07 Anti-Lock Braking

8/13/2019 07 Anti-Lock Braking

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ME 360/390 Prof. R.G. LongoriaVehicle System Dynamics and Control

Department of Mechanical EngineeringThe University of Texas at Austin

Overview of content

Feedback control of a single-braking wheel to

avoid lock-up (slip regulation)

Anti-lock brake system (ABS) model and

simulation


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Recall the rotational dynamics of a wheel in traction or braking,

Brake application

w w d loss t b

d loss tx b

I T T T T

T T r F T

=

=

Road loads

x x tx

W

g

p m a F= =

Control problem: forward velocity control

by controlling Ftx and/or Tb

Must consider vertical loads and the pitchequation (for load transfer).

Refer to Wong from Eqs. 3.47 and 3.48.

xv

w

W

txF

dT

zF

x

z

bT

Recall dynamics of a braking tire


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Brake application

w w d loss t b

d loss tx b

I T T T T

T T r F T

=

=

For accelerating or

braking,

T

r. .

10

R

1

II

Brake

bT

txr Fd lossT T

resF

Road loadsx x tx

W g

p m a F= =

txF

zF

GaP

In this first example, just set Tb; later we may

examine hydraulically-actuated and/or feedback-

controlled (ABS).

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Bond graph for traction/braking


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These systems monitor operating conditions and

modify the applied braking torque by modulating the

brake pressure. The systems try to keep tires operating within a

desired range of skid, and by preventing wheel lock-

up during braking they can help retain steerability andstability

Anti-lock braking systems are closed loop control

systems within the braking system.

Anti-lock braking systems


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Bosch (1999)

Wong (1993)

Trying to control slip

in a desirable range

is complicated by

changing road

conditions.

Desired range of slip or skid


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ME 360/390 Prof. R.G. LongoriaVehicle System Dynamics and Control Department of Mechanical EngineeringThe University of Texas at Austin

Concept dates back to early 1900s, and first

patent went to Bosch in 1936.

Concept is now well established.

Figures from Gillespie (1992)

ABS background


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This figure shows results for a

heavy vehicle braking on wetpavement.

The cycle of reducing and

restoring pressure can be

repeated from 5 to 16 times

per second.

The ABS operation is usually

deactivated once the vehicleslows to about 2 or 3 mph.

Example ABS operation

Heavy Vehicle with Pneumatic Brakes (Wong, 1993)


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1

2

( ) 0( )

( ) 0

M e tm t

M e t

>=


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Sometimes there may be some hysteresis in an on-off controller. The range that the errorsignal must go through before actuating either way is called the differential gap.

The hysteresis may be unintentional (caused by

friction or gap in the mechanism), or it may bedesigned into the control action.

One reason to purposefully include hysteresis in an

on-off controller is to slow down the switching

between the two-states.

Switching too often can lead to reduced life in the

control actuating element. A differential gap,

however, will cause the output to have some

oscillations, the amplitude of which can be reducedby decreasing the gap.

Ogata (1978)

Typical oscillations, as inducedby on-off control of tank level.

On-off controller issues


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The bang-bang principle of control says that a system being operated under limited

power can be moved from one state to another in the shorted time possible by at all times

utilizing all available power. This was hypothesized and proven experimentally and

theoretically long ago.

Not all bang-bang implementations guarantee time-optimal control, of course, but for

certain systems this is the case.

For,1

1 0

( )n n

n n

Y bG s

U a s a s a

= =+ + +

With the input,

It can be proven that bang-bang gives time-optimal control, and you can reach a desired

state in at most n-1 switches.

o ou u u

Bang-bang control


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On-off control of a pressure modulator

Can use a simple two-position controller, maybe with some gap.

Reference slip

Measured (estimated) slip

se 1

2

0( )

0

s

p

s

u eu t

u e

>=


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Studying implementation in simulation

1. Whats the simplest ABS model you can build that gives

realistic results? Likely you need some minimal hydraulics.

Just having a pure on/off gives unrealistic effects.

2. Do you need to include hydraulics? I think the true natureof ABS requires inherent dynamics that arise from the

hydraulic components.

3. What is the simplest model that includes some hydraulics?

Need to at least have the pressure build-up. The lag due tothe lines can actually be removed.

4. Do you need to implement a differential gap? Probably not.

I think the hydraulics adds the true delay effect and is more

realistic.


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Case 1: Simple on/off, no hydraulics, no differential gap

Is it sufficient to representan ABS controlled braking

system with just on/off

control?

% determine slip state

s1 = (Rw*omega-V)/max(Rw*omega,V);mu_ex = 0.7; % for extrapolation, keep mu at nominal value

mu1 = sign(s1)*interp1(slip,mu,abs(s1),'linear',mu_ex);

Ftx = +mu1*mv*g/Nw;

Tt=Rw*Ftx;

% Feedback for ABS% slip_ref is the specified slip reference value

% Note, routine above gives + or - slip.

slip_error = slip_ref - abs(s1);

% Control: if slip is less than the reference value, increase pressure

% if slip is greater than reference value, decrease pressure

UP = sign(slip_error);

Kf = 6.895e5; % this is 100 psi expressed in Pa

% ideal pressure on/off

Pb = Kf*0.5*(UP+1);

% Assume same pressure goes to all Nw tiresTb=G*abs(Pb);

% Net torque on one wheel

T = Td - Tb*sign(omega) - Tt;

This sets the pressure

on and off.


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So simple on/off works to

control the slip at thereference level, keeping

wheel from locking up.

However, the brake torqueis turned on and off at a

very high rate.

This is not realistic or

practical.

The green graph showsthe high frequency brake

torque pulsing.

Slip maintained at desired level

Stopping distance

No lock up to a stop


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Stockel, et al

-Proportioning

-Metering-Pressure differential

Basic brake actuation system, showing hydraulic lines, etc.


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We dont really have many details, so lets construct a simple hydraulic line. Assume there is an input

pressure commanded by your controller, Pd. This pressure would travel through a hydraulic line having

some inertia, I, some resistance, R, terminated by a piston actuator, where pressure might build up to Pb.

This pressure would induce a normal force, Fb, on disc brake caliper. This is illustrated below:

Note: This shows how

the hydraulic circuit

would couple into the

braking wheel bond

graph.

Model of a simple hydraulic line and pressure build-up

Hydraulic system

Rotating wheel

b b bP F T NOTE:


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Well model the line with a hydraulic lag, so the pressure at the end of that line would then build up

through an integrator (this is like a hydraulic capacitor). It is good practice to include a saturation tolimit the pressure output, Pb, to some maximum value. This Pb pressure sets the disc brake force.

In a block diagram form, we can show the progression as follows.

Integrating into a system model

But we want to write code, so we convert the transfer functions shown into ODEs.


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Deriving the hydraulic model equations

( )11

lag lag

lag lag lag d

d lag

P KP s K P

P s

= + =

+

Begin with the hydraulic line, and convert the transfer function to time-domain ODE:

( )1

or:

lag lag lag lag d

lag lag lag lag d

lag lag lag d

lag

sP P K P

P P K P

P P K P

+ =

+ =

= +

Here is one more ODE to add toyour model. This is the ODE

for Plag. Pd is the pressure at the

entrance to the line.

Now, this pressure at the end of the line feeds into the brake actuator cylinder, which is

being modeled as a simple capacitor, and the pressure is given by,

1

1or:

b lag b b lag

b

b lag

b

P P sP Ps

P P

= =

=

Another ODE. This is the ODE

for Pb. Plag is found from the

other ODE above.

Once you have Pb, you can compute the brake torque: abs( )b bT G P=


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Deriving the hydraulic model equations (cont.)

The last piece is setting the drive

pressure at the inlet.

Were scaling this value by Klag in this

model, so well let Pd be the output

from the sign function, i.e.,

1,slip_error 0

sign(slip_error) 0,slip_error 0

1,slip_error 0

dP UP

+ >

= = = =

>

slip_error = slip_ref - abs(s1);

UP = sign(slip_error);

Klag = 6.895e5; % this is 100 psi expressed in Pa

Plagdot = (-Plag + Klag*UP)/tau_lag; % lag

% if Pb exceeds Pbmax, saturate

if (Pb>Pbmax | Pb


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Slip varies about desired level

Stopping distance

No lock up to a stop

This is pulsing torque (due to pressure)

Here is a result from using

the ABS simulation withhydraulics.

The brake torque varies in

a pulsing manner asexpected.

This is more realistic and

practical.


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References

R. Bosch, Driving Safety Systems, SAE, 1999.

T. Gillespie, Fundamentals of Vehicle Dynamics, SAE, 1992.

H. Heisler, Vehicle and Vehicle Engine Technology, SAE, 1999. D.B. Maciuca, Brake Modeling and Control, Ch. 12 in

Intelligent Vehicle Technologies, SAE, 2001.

Steeds, W., Mechanics of Road Vehicles, Illiffe and Sons,

Ltd., London, 1960.

J.Y. Wong, Theory of Ground Vehicles, Wiley-Interscience,

2001.


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References

R. Bosch, Driving Safety Systems, SAE, 1999.

T. Gillespie, Fundamentals of Vehicle Dynamics, SAE, 1992.

H. Heisler, Vehicle and Vehicle Engine Technology, SAE, 1999.

D.B. Maciuca, Brake Modeling and Control, Ch. 12 in Intelligent Vehicle

Technologies, SAE, 2001.

Steeds, W., Mechanics of Road Vehicles, Illiffe and Sons, Ltd., London,

1960.

M.W. Stockel, M.T. Stockel, and C. Johanson, Auto Fundamentals, TheGoodheart-Willcox Company, Inc., Tinley Park, IL, 1996.

J.Y. Wong, Theory of Ground Vehicles, Wiley-Interscience, 2001.

07 Anti-Lock Braking

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