Stability at the Limits
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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)
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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
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tire curvemaximum tire grip
Linear Saturation Loss of control
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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
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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
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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
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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
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vehicle modelBicycle model
2 states: β and r Nonlinear tire model
(Dugoff) Steer-by-wire
Assume Small angles Ux constant
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equations of motionSum forces and
moments:
Dugoff tire model:-C
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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
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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)
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getting the data
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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
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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)
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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
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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:
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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
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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
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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
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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
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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
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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
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controller validation Simulate control system on more complete
vehicle model
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validation results II input: ramp steer from 0 to 5° at 45 mph in 0.5 s
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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
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z
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zf
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rrmVbC
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mVF
r
122
22
2
2
4
4
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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
IaC
mVCF
mVC
rVIaC
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IFb
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ICFb
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new inputs
Define new inputs v1 and v2
to represent system as
fVrav
11
rVrbv
12
uxgxfx )()(
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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
,)(
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,0)(
)()(
1
,)(
)()(
2
)()(
)()(
2
)()()(
xhyuxgxfx
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driver saturation algorithm
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Front steering only approach Model Fyf as: Substitute into system equations:
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Tracking yaw rate Choose new input
cr = 200c = 50
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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
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Experimental Tire Curve P1: Ramp steer from 0 to 9° in 24 s at V = 31 mph
shad_2004-12-11_l.mat
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questions?
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overview
Motivation Background Controller design
Feedback linearization Driver input saturation
Validation on complex model Conclusions
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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
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Linear tire model
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Nonlinear tire model
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comparing vehicle responses Ramp steer to from 0 to 4o at 45 mph in 0.5 s
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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
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