Stability at the Limits

2005 ASME IMECE November 10, 2005 Dynamic Design Laboratory

Stability at the Limits

Yung-Hsiang Judy HsuJ. Christian Gerdes

Stanford University

2005 ASME IMECE 2 Stanford UniversityDynamic Design Laboratory

did you know… Every day in the US, 10 teenagers are killed in

teen-driven vehicles in crashes1

Loss of control accounts for 30% of these deaths Inexperienced drivers make more driving errors,

exceed speed limits & run off roads at higher rates In 2002, motor vehicle traffic crashes were the

leading cause of death for ages 3-33.2

To understand how loss of control occurs, need to know what determines vehicle motion

1 National Highway Traffic Safety Administration. Traffic safety facts (2002)2 USA Today. Study of deadly crashes involving 16-19 year old drivers (2003)


motion of a vehicle

Motion of a vehicle is governed by tire forces

Tire forces result from deformation in contact patch

Lateral tire force is a function of tire slip

SIDE VIEW

Ground

BOTTOM VIEW

Contact Patch

Fy


tire curvemaximum tire grip

Linear Saturation Loss of control


vehicle response Normally, we operate in linear region

Predictable vehicle response But during slick road conditions,

emergency maneuvers, or aggressive/performance driving Enter nonlinear tire region Response unanticipated by driver


loss of controlImagine making an aggressive turn If front tires lose grip first, plow out of turn

(limit understeer) may go into oscillatory response driver loses ability to influence vehicle motion

If rear tires saturate, rear end kicks out (limit oversteer) may go into a unstable spin driver loses control

Both can result in loss of control


overall goals

We’d like to design a control system to Stabilize vehicle in nonlinear handling

region Make vehicle response consistent and

predictable for drivers Communicate to driver when limits of

handling are approaching


Outline

1. Identify tire operating region Vehicle/Tire models Tire parameter estimation

2. Produce stable, predictable response Feedback linearizing controller Driver input saturation Simulation results


vehicle modelBicycle model

2 states: β and r Nonlinear tire model

(Dugoff) Steer-by-wire

Assume Small angles Ux constant


equations of motionSum forces and

moments:

Dugoff tire model:-C


tire estimation algorithm

Find f: use GPS/INS Find Fyf: SBW motor

give steering torque

Estimate C f and LS fit to linear tire model NLS fit to Dugoff model Compare residual of fits to tell us if we’re in the

nonlinear region estimate


tire parameter estimation

26 28 30 32 34 36 38 40 42-20

-10

0steering angle

(d

eg)

26 28 30 32 34 36 38 40 420

5

10

15

front slip angle

f (d

eg)

26 28 30 32 34 36 38 40 42-8000-6000-4000-2000

0

front lateral forceF

yf (N

)

time (s)


getting the data


0 2 4 6 8 10 12 14 16 18-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

side

forc

e -F

yf (N

)

slip angle f (deg)

estimation technique


parameter estimates Begin estimating after entering NL region C f estimate is steady

26 28 30 32 34 36 38 40 421

1.02

1.04

1.06

1.08

es

timat

e

26 28 30 32 34 36 38 40 42

8.5

9

9.5

10x 10

4

C

f est

imat

e

time (s)

26 28 30 32 34 36 38 40 420

0.5

1

1.5

2

2.5

x 107 Incremental Fit Error

MS

E (N

2 )

els

enls

26 28 30 32 34 36 38 40 420

1

2

x 107 Incremental Fit Error Difference

MS

E d

iffer

ence

(N2 )

time (s)


controller design

Desired vehicle response Track response of bicycle model with linear tires Be consistent with what driver expects

When tires saturate, compensate for decreasing forces with steer-by-wire input

One input f; two states ,r Could compromise between the two Or, track one state exactly


feedback linearization (FBL)

Nonlinear control techniqueApplicable to systems that look like:

Use input to cancel system nonlinearities.In our case,

Apply linear control theory to track desired trajectory:


FBL in action Ramp steer from 0 to 4o at 20 m/s (45 mph) in 1 s Controller results in exact tracking of linear tire model yaw

rate trajectory


FBL in action Ramp steer from 0 to 6o at 20 m/s (45 mph) in 1 s FBL works well up to physical capabilities of tires


driver input saturation Road naturally saturates driver’s steering

capability often unexpectedly Here, we safely limit steering capability in

a predictable, safe manner Why do we need it?

Prevents vehicle from needing more side force than is available

Keeps vehicle in linearizable handling region Saturation algorithm

If < th, driver commands are OK If ¸ th, gradually saturate driver’s steering

capability


overall control system Ramp steer from 0 to 6° at 20 m/s (45 mph) in 1 s

Tracks linear model yaw rate, then saturates input Reduced sideslip


design considerations

Relative importance of vs. r Which produces a more predictable

response? Could add additional input to track

and r differential drive rear steering


conclusions Overall approach

1. Sense tire saturation and actively compensate for them with SBW inputs Algorithm can characterize tires (C, ) using GPS-

based f and estimates of Fyf, 2. Make vehicle response more predictable

Up to capabilities of tires, controller tracks linear yaw rate trajectory

Reduces sideslip Current work

Estimate C, on board in real-time Implement overall controller on research vehicle



controller validation Simulate control system on more complete

vehicle model


validation results II input: ramp steer from 0 to 5° at 45 mph in 0.5 s


4 casesCase 1: Both tires are linear (f ¸ 1 and r ¸ 1)

Case 2: Both tires saturating (f < 1 and r < 1)

r

f

z

r

z

f

rf

z

rf

z

fr

frrf

IbC

IaC

mVC

mVC

rVI

bCaCI

aCbCmV

aCbCmV

CC

r

22

21

r

z

r

zf

nff

r

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r

z

nff

rrnff

v

IbC

ICFa

mVC

mVCF

rVIbC

IbC

IFa

rrmVbC

mVC

mVF

r

122

22

2

2

4

4


4 casesCase 3: front is nonlinear, rear is linear (f ¸ 1 and r < 1)

Case 4: front is linear, rear is nonlinear (f ¸ 1 and r < 1)

222

22

2

2

4

4v

ICFb

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new inputs

Define new inputs v1 and v2

to represent system as

fVrav

11

rVrbv

12

uxgxfx )()(


More general form of FBL

SISO algorithm:

wxhLxhLL

u

xhLLif

uxgxhL

xfxhL

y

xhLif

wxhLxhL

u

xhLif

uxgxhxf

xhy

ffg

fg

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g

xhL

f

g

fg

g

xhLxhL

fgf

gf

)()(

1

,)(

)()(

,0)(

)()(

1

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)()(

2

)()(

)()(

2

)()()(

xhyuxgxfx


driver saturation algorithm


Front steering only approach Model Fyf as: Substitute into system equations:


Tracking yaw rate Choose new input

cr = 200c = 50


Estimating Cf

1. Find f: Use GPS/INS to measure r and f and estimate

2. Find Fyf: Estimate tm from steering geometry, model tp as

and use disturbance torque estimate from SBW system to find Fyf

3. Estimate : Using least squares


Experimental Tire Curve P1: Ramp steer from 0 to 9° in 24 s at V = 31 mph

shad_2004-12-11_l.mat


questions?


overview

Motivation Background Controller design

Feedback linearization Driver input saturation

Validation on complex model Conclusions


steer-by-wireRemoves mechanical linkage between steering wheel and road wheels

electronically actuate steering system separately from driver’s commands

decouple underlying dynamics from driver force feedback

Conventional steering Steer-by-wire


Linear tire model


Nonlinear tire model


comparing vehicle responses Ramp steer to from 0 to 4o at 45 mph in 0.5 s


tire estimation algorithm

Find f: GPS/INS measures , r, V

Find Fyf: SBW motor give steering torque

Estimate C f and from (Fyf, f) data LS fit to line NLS fit to Dugoff

Compare fit errors to tell us if in nonlinear region

Stability at the Limits

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