Coordination of Steering and Individual Wheel Braking Actuated Vehicle Yaw Stability Control

8/9/2019 Coordination of Steering and Individual Wheel Braking Actuated Vehicle Yaw Stability Control

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1 -

Figu re

1:

model regulator based vehicle dynamics steering

control

in Section 2. Model regulator based steering control

is considered in Section

3.

Model regulator based in-

dividual wheel braking control is presented in Section

4. Section presents strategies for combining the tech-

nologies in the previous two sections for coordinated

steering and individual wheel braking action. Finally,

the paper ends in Section

6

with some conclusions and

discussion.

2

Vehicle Model

A two track model that neglects roll and pitch motions

is used here as it is the simplest model that can accom-

modate steering and individual wheel braking action.

Th e model presented here is similar to the mathemati-

cal model considered by [ l l 121. The geometry of the

double track model is shown in F i g 2 Since the roll and

1 1

t

Y

Figure 2: vehicle model

pitch motions have been neglected, dynamic equations

aregivenas, i =

1 , 2 , 3 , 4 ) ,

4

m a z

TU,)

= ~ ( F z , c o s 6 ,F,isin&)

m a,+rv,) = ~ ( F z i s i n 6 i + F , i c o s & )

(2)

(1)

i= 1

4

i l

4

Z,? = ~l , i (F , ; cos6 i -Fy ,s in6 , )

(3)

I (F,, sin

6,

+ F,i cos6,

+

M ;

where

1,i

=

lr3

=

-lW/2

,

Z2=

r 4

=

1,/2

11

=

ly*

= f . ly3 = 1.4

= -1

61 =

62

= 6/ ,63

=

6 = 0

for front wheel steering vehicle, where the parameters

are defined in Table 1 .

Table

1:

List of physical meanings of two track vehicle

model

The dynamics

of

each tire is modelled

as

4)

. - T . - R F

w - ,

e i

= 1 , 2 , 3 , 4

where

Zw

is the moment

of

inertia of each tire about

its axis of rotation,

wi is

its angular speed,

Re

is the

effective tire radius and T, is the braking or traction

moment. The tire center speeds are,

1

=

Uz

Tlw/2)zf

U, T I , ) ; (5)

U3 = Uz - l,/2)7+ U - ) j 7)

U4

=

Uz

Tlw/2)

a (Uy

Tl,)j

(8)

uz = U= /Z) zf U, rlf); (6)

The tire side slip angles are given by,

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The longitudinal wheel slip ratio for each tire is given,

model regulator (also called the disturbance observer).

The simplified tire model of Dugoff (see [13]) is given

by,

F Z i

= JiCzg~; (14)

F,, - = fiC,.@i (15)

where CZiand

C ;

are the longitudinal and cornering

stiffness of the t ire, respectively. The coefficients f i in

Eq.14 and Eq.15 are determined by using,

where

FR,= J C, ,S;)~ C,,ai)

(17)

Note that the ti re model given in equations

Eq.14

thru

Eq.17

is

a simple model that can represent tire forces

under combined steering and braking action.

3 Steering Based Implementation

The steering based model regulator for VDC is pre-

sented in this section in accordance with the earlier.

work of

t h e

authors in [9, o]. The linearized version

of the two track model with only front wheel steering

actuation for yaw stabilization is expressed as,

T

=

G,tSf +GdM,i

18)

= ( G , s t ( l + Anst))6f GdMwi

where GS t , d and Gnatare the steering command,

disturbance rejection and nominal steering command

functions, respectively. The terms due t o the model

uncertainty Anst and disturbance GdM,, are treated

as

an extended disturbance e in model regulator design

T

=

Gnat6f

Gn s t A d f

+Gdhfw, (19)

= G, f

+ e

e = T-Gn,t6f (20)

The aim

of

the model regulator based steering con-

troller is to regulate,

T = G n d f (21)

regardless of the model uncertainty and yaw distur-

bance moment. Interested readers are encouraged to

refer to [5, 6, 71 for more detailed information on the

-

The new control signal 61. given by,

will result

in

the achievement of the goal in Eq.21

The above control equation is modified by multiplying

the feedback quantities on its right hand side by the

tunable low pass

filter

Q to obtain

as the implementation equation (see for instance,

Fig.1). Q is used for achieving robustness

of

stability

and for making

/Gnat

causal.

The two track model can be linearized into a linear

parameter varying model. The timevarying parameter

is the vehicle longitudinal speed

U=.

Parameter space

methods are then customarily used to design robust

velocity scheduled controllers,

(12,

3,

91)

and they

are

illustrated in Fig.1

as GnSt(s , v , ) .

4

Individual Wheel Braking Based

Implementation

The linearized version

of

the two track model with the

individual wheel braking action can be expressed.as,

- .

( 141~

T

= G;bTi+GdMw;

(24)

= (Gnib(1+ ))Ti + G ,i

where

Gib, Gn;b

and

A;,

are the braking command,

desired braking command transfer functions and the

multiplicative model uncertainty, respectively. Similar

P-

Figure

3:

Model regulator based individual wheel braking

controller

derivations lead t o the individual.wheel braking model

regulator implementation equation,

(25)

i T,;

-

TQ

QT;

Gnib

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where Tniis the new input equal to zero. The transfer

function is given by Eq.25 and its block diagram is

given in Fig.

3.

5

Combined Action, Coordination and

Simulation Stu dy

the^ basic method of combining steering and individ-

ual wheel braking compensation may be to use a single

common model regulator and to consider prespecified

proportion of the generated control inputs. The gener-

ated control input is given,

G

U

=

--r+Qu

a = 7

Ti = ( I - y ) u

where steering wheel command is assumed to be

zero.

The block diagram representation of

Eq.26

is shown in

Fig. 4. Several coordination strategies for distributing

control between steering and individual wheel braking

actions are possible. The most obvious strategies are

to use,

1. y= 1 i.e. only steering control in cer tain chosen

speed ranges

2. y

=

0 i.e. only individual wheel braking in certain

chosen speed ranges other than those in case

1

3. 0 < y < 1

i.e. combined steering and individual

wheel braking control

Simulation results for

all

three cases are shown in Fig.5

IMi

I

Figure

4:

Combined controller with actuation proportion-

ing

thru Fig.7. In the first simulation scenario (see Fig.

5 ,

only steering action

(y = 1)

is considered. The

responses to a step steering input of 6,

=

Y and side

wind moment of

M i =

1000 N.m. are given. The

desired transfer function from steering input to yaw

rate is specified as,

where

K n v z ) = Gsr s,vz)l,,o (28)

is the static gain at the longitudinal speed of U, =

30m/sec. Eq.27 is considered in all of the simulation

scenarios. A simple cruise control system is integrated

into the two track model t o keep constant longitudinal

velocity of the vehicle. Note tha t both t he linearized

version of the two track model and the controller used

are linear parameter varying

LPV).

The varying pa-

rameter is vehicle longitudinal speed U= that is used to

implement the controllers Eq.22, Eq.25 and Eq.26

as

continuous gain scheduling controllers. The

Q

filter is

chosen

as

1

& = -

TQS 1

with ~ = 0 . l

ec

is chosen thru all the simulation sc e

narios. An investigation of the simulation result in Fig.

5 shows tha t good steering command following and yaw

disturbance moment rejection are achieved in case

1

(7 =

1).

Individual wheel braking control ( y

= 0)

with a side

wind moment disturbance of 1000Nm is considered in

th e simulation scenario given by Fig.6. The desired

transfer function from the brake input to the yaw rate

output is specified

as

having the same form and numer-

ical parameters as given in Eq.27. An investigation of

Fig3

shows tha t good distu rbance rejection has been

achieved even though the results are not

as

good as

those in Fig.5 realized with steering actua tion. Bet-

ter tuning of the controller

for

=

0

would obviously

improve those results. The presence of a first order

braking actuator model with

0.1

sec time constant in

the simulation scenarios also degrades performance.

The yaw moment disturbance rejection properties of

th e individual wheel braking based implementation can

easily be improved by incorporating some simultaneous

steering action. Fig.7 displays the yaw moment distur-

bance rejection responses for y=O.l,

0.2

and 0.3. More

steering action is added progressively to complement

an individual wheel braking based yaw stability con-

troller. One can observe th at increasing the proportion

of steering action may improve the overall disturbance

rejection property.

The coordination study reported here is preliminary

in nature. Results not reported here indicate th at the

ideal steering

to

individual wheel braking action p r e

portioning depends on vehicle longitudinal speed.

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lnn1U

Figure

5: Steering based responses

7

1)

5 a ; ; s m

6 Conclusions

-

q

Figure 7: Combined action responses

References

[I] T. Acarman,

Y .

Pan and U . Ozguner. A Con-

trol authority transition system for collision and acci-

dent avoidance. Journal

of

Vehicle System Dynamics,

The Special Issue on Intelligent

ansportation

terns, vo , 39, no, 2, pp. 149-187,

2o03,

A revised model regulator

hss

been formulated here for

coordinated use of steering and individual wheel brak-

i n v art,llat,ionwit,h t,he

aim

of achievinc better vehicle

-

__e__ II ~~ ....~. ~ ~ ~ . .

yaw

stability control. The use

of

steering and individ-

ual wheel braking action have been treated separately

first. Then, some possible strategies .for the combined

coordinated use of both actions in a model regulator

121 J. Ackermann, P. Blue, T Biinte, L. Giiveq, D .

Kaesbauer, M. Kordt,

M.

Muhler, D. Odenthal. Ro-

bust Control, the Structural Approach. Springer Verlag,

London, 2002.

ther studies that consider the combined implementa-

[S) B. Aksun Guvenq,

L.

GiivenG. Robust model reg-

tion and proportioning

of

individual actions are under- ulator based vehicle dynamics cont.ro1 using individual

way. wheel braking. ESDA 2002, the 6th Biennial Con-

292


6/6

fewnce on Engineering Systems Design and Analysis,

Istanbul, 2002.

[SI

L.

Giivens, K. Srinivasan. Friction compensation

and evaluation for

a

force control application.

J.

of

Mechanicnl Systems and Signal Processing,

vol.

8, no.

6, pp. 623-638, 1994.

[6]

K.

Ohnishi.

A

new servo method in mechatronics.

Puns. Japanese Soc. Elect. Eng, vol. 107-D, pp. 83-86,

1987.

[7]

T.

Umeno and

Y .

Hori. Robust speed control of

dc servomotors using modern two degrees-of-freedom

controller design. IEEE hnsact ions on Industrial

Electronics,

vol 38,

no.

5,

pp.

363-368, 1991.

[8] A.T. van Zanten,

R.

Erhardt and

G.

Pfaff. VDC,

the vehicle dynamics control system of Bosch. SAE

paper, No. 950759, 1995.

191 B. Aksun Giivenq,

L.

Giiveng. Robust steer-by-

wire control based on the model regulator. Jo in t

IEEE

Conference on Control Applications and IEEE Con-

ference

on

Computer Aided Control System Design,

Glasgow, pp. 435440, 2002.

[lo] B. Aksun Giivenq,

L.

Giivens. The limited inte-

grator model regulator and its use in vehicle steering

control.

Thrkish

Journal of Engineering and Environ-

mental Sciences,

,

pp.

473-482, 2002.

[ l l ]

S.

Horiuchi, K . Okada and

S.

Nohtomi. Effects

of integrated control of active four wheel steering and

individual wheel torque on vehicle handling and sta-

bility - A comparison of alternative control strategies.

The Dynamics of Vehicles on Roads and on Iltncks,

Supplement t o Vehicle System Dynamics ,Vol. 33,

R.

Frohling (Editor), 2000.

[12]

U . Kiencke,

L.

Nielsen. Automotive Control Sys-

tems for Engine, Driveline and Vehicle. Springer Ver-

lag (SAE), Berlin, 2000.

1131 H. Dugoff,P. Fancher and L. Segel. An analysis

of tire traction properties and their influence on vehicle

dynamic performance. SAE paper

no. 700977, 1970.

[14] L. Giivenq and J.Ackermann. Links between the

parameter space and frequency domain methods

of

ID.

bust control. International Journal of Robust and Non-

linear Control, Vol. 11 , no. 15, pp. 1435-1453, 2001.

[15] B. Aksun Giivenq, L. Giivens and E.S. Oztiirk

hlodel regulator based individual wheel braking .con-

trol.

IEEE

Conference on Control Applications,

Istanbul, accepted for publication,

2003.

Appendix

The numerical values of he two track vehicle model

used

in

the

simulation and design studies are

chosen

=,

Table 2: numerical values of vehicle and tire model pa-

rameters

Acknowledgement

The authors would like to thank Tevfik Yigit and Eyiip

Serdar Oztiirk for their help in implementing the sim-

ulation model.

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Coordination of Steering and Individual Wheel Braking Actuated Vehicle Yaw Stability Control

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