Introduction ABS

Nederkoorn, C.J.S.

Published: 01/01/1993

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C.J.S. Nederkoorn University of Techology Eindhoven

Reportnumber: 93-096

July 9,1993

Summary

In this report a short introduction in ABS is given. After some simplifications and special conditions we build a simple %-cx model. With this model for the car and the tyre model of Pacejka we make simulations. In these sinulations we u e able to view the

effects of different controllers on the input Mbr , and the main variable: the relative slip A.

1 Introduction .................................................................................................. 1

2 Purposes and problems ................................................................................ 2

3 Modeling ....................................................................................................... 4

3.1 Tyre model ............................................................................................... 4

3.2 Mechanical model ................................................................................... 7

3.3 Feedback linearisation ............................................................................ 10

................................................................................................... 12 4 !3 Li, 3!2tic=ns

4.1 Saw-toothed moment .............................................................................. 12

4.2 Control h to a desired value .................................................................... 14

4.2.1 Sign control ......................................................................................... 14

4.3 Comparison of energy ............................................................................. 16

4.4 Effect of 8, ............................................................................................... 17

5 simulation program ..................................................................................... 18

4.2.2 Proportional control ........................................................................... 15

Biba?sgsa .hy ..................................................................................................... 21

.... .......... _- . ............. .... ..__ .. ........

I

Introduction ABS Chapter 2

2 Purposes and problems

m€ first p b k m we cm..ae acrcss with A B S is tQ dekmh-e the desird behaviour of the system. Furthermore this system should never be inferior to a conventional system.

The most important features are summarized below: Maneuverability is preferred to a short braking distance.

One should be able to correct swerves, caused by different brake forces at the tyres, by steering. ABS must function &om very high speeds down to low speeds where safety is not relevant any more.

ABS should adapt fast to the variations of traction (i.e., force and torque the tyres can practice on the car). Especially at rough surfaces, during aquaplaning, at ice or snow, the

system should function well. Excitation of other parts of the car must be avoided. When the system fails, the system should switch over to the conventional system and generate a warning to the driver.

The system should be cost-effective, light and compact.

The system should be connectable to existing hydraulic energy supplies

(e.g., for servo-assisted brakes and steering).

The complications that occw are: Adaptation times that arise during the variations of traction can lead to

unexpected phenomena. The interaction process between road and tyre is unstable for large slip

CseffiCieIltS. The input signals are physical limited, the number of measurements has to

be restricted on cost-effective viewpoint.

2

Introduction ABS Chapter 2

Interaction between tyre and road is strongly non-linear and is influenced by longitudinal and side slip, weight on the tyre, moment of inertia of the lyre, ste3rii.g dge, etS.

Before we can continue with modeling of the brake process, we first need to

know what the purpose of A B S is. The main goal of ABS is to supply a control law, which controls the torque on

the tyres in such way that the car stops as quickly as possible, maintaining

maneuverability.

Maneuverability implies the possibility to change direction, between limits

around the driving direction. Besides the force needed for braking the car, there has to be a torque that turns the car round its center of gravity. Looking at a %-

car model (bicycle model) one can see that maneuverability is possible when the tyres can supply longitudinal forces as well as side forces.

Our purpose is to get a first insight into the relevant parameters. Also we want to be aware of the simplifying conditions and assumptions we've macle to obtain this vision.

vehicle meed

J reor wheel vehicle -

F E O i

F front

3

Introduction ABS Chapter 3

3 Modeling

The interadon of forces between tyre and road is mainly a function of the relative slip A. In our situation of braking 3L is defmed as:

v f : speed of the car with regard to the road

v, : speed of the wheel with regard to the road with:

Note that A must have a value between O and 1.

We can split the forces of the tyres in longitudinal and side forces. The longitudinal friction coefficient, Corn here called the brake coefficient p, and the side fiction coefficient p, as functions Com A, look as follows:

I .2 0.3 0.4 0.5 0.6 0.7 0.6 0.B 1

Y

Introduction ABS Chapter 3

De highest value of p=pm , occur at h=hk As we can see, p, has an acceptable value for this value of h (around the 90 % of its maximimum). m e a A ixreases, p, decreases mwmds zero md the side ~QXXS deEverd by

the tyres are insufficient to supply a torque that is big enough to control the

maneuverab~îy of the car. The value of A, differs from situation to situation, but dependent on the sort of

tyre, surface of the road, steering edge, its value will be between 0.08 and 0.30. The value of p,, can vary among f0.9 for ideal circumstances and 0.2 for an

icy road (p 2). When we want to approach p as a function of h we can formulate:

Besides the fact that the outline of this formula doesn't look like its real outline,

there are some applications when we wish to use eventually the derivative of p (A) for h=hk More elegant and numerical convenient is to use the tyre model

of Pacejka. This is the so called magic formula (it means just a formula) where one can choose parameters (A, B, C, D) so p(h) resembles (p I ) .

-

5

Introduction ABS Chapter 3

v = (1 - E ) * h + E í B * arctan(B*h) withv: help variable p = D * sin(C * arctan(B * v ) ) u3 -

As one can see in (f3), D is equal to the maximum value of ~ ( h ) = ~ , - .

From here we make no diference between p and the approach of v, p. To get familiar with the relevant parameters that describe the brake process,

we build a numerical model. As we've seen before, maneuverability is #g.KlmnteeCi WheE ?L %cs ïuu;;u ?Lk. If \vu wmt tv smdy SSGIlg hrzlng, we can

neglect ~ W S C ~ ~ I X I I S ~ Q I I ~ effkts by looking only at A, so for our purpose a one

dimensional model will be sufficient. We assume the following:

o When h temporarily exceeds A k , we assume that the car brakes in a

straight line. We don't take maneuverability into account. We look only at one wheel, we use a one dimensional model. The brake coefficient may vary but place and value are known by the

system. The steering edge does not infiuence the brake coefficient. In practice there must be an algorithm to calculate y(h) from measurements. The side friction coefficient is acceptable at h, for any situation.

The starting vehicle speed is 5 [ d s ] .

Vehicle speed below 0.2 [ d s ] is not of importance for safety. We don't take the hydraulics into account. Instead, we restrict the

derivative of Mbr to 20.000 [Nm/s], equivalent with the performance of a common hydraulic system.

6

Introduction ABS Chapter 3

3.2 Mechanical model

mass Vf

I

With the preceding assumptions and simplifications a %-car model will do for

us. This mede! mfisists vf a m m IFnl! s? mmslerr wheel with moment of inertia 0 . On the wheel operate the input M,, controlled by the A B S and the torque

caused by friction on the road (p 3). Formulating the equations of motion yields:

(f4)

(f5)

1.

0

mass * v f = -p (h ) * No . e, * o = M*r- Mbr

and:

(f6) h = 1 - o r r i v f

cf7)

va) mass * g = No

Mwr = p(hL) * r * No

with: A NO r

g

as defined in c f l )

weight of the mass

radius of the tyre gravity acceleration

Chapter 3

0:'

We have chosen the parameters in the magic formula (f 3) B=.104, C=1.27, E=- 1.61.

@i is chûsm kmeen 0.2 mei 0.9 dependeat sa the simldation (for D represents the maximum brake coefficient).

The slip condition of a wheel is divided in 4 phases (p 4) :

lSt phase: phase without braking. A and p equal zero, there is no brake

action. 2nd phase: stable phase. A lies in between zero and i k , p lies in bemeen

zero and CL,,. 3rd phase: instable phase. A lies between i k and 1, p between p,, and pg . In this phase a wheel will lock very soon, dependent on his moment of inertia. 4th phase: A equals 1 and p equals pg . A wheel stays in the 41h phase until the input Mbr decreases under p,*r*N,.

F e ~ 0.5

0.2

0.1

Tne equations of motion in phase 4 ctfffeï 5ûnì phase 2 s ~ d phase 3. I:: this

phase the wheels are locked so u) and o equal zero and A equals 1. This

yields:

8

Introduction ABS Chapter 3

Last formula implicates that Mbr must equal M,, and thus an input controlling

a higher value has no practical sense.

Introduction ABS Chapter 3

3.3 Feedback linearisation

( f l0 ) y = h

With: U = MJt) = input

x = state vector -

. Soweobtain: (f13) x = f ( x , p ) - - -

(f14) Y = - - g o )

If we want h to be í ik we get: b b

(f16) yd = h, = 0

with (f13) and linearising around an (unknown) operating point we obtain:

( f18) y = a ( x ) - + p(x)* - U b

With a aná unknown Íùnctions.

Introduction ABS Chapter 3

When we choose the input U as follows:

where the

trajectory.

- - ISt component (zero in OUT case) takes care of the following of the The 2nd component with U, free to choose should take care of the

convergence of y to yd. From (f 19) we get: e e

(f20) U, = yd - y = e and: (f21) e = yd - y

If

e>O then

e<O then

So we can choose U, :

e g 0

e 2 0

(g 4) Wheel speed v,(t) - and vehicle speed vkt} - -

I

o 0.4 0.5 0.6 0.1 0.2 0.3

time

I I I I

O 0.4 0.5 0 6 0.2 0.3 0.6

0.í

time

Simulation with a saw-toothed input M,, During the first peak h just doesn’t get the value 1. The second and third peak show that h =1, so the wheels are locked. During the first decrease of Mbr p increases. Then the first decrease takes place (the wheel is in the 3rdphase). The following increase is caused by the jump of Mbr back to 8OO[Nm]. The wheel is then accelerated by the speed of the mass.

time

Introduction ABS Chapter 4

4 Simulations

Now we wp?! !mk fer three WIXI-Q~~I-S of the input Mbr and analyse the

behaviour of the wheel.

4.1 Saw-toothed torque

The pr0gra.111 structure for this input looks as follows:

Torque

M b r ( t = û ) = lOOO[Nm] Mbr = Mbr + 20OO*At A t i s the integration time

if Mbr greater then 1300[Nml Mbr 8 0 0 [Nml

return

When we control the input in this way, we obtain a clear view of the different phases of slip the wheel is going through. After the moment the input is activated, the wheel comes into the 2nd phase. A and p increase both and the

mass speed vf will diminish proportional to p. After a little while the wheel

comes into phase 3 and A increases very fast. Looking at of 5) we see that or

will decrease fast, for:

The difference between M,(?L) and Mbr(t) is extra great because of the decreasing ~f M&). Dependent on the radial speed of the wheel, the wheel

comes into phase 4. During the first "tooth' of the input M,, this doesn't happen. When the input jumps back to 8OO[Nm], the speed of the wheel

increases fast, so the wheel comes into phase 2. Here we can find an

Introduction ABS Chapter 4

equilibrium between h and input Mbr . This process will repeat during the

second "tooth", but the speed of the wheel is now lower. The wheel comes now into phase 4. The input now is higher t . 3 ~ what's possible in redity, because of the locking wheels (a negative wheel speed is not possible in this case). After the torque jumps back again? the wheel speed increases and the speed of the mass and wheel will finally go to zero.

Remarks:

In phase 4 a bigger torque than possible is send. The program takes this in consideration and calculates with the biggest possible torque. During the locking of the wheels, the decline of the vehicle remains almost constant. The change of the input Mbr is absorbed mostly by change in A (dependent on the moment of inertia and or ). The value of h(t) shows that

we can say little about maneuverability. This way of braking (a sort of pumped braking) gives an almost maximum of decrease, but no guarantee for maneuverability.

I3

01 I a 0.1 0.2 0.3 a.4 0.5 0.6 0.7 0.8

time

5 I

'"i : - - I

I I ' I

01 I O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

time

(g8) vht) - - and v,(t)-

Results of simulations with a sign control. The y e a h irc ?L. ( t ) ai2 cau~ed by the jwzps cf p (t) from 0.9 to 0.2 and back (actually D in the magic formula (f3) jumps when t=0.3 and t=0.6). Through the character of the control there arises a vibration around the equilibrium (A +u h k), that could influence the comfort in the vehicle. The performance of the system depends on the quickness that Mbr can change. The quality of the controller depends onthetimeh + h k .

u - a 0.1 0.2 0.3 0.4 0.5 0.6 0.7

time

(g 9) Again h (t). This time p max jumps from 0.9 to 0.6 and back. h stays below 1.

Introduction ABS Chapter 4

4.2 Control h to a desired value

The stnicture of the first controller we take a look at, looks as follows:

Torque

if (3c<O. 15) h b r = 2 0 . 0 0 0 [Nm/s] *At (Maximum change torque)

if (A>O. 15) h b r = -20.000 [Nm/sl *At

if(Mb,<O) M b r = 0 (Negative torque not possible)

return

We assume that p(A) is known and doesn't change. This means that we can control A to an adjusted value, for example O. 15. Dependent on the sign of the

expression (0.15-A) the torque increases (sign positive) or decreases (sign

negative), both with maximum change. Good functioning during variable brake coefficients is demanded, so we vary the brake coefficient between 0.9 and 0.2.

5 I I I I I I I I I

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 a.a 0.8

time

( g í 3 ) vbt) - - andv,(t)-

Results of the simulations with a proportional control. At this control p mm also jumps from 0.9 to 0.2 and back. AL1 of the plots show an even passage regarding to the sign contol. The efforts of the controlaroundh =h is Less so h = h k

will not be achieved fast. Underneath h (t) is plotted again with p,, between 0.9 and 0.6.

O' l a 0.1 0.2 0.3 0.4 0.5 0.6 n.]

t i e

Introduction ABS Chapter 4

4.2.2 Proportional control

Torque

if (i<0.15) L\Mbr = 133.333[Nm/sl * (0.15-1) * At if (i>o.15) b b r = -35.000[Nm/s] * (A-0.15) * At if (Mbr< o) M b r = 0 (Negative torque not possible)

return

. Dependent on the value of (0.15-A,) we change the torque, limiting M , to 20.000 cNm/s] (this explains the choice of the values in the controller). As A is closer to the desired value A,, the torque changes less.

M

dillcnncc energy mu 0.9/0.2 450, ' * - - - - _

400

350

-

, , c--_

- 3 s o l

; 200

150

,

,J o 0.2 0.4 0.6 0.8 1 1.2 i . 4 1.8

aidanoe

(g 17) A-Energy(xf) (pm 0.9 or 0.2)

-

- - - _ _ _ 50

8

diiionse

(g 15) Energyfxf) (p- O 9 or 0.2)

aifierensc energy rn" o 9/0 6 300

250

O J , \

o o 2 o 4 0.1 0.0 1 1.2 1 4 1.1

didmsc

(g 16) Energy(xf) (p- 0.9 or 0.6)

Reference for the energy as function of the distance xf during the simulations. Figure (g 15) shows the simulation where p,, jumps between 0.9 and 0.2. Figure (g 16) shows the simulation where p,, jumped between 0.9 and 0.6. In contrast with the previous simulations when p,, is a function of time, the jumping of p,, is related to the done distance of the vehicle for a fair comparison. Energy O[J] implicates a vehicle speed of O[mls]. The energy must decrease continuously for the system is pure dissipative.

I - - -

Introduction ABS Chapter 4

4.3 Comparison of energy

Vi%en we want to make a first esmpzu-isun ktweez the esnsders, we cm use an energy criterion. A simulation where we hold p on pm is useful as a

reference trajectory. The energy as a function of time is plotted for p between 0.9 and 0.2 and p between 0.9 and 0.6. For the differences between the

energies are small, we've printed the differences separately. A logical conclusion is to try a sign controller that is proportional around Ak (between 0.1 and 0.2). The energy of this controller is also compared with the other controllers. When p- switches between 0.9 and 0.2, the overall performance is better. When pm switches between 0.9 and 0.6 however, the

sign controller performs better with respect to brake distance.

The structure of the sign/proportional controller looks as follows:

Torque

i f ( A < O . l O ) b4br = 2OOOO[Nm/sl * At i f ( b 0 . 2 0 ) b4br = -20000[Nm/sl * At i f ( ~ < 0 . 2 0 & & A > 0 . 1 0 ) ~ b M b , = 400000* ( 0 . 1 5 - A ) *At

if (Mbr< o) M b r = 0

return

(Negative torque not possible)

cnmpati*an belieen moment. af inertie

o 0.1 0.2 0.3 Q.4 0.5 I

time

(g 19) Moment of inertia 0.25 { - ) and 0.75 { - - ) with saw-toothed torque

lima

(g 20) Moment of inertia 0.25 { -) and 0.75 ( - - ) with aproportional controller

Introduction ABS

~

Chapter 4

4.4 Effect of 0 i

The &iereme k m e n Mb,. md -Uwr resds in ;ui angular acceleration formulated by u5). When the wheel is inphase 3 and the torque increase is not large enough, the resulting torque can grow because of the shape of p(A). From

this point of view we can see the wheel (and its moment of inertia) as a kind of

buffer. So a large torque of inertia is good for a low o and consequently for a low A. The disadvantage of a large moment of inertia is the chance on a shaky h(t). It results in shaking of h(t) and the input Mbr(t). The differences caused

by the moment of inertia are printed on the opposite page.

.

Introduction ABS Chapter 5

5 Simulation program B

Underaeath is writte~ the sm~ctwe of the body ~f the simidation program in ANSI-C. The routines for the controller Torque are described in the previous sections. When the program switches ftom equations of motion I cf4) and US) to equations of motion II c f 4a) and cf 5a) A must be 1. Therefore a new At

must be calculated. After the integrationstep, A will then be 1.

bold : program modules

i t a l i c : program commands

normal: program module name or variables

italic: comments

Main program

Star tup Preparations, opening files

Get parameters Import of parameters

While t < tend and v > O . 2 simulation conditions

get D ( X f , t ) calculating friction as a function of the

done distance or time

onest ep calculate one step

archive to file write results to afile

end

exit

Introduction ABS Chapter 5

Ones t ep

I f p h a s e 2 / 3 t hen model-magic

If p h a s e 4 then model-lock

c a l c u l a t e t o rque

c a l c u l a t e mu

return

Torque

( U s e r de f ined )

return

Described in the previous chapters

Mu

Magic formula (A,D (t) )

return

Model - magic c a l c u l a t e equat ions of motion i Cf4)andIfS)

i n t e g r a t e w i t h E u l e r

c a l c u l a t e íì

i f A 2 1

c a l c u l a t e new At new A tso next A will be 1

r e c a l c u l a t e equat ions of motion I

set p h a s e 4

return

Chapter 5

Model - lock

c a l c u l a t e equa t ions o f motion I 1 Locking wheels, If4a)

and If5a)

i n t e g r a t e w i t h E u l e r

c a l c u l a t e A

i f A < l

c a l c u l a t e new A t

r e c a l c u l a t e equat ions of motion I1

set p h a s e 2 / 3

new A tso next h will be 2

return

Introduction ABS Chapter 5

Bibliography

[I] SOS& Tech. Berichte 7 (1980) 2: Antiblockiersystem (ABS) fur personenkraftwagen

121 Vehicle System Dynamics Volume 20, Number 3-4: Shear Force Development .... : page 159