Art. Ackermann Bunte3

8/10/2019 Art. Ackermann Bunte3

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YAW DISTURBANCE ATTENUATION BY ROBUST

DECOUPLING OF CAR STEERING

J. Ackermann T. Bunte

Deutsche Forschungsanstaltfur Luft- und Raumfahrt (DLR) Oberpfaffenhofen

Institut fur Robotik und Systemdynamik82234 Wessling, Germany

e-mail: [email protected]

Abstract: Robust decoupling of the lateral and yaw motions of a car has beenachieved by feedback of the integrated yaw rate into front wheel steering. In thepresent paper the yaw disturbance attenuation is analyzed for a generic single-trackvehicle model. The frequency limit, up to which yaw disturbances are attenuated,is calculated. For specific vehicle data, it is shown that this control law significantlyreduces the influence of yaw disturbances on yaw rate and side-slip angle for lowfrequencies. This safety advantage is experimentally verified for -split braking.

Keywords: Automotive control, decoupling problems, robust control, disturbancerejection, sensitivity functions

1. INTRODUCTION

Consider the vehicle of Fig. 1.

The input to the system is the front-wheel steeringangle f, and the output is the yaw rate r =

measured by a gyro. is the heading anglebetween an inertial coordinate system (x0, y0) andthe chassis coordinate system (x, y). Uncertainparameters are

(1) the velocityv = |v| >0,(2) the cornering stiffness of the tires,

(3) the masses mf and mr at the front andrear axles; it is assumed, however, that thelocation of the center of gravity (CG) at adistance f from the front axle and r fromthe rear axle is known, i.e. mff = mrr.The wheelbase is = r+ fand the vehiclemassm = mr+ mf. The assumed ideal massdistribution with mr =mf/, mf =mr/implies the moment of inertia Jz w.r.t. avertical axis through the CG as

Jz = mr2r+mf

2f =mrf. (1)

For the case of non ideal mass distributionthe reader is referred to (Ackermann, 1994).

For small sideslip angle and small steeringangle fthe single-track model of car steering is(Mitschke, 1990):

mv(+r) = Fy+ Fd

Jzr = Mz+Md.(2)

The lateral force Fy and the torque Mz arounda vertical axis through the CG are generated

by side forces of the tires. Also, a disturbance

Fig. 1. Vehicle with ideal mass distribution


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force Fd and a disturbance torque Md act onthe vehicle. Examples are crosswind, lateral roadslope, flat tires, and asymmetric braking forceson a slippery road. In particular, an unexpecteddisturbance torque Md can lead to dangerousdriving situations because of the reaction timeof the driver, followed by overreactions. Time-delay and high gain may even cause instability ofthe driver-vehicle feedback system. An automaticcontrol system with feedback of the yaw rate r

can react faster and more precisely than a humandriver (Ackermann et al., 1993; Ackermann etal., 1996). By integration of r it also providesa relative direction reference with respect to theinitial heading of the car before the disturbance.The driver does not have to worry about theautomatically controlled yaw motion. He only hasto command a lateral acceleration to keep thecar, considered as a mass point, on his plannedpath. He can easily compensate for the lateraldisturbance forces Fd, e.g. from a lateral slope ofthe road; therefore Fd = 0 was set for simplernotation.

The next sections show the derivation of the dis-turbance transfer function fromMdto the sideslipangle at the front axle and to the yaw rate, first inSection 2 for a conventional car, then in Section3 for a robustly decoupled car. In Section 4 thesteady-state effects and in Section 5 the ratio ofthe transfer functions (sensitivity function) areanalyzed, and the frequency limit of disturbanceattenuation is calculated. Section 6 gives an ex-ample, and Section 7 shows experimental results.

2. YAW DISTURBANCE TRANSFERFUNCTION OF THE CONVENTIONAL CAR

Equations (1) and (2) withFd= 0 may be written

mv(+r)

mrfr

=

Fy

Mz

+

0

Md

. (3)

The steering forceFyand torqueMzare generatedby the lateral tire forcesFf(f) and Fr(r) via

FyMz

= 1 1

f r

Ff(f)Fr(r)

. (4)

The tire forces depend on the tire sideslip anglesf and r, as illustrated in Fig. 2 for the frontwheel. The local velocity vector vf forms the(chassis) slip angle fwith the car body and thetire slip anglefwith the tire direction; thus

f = f f

r = r r.(5)

Fig. 2. Variables of the tire model

In this paper rear wheel steering is not used,i.e. r 0. The front sideslip angle f will beintroduced as a state variable. The sideslip anglesat the CG () and at the rear axle (r) are relatedtofby the kinematic relations for small angles

= ff

vr

r = f

vr.

(6)

The tire force characteristics are linearized as

Ff(f) = cff

Fr(r) = crr(7)

where the cornering stiffnesses cf and cr are un-certain parameters that vary with the road tirecontact. It is assumed that cf = cf0, cv = cr0where cf0 and cr0 are nominal values for the dryroad and [ ; 1] is an uncertain parameterwith > 0. Further uncertain constant param-eters are the vehicle mass m [m ; m+] andvelocity v [v ; v+]. The model (3), (4) withthe above equations substituted becomes

mv(f fv r+r)mrfr

= (8)

=

1 1

f r

cf(f f)

cr(f+

vr)

+

0

Md

and, solving for f and r,

f

r

=

1

mfv 0

1mf 1mr

cf(f f)

cr(f+

v r)

(9)

1

0

r+

1

mfv

mr+mf

mrmf2

Md

where the front and rear masses mf = mr/andmr = mf/have been substituted to replacem, r and f. The state equations of the systemare then


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fst,dec= Bdec(0)

Ddec(0)mf=1

cr(18)

rst,dec= Rdec(0)

Ddec(0)mf= 0.

The yaw rate goes to zero, i.e. the car has aconstant heading angle as a steady-state responseto a step yaw disturbance.

5. DISTURBANCE ATTENUATION RATIOS;FREQUENCY LIMIT

Define

(s) =Bdec(s)D(s)

Ddec(s)B(s)

r(s) =Rdec(s)D(s)

Ddec(s)R(s) =

sD(s)

Ddec(s) (19)

(s) =1 cf

mrvsB(s) r(s)as disturbance attenuation ratios (sensitivity func-tions). The influence of the yaw disturbances onf or r is reduced for all frequencies for which|(j)| .

6. EXAMPLE

Consider a car with the following parameters

f= 1.514m, r = 1.323m, m= 1916kg,

cf= 49400N/rad, cr = 103800N/rad

and uncertain velocity v [15 ; 220] km/h.

Fig. 3 shows the disturbance attenuation at fivevelocity values. The frequency limit in the lowerfigure is between 0.6 and 1 Hertz.

7. EXPERIMENTAL RESULTS

A periodic disturbance torque Md is not typicalfor car driving. A much harder test is a stepdisturbanceMd(s) = M /s. This is generated ex-perimentally in the split test shown in Fig. 4.The left-hand column of pictures, taken at 1-second intervals, shows the experiment with a con-

ventional car. The car drives with its right-handwheels (left-hand side of the figure) on water-flooded tiles with a friction coefficient 0.1,and with the left-hand wheels on wet asphaltwith a friction coefficient 0.9. At an initialvelocity of 80 km/h the driver brakes and keepshis steering wheel straight. In the second picturethe beginning of skidding is recognizable; only onesecond later the car has all four tires on asphalt ina safe situation. In normal driving on an icy road ittakes the driver just this second to react, and thatmay be too late. The right-hand column shows the


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Fig. 4. -split-braking at 80 km/h, left: conventional vehicle, right: robustly decoupled vehicle


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0 2 40

20

40

60

v/[km/h]

conventional car

0 2 40

20

40

60decoupled car

0 2 46

4

2

0

2

4

f

[]

,

L

[](

)

0 2 46

4

2

0

2

4

0 2 410

0

10

r[/sec]

0 2 410

0

10

0 2 4

1

0

1

2

ayCG

[m/sec

2]

t/sec

0 2 4

1

0

1

2

t/sec

Fig. 5. Experimental data for-split braking withinital velocityv 50km/h

same experiment with the robustly decoupled ve-hicle. In the second picture the situation is similarto that of the conventional car. But now the yawrate is measured and the controller steers the carquickly back to the original heading angle. Thecar remains straight with yaw rate back to zero,

see (18), until the complete stop. This spectac-ular safety improvement is not only obtained atnominal speed, load and friction coefficients butwith perfect robustness against changes in theseparameters.Fig. 5 shows a comparison of experimental data

for -split braking for the conventional car (left)

and for the decoupled car (right). The velocity vshows the occurrence of the Md-step by brakingand the speed reduction. The angleL(handwheelangle steering gear ratio) is kept close to zeroin both experiments, in the decoupled car a frontwheel steering angle f from the controller (13)occurs in response to the initial peak in the yawrate r. In contrast to the conventional vehicle,feedback quickly reduces r and changes its signsuch that (t) = (t0) according to (17). The

lateral acceleration ayCG = v

+r

at the cen-ter of gravity shows a significant reduction by thedecoupling controller.

8. CONCLUSIONS

The robustly decoupling control law for cars at-tenuates yaw disturbances, e.g. from crosswinds,flat tires, or asymmetric braking forces for lowfrequencies. The frequency limit is calculated forgeneric vehicle parameters. The effect for a spe-cific car is shown by the calculated disturbanceattenuation ratio and by-split braking tests.

9. REFERENCES

Ackermann, J. (1994). Robust decoupling of carsteering dynamics with arbitrary mass distri-bution. In: Proc. American Control Confer-ence. Vol. 2. Baltimore, USA. pp. 19641968.

Ackermann, J., A. Bartlett, D. Kaesbauer,W. Sienel and R. Steinhauser (1993). Robustcontrol: Systems with uncertain physical pa-

rameters. Springer. London.Ackermann, J., T. Bunte, W. Sienel, H. Jeebe

and K. Naab (1996). Driving safety by robuststeering control. In: Proc. Int. Symposium onAdvanced Vehicle Control. Aachen, Germany.

Mitschke, M. (1990).Dynamik der Kraftfahrzeuge.Vol. C. Springer. Berlin.

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