-up. When the wheel slip reaches the range where the coefficient of adhesion is maximized, the ABS system takes control from the driver (point B). The ABS system releases the brake torque before Figure 3: Front wheel free body diagram; for rear wheel F_{bf} becomes F_{br}.

Figure 4: ABS operating modes.

Figure 5: A vehicle free body diagram.

the wheels lock-up (point C) and the wheels begin to pick up speed again. As the wheel speed increases, the wheel slip increases until its slip falls in the maximum brake force region (point B). The ABS system then repeatedly applies the brake torque and releases the brake to keep the wheel slip within the optimal slip region (point B) near the peak of the μ-slip curve.

An ABS model must include the wheel dynamics and road surface for accurate results. Here, we develop a simple model of an ABS for straight line braking with zero steering wheel input. In our model, we neglect pitching motion and consider motion only along the longitudinal axis. Our development follows that of Will and Żak [6]. A straight line braking vehicle free body diagram is shown in Figure 5. The masses m_{s}, m_{uf}, and m_{ur} are the vehicle’s sprung, front unsprung, and rear unsprung masses. The front and rear tractive forces are denoted F_{tf} and F_{tr}, respectively. Tractive force is the force developed at the road-to-tire interface that provides traction for either accelerating or braking. Tractive force is a function of the tire normal load and the road coefficient of adhesion.

The road coefficient of adhesion model we use was adapted from the data obtained by Layne, Passino, and Yurkovich [2] in their fuzzy learning control for antilock brake systems. The coefficient of adhesion μ versus wheel slip λ curves for asphalt, wet asphalt, and ice surfaces are shown in Figure 2. This is the surface model we use in later simulation studies. We denote the adhesion coefficient of front wheels by μ(λ_{f}) and that of the rear wheels by μ(λ_{r})

We now derive the equations for the total tractive force. From the vehicle free body diagram, shown in Figure 5, we write the equation of the total tractive force F_{tot} as

F_{tot} = F_{tf} + F_{tr},  (2)

where

F_{tf} = μ(λ_{f})F_{zf},
F_{fr} = μ(λ_{r})F_{zr}.

Upon substituting the above into (2), we obtain the expression for the total tractive force,

F_{tot} = μ(λ_{f})F_{zr} + μ(λ_{r})F_{zf}.  (3)

Let

a = distance from sprung mass to front wheels
b = distance from sprung mass to rear wheels
h_{cg} = sprung mass height
h_{f} = front unsprung mass height
h_{r} = rear unsprung mass height
m_{s} = sprung mass
m_{uf} = front unsprung mass
m_{ur} = rear unsprung mass
m_{tot} = m_{s} + m_{uf} + m_{ur}.

The normal forces acting on both the front and rear tires have two components. One component is due to the longitudinal weight transfer due to braking or acceleration. The other component is due to the static distribution of the masses. In our derivations of the expressions for the above mentioned forces, we follow arguments of Nalecz and Bindemann [4, 3], and Will and Żak [5]. We first concern ourselves with the components due to the longitudinal weight transfer. Consider the diagram shown in Figure 5. Summing the moments about the line passing through the points of contact of the rear tires, and dividing the resulting equation by the wheel base a + b yields

F_{zf1} = (m_{uf}h_{f} + m_{s}h_{cg} + m_{ur}h_{r}) \frac{ẍ}{a + b}.  (4)

Similarly, summing the moments about the line passing through the points of contact of the front tires and performing simple manipulations we obtain

F_{zr1} = − (m_{uf}h_{f} + m_{s}h_{cg} + m_{ur}h_{r}) \frac{ẍ}{a + b}.  (5)

We denote the components of normal forces due to the static distribution of the masses by F_{zf2} and F_{zr2}. The static force acting on the front tires is

F_{zf2} = \frac{b}{a + b}m_{tot}g.  (6)

The static force acting on the rear tires is

F_{zr2} = \frac{a}{a + b}m_{tot}g.  (7)

The total normal force acting on the front tires is

F_{zf} = −F_{zf1} + F_{zf2}
= − (m_{uf}h_{f} + m_{s}h_{cg} + m_{ur}h_{r}) \frac{ẍ}{a + b} + \frac{b}{a + b}m_{tot}g.  (8)

Similarly, the total normal force acting on the rear tires is

F_{zr} = −F_{zr1} + F_{zr2}
= (m_{uf}h_{f} + m_{s}h_{cg} + m_{ur}h_{r}) \frac{ẍ}{a + b} + \frac{a}{a + b}m_{tot}g.  (9)

Let

m_{eq1} = \frac{bm_{tot}}{a + b},
m_{eq2} = \frac{h_{s}m_{s} + h_{r}m_{ur} + h_{f}m_{uf}}{a + b} ,
m_{eq3} = \frac{am_{tot}}{a + b} .

Then, equations (8) and (9) can be represented as

F_{zf} = gm_{eq1} − m_{eq2}ẍ,
F_{zr} = gm_{eq3} + m_{eq2}ẍ.

Hence, the total tractive force is

F_{tot} = F_{zf} + F_{zr}
= μ(λ_{f})(m_{eq1}g − m_{eq2}ẍ) + μ(λ_{r})(m_{eq3}g + m_{eq2}ẍ).  (10)

Then, the equation of motion of the vehicle undergoing a braking maneuver is

m_{tot}ẍ = −F_{tot}.

Hence,

ẍ = −g \frac{μ(λ_{f})m_{eq1} + μ(λ_{r})m_{eq3}}{m_{tot} − μ(λ_{f})m_{eq2} + μ(λ_{r})m_{eq2}} .  (11)

Next, we model the front and rear wheel dynamics using the free body diagram shown in Figure 3. Let T_{e} be the engine torque, F_{bf} and F_{br} denote the front and rear brake forces, and R_{b} be the mean radius of the brake lining contact. It is assumed that the vehicle is front wheel driven, and hence T_{e} will only appear in the front wheel dynamics equation. We model the front and rear wheel dynamics as:

2J_{f} ω̇_{f} = −F_{bf}R_{b} + μ(λ_{f})F_{zf}R_{w} + T_{e}  (12)
2J_{r}ω̇_{r} = −F_{br}R_{b} + μ(λ_{r})F_{zr}R_{w}.  (13)

Substituting equations (8) and (9) into (12) and (13), we get

ω̇_{f} = \frac{1}{2J_{f}} (−F_{bf}R_{b} + μ(λ_{f})m_{eq1}R_{w}g − μ(λ_{f})m_{eq2}R_{w}ẍ + T_{e})  (14)
ω̇_{r} = \frac{1}{2J_{r}} (−F_{br}R_{b} + μ(λ_{r})m_{eq3}R_{w}g + μ(λ_{r})m_{eq2}R_{w}ẍ) .  (15)

We assume that a single brake force actuator supplies the braking forces F_{bf} and F_{br}. This actuator powers both the front and rear brakes. Let the brake stiffness coefficient for the i-th wheel be denoted by C_{i} where i=1, 2 corresponds to the left and right front wheels, while i=3, 4 corresponds to the left and right rear wheels. Then, we model the front brake force actuator as

F_{bf} = μ_{b}(C_{1} + C_{2})u,  (16)

and we model the rear force actuator as

F_{br} = μ_{b}(C_{3} + C_{4})k_{br}u,  (17)

where u is the brake actuator displacement, μ_{b} is the brake pad coefficient of friction, and k_{br} is the brake displacement proportionality constant. In a manual four-wheel brake system, the rear brake force is a fraction of the front wheel brake force. Typical values for k_{br} are in the range [0.2, 0.6]. The reason for proportioning the rear brake force in a brake system is to prevent the rear wheels from lock-up, even when front brake lock-up occurs. Rear wheel lock-up destabilizes the vehicle so that no steering input can stabilize it. However, when the front brakes lock-up, the vehicle still can be controlled to a certain degree. We now write the expressions for the front and rear wheel dynamics. Substituting (16) into (14), we obtain an equation that models the front wheel dynamics. We have,

ω̇_{f} = \frac{1}{2J_{f}} (−μ_{b}(C_{1} + C_{2})R_{b}u + μ(λ_{f})m_{eq1}R_{w}g − μ(λ_{f})m_{eq2}R_{w}ẍ + T_{e}) .  (18)

Similarly, substituting (17) into (15) yields an equation that models the rear wheel dynamics. We have,

ω̇_{r} = \frac{1}{2J_{r}} (−μ_{b}(C_{3} + C_{4})R_{b}k_{br}u + μ(λ_{r})m_{eq3}R_{w}g + μ(λ_{r})m_{eq2}R_{w}ẍ) .  (19)

Let x_{1} = x, x_{2} = ẋ_{1}, x_{3} = ω_{f} , and x_{4} = ω_{r}. Then, we can represent equations (11), (18), and (19), that model a braking vehicle, in the state space format as

\begin{matrix}ẋ_{1} & = & x_{2} \\ ẋ_{2} & = & −g \frac{μ(λ_{f})m_{eq1}+μ(λ_{r})m_{eq3}}{m_{tot}+μ(λ_{f})m_{eq2}−μ(λ_{r})m_{eq2}} \\ ẋ_{3} & = & \frac{1}{J_{f}} (−μ_{b}C_{1}R_{b}u + μ(λ_{f})m_{eq1}R_{w}g − μ(λ_{f})m_{eq2}R_{w}ẍ + T_{e}) \\ ẋ_{4} & = & \frac{1}{J_{r}} (−μ_{b}C_{3}R_{b}k_{br}u + μ(λ_{r})m_{eq3}R_{w}g + μ(λ_{r})m_{eq2}R_{w}ẍ) . \end{matrix} \}  (20)

The output variable is

y = [ \begin{matrix}x_{1} \\ x_{2} \\ λ_{f} \\ λ_{r} \end{matrix} ] = [ x_{1} ; x_{2} ; \frac{x_{2}−R_{w}x_{3}}{x_{2}} ; \frac{x_{2}−R_{w}x_{4}}{x_{2}} ] .  (21)

The vehicle data that we use in the simulation experiment is shown in Table 1. Upon substituting the data into equations (20) and (21), we get

\begin{matrix}ẋ_{1} & = & x_{2} \\ ẋ_{2} & = & \frac{−7574.0μ(λ_{f})−7140.7μ(λ_{r})}{1500+727.905(μ(λ_{f})−μ(λ_{r}))} \\ ẋ_{3} & = & 726.237μ(λ_{f}) − 69.793μ(λ_{r})ẍ − 31264.0u \\ ẋ_{4} & = & 683.974μ(λ_{f}) + 69.793μ(λ_{r})ẍ − 15623.2u, \end{matrix} \}  (22)

and

y = [ x_{1} ; x_{2} ; \frac{x_{2}−0.326x_{3}}{x_{2}} ; \frac{x_{2}−0.326x_{4}}{x_{2}} ] .  (23)

We now test the above brake system model by simulating an emergency brake situation. The vehicle is traveling with constant speed, ẋ = 20 m/s, on a dry asphalt road before the brakes are applied. We first validate our brake model by simulating the situation when the brakes lock the wheels. Specifically, we apply a step brake force input of 1500 Nm and this input is held for 50 milliseconds and then stepped back to zero brake torque. The step brake input simulation result is shown in Figure 6. The figure shows that the front wheels lock-up Table 1: Antilock brake system vehicle data.

g  9.81 m/s^{2}
a  1.186 m
b  1.258 m
h_{s}  0.6 m
h_{f}  0.30 m
h_{r}  0.30 m
m_{tot}  1500 kg
m_{s}  1285 kg
m_{uf}  96 kg
m_{ur}  119 kg
μ_{b}  0.90
J_{f}  1.7 kg m^{2}
J_{r}  1.7 kg m^{2}
R_{b}  0.163 m
R_{w}  0.326 m
T_{e}  0.0 Nm
C_{1} = C_{2} 362.3 kN/m
C_{3} = C_{4} 362.3 kN/m
k_{br}  0.50

Figure 6: Response of the proposed brake system model to an emergency brake input. 11 and the wheel skids at zero velocity until brake force is released. When the brake force is released, the wheel accelerates to the velocity of the vehicle. The vehicle decelerates during the brake application. The response of the proposed brake system model, shown in Figure 6, is in agreement with the dynamic behavior of an actual brake system.

References

[1] T. D. Gillespie. Fundamentals of Vehicle Dynamics. Society of Automotive Engineers, Inc., Warrendale, PA, 1992.

[2] J. R. Layne, K. M. Passino, and S. Yurkovich. Fuzzy learning control for antiskid braking systems. IEEE Transactions on Control and Systems Technology, 1(2):122–129, June 1993.

[3] A. G. Nalecz and A. C. Bindemann. Analysis of the dynamic response of four wheel steering vehicles at high speed. Int. J. of Vehicle Design, 9(2):179–202, 1988.

[4] A. G. Nalecz and A. C. Bindemann. Investigation into the stability of four wheel steering vehicles. Int. J. of Vehicle Design, 9(2):159–178, 1988.

[5] A. B. Will and S. H. Żak. Modeling and control of an automated vehicle. Vehicle System Dynamics, 27(3):131–155, March 1997.

[6] A. B. Will and S. H. Żak. Antilock brake system modelling and fuzzy control. Int. J. of Vehicle Design, 24(1):1–18, 2000.