Stability at the Limits
Yung-Hsiang Judy Hsu
J. Christian Gerdes
Stanford University
2005 ASME IMECE November 10, 2005 Dynamic 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)
2005 ASME IMECE 2 Stanford University
Dynamic Design Laboratory
motion of a vehicle
SIDE VIEW
Motion of a vehicle
is governed by tire
forces Contact Patch
Tire forces result Ground
from deformation in BOTTOM VIEW
contact patch a
Lateral tire force is
a function of tire
slip
Fy
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Dynamic Design Laboratory
tire curve
maximum 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
2005 ASME IMECE 5 Stanford University
Dynamic Design Laboratory
loss of control
Imagine 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
2005 ASME IMECE 6 Stanford University
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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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Dynamic Design Laboratory
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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Dynamic Design Laboratory
vehicle model
Bicycle model
2 states: β and r
Nonlinear tire model
(Dugoff)
Steer-by-wire
Assume
Small angles
Ux constant
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Dynamic Design Laboratory
equations of motion
Sum forces and
moments:
Dugoff tire model:
-Ca
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Dynamic Design Laboratory
tire estimation algorithm
Find af: use GPS/INS
Find Fyf: SBW motor
give steering torque
Estimate Ca 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
2005 ASME IMECE 11 Stanford University
Dynamic Design Laboratory
tire parameter estimation
steering angle
0
(deg)
-10
-20
26 28 30 32 34 36 38 40 42
front slip angle
15
a f (deg)
10
5
0
26 28 30 32 34 36 38 40 42
front lateral force
0
-2000
Fyf (N)
-4000
-6000
-8000
26 28 30 32 34 36 38 40 42
time (s)
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Dynamic Design Laboratory
getting the data
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Dynamic Design Laboratory
estimation technique
9000
8000
7000
6000
side force -Fyf (N)
5000
4000
3000
2000
1000
0
-1000
0 2 4 6 8 10 12 14 16 18
slip angle a f (deg)
2005 ASME IMECE 14 Stanford University
Dynamic Design Laboratory
parameter estimates
Begin estimating after entering NL region
Ca f estimate is steady
7
x 10 Incremental Fit Error
1.08
2.5 els
1.06
estimate
2 enls
MSE (N2)
1.04 1.5
1
1.02
0.5
1
26 28 30 32 34 36 38 40 42 0
26 28 30 32 34 36 38 40 42
4
x 10 7
Incremental Fit Error Difference
10 x 10
MSE difference (N2)
Ca f estimate
9.5 2
9
1
8.5
26 28 30 32 34 36 38 40 42 0
26 28 30 32 34 36 38 40 42
time (s)
time (s)
2005 ASME IMECE 15 Stanford University
Dynamic Design Laboratory
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
2005 ASME IMECE 16 Stanford University
Dynamic Design Laboratory
feedback linearization (FBL)
Nonlinear control technique
Applicable 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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Dynamic Design Laboratory
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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Dynamic Design Laboratory
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
2005 ASME IMECE 19 Stanford University
Dynamic Design Laboratory
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 a < ath, driver commands are OK
If a ¸ ath, gradually saturate driver’s steering
capability
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Dynamic Design Laboratory
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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Dynamic Design Laboratory
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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Dynamic Design Laboratory
conclusions
Overall approach
1. Sense tire saturation and actively compensate
for them with SBW inputs
Algorithm can characterize tires (Ca, ) using GPS-
based af 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 Ca, on board in real-time
Implement overall controller on research vehicle
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Dynamic Design Laboratory
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Dynamic Design Laboratory
controller validation
Simulate control system on more complete
vehicle model
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Dynamic Design Laboratory
validation results II
input: ramp steer from 0 to 5° at 45 mph in 0.5 s
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Dynamic Design Laboratory
4 cases
Case 1: Both tires are linear (f ¸ 1 and r ¸ 1)
Caf Car Car b Caf a Caf Car
1
mV mV 2 mV
C a mV f
Car b r
r Car b Caf a Caf a 2 Car b 2 r af
Iz Iz
Iz I zV
Case 2: Both tires saturating (f < 1 and r < 1)
f Fnf Car C b 2 Fnf
f
2
Car
mV
ar 2 r r
mV v1
a2f 2
4C mV
a F mV mV
r
bCar Car b 2 a f Fnf Car b r
f nf
r
Iz Iz I zV 4Caf I z
Iz
2005 ASME IMECE 27 Stanford University
Dynamic Design Laboratory
4 cases
Case 3: front is nonlinear, rear is linear (f ¸ 1 and r < 1)
r Fnr Caf C af a Caf r2 Fnr
2
r r
mV
2
4Car mV f
b F
mV mV mV
b r2 Fnr v 2
2
r C a
r af
aCaf C af a 2
I
r nr
z Iz I zV Iz 4Car I z
Case 4: front is linear, rear is nonlinear (f ¸ 1 and r < 1)
2 Fnf
f Fnf r Fnr f
2
r2 Fnr
2
r
4Car mV v1
a F b F a2f 2
mV 4C mV
a f Fnf 2
r
f nf r nr b r2 Fnr v 2
4C I
Iz
af z 4Car I z
2005 ASME IMECE 28 Stanford University
Dynamic Design Laboratory
new inputs
Define new inputs v1 and v2
1 1
v1 v2
ra
rb
r
f
V V
to represent system as
x f ( x) g ( x) u
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Dynamic Design Laboratory
More general form of FBL
h h
y
f ( x) g ( x)u
x x
SISO algorithm: L f h( x) Lg h ( x )
if L g h( x) ,
x f ( x) g ( x)u
y h( x )
u
1
L g h( x )
L f h( x ) w
if L g h( x) 0,
L f h L g h
y f ( x) g ( x)u
x
x
L2f h ( x ) Lg L f h ( x )
if L g L f h( x) ,
u
1
L g L f h( x )
L2f h( x) w
2005 ASME IMECE
30 Stanford University
Dynamic Design Laboratory
driver saturation algorithm
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Dynamic Design Laboratory
Front steering only approach
Model Fyf as:
Substitute into system equations:
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Dynamic Design Laboratory
Tracking yaw rate
Choose new input
cr = 200
c = 50
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Dynamic Design Laboratory
Estimating Caf
1. Find af: 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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Dynamic Design Laboratory
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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Dynamic Design Laboratory
questions?
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Dynamic Design Laboratory
overview
Motivation
Background
Controller design
Feedback linearization
Driver input saturation
Validation on complex model
Conclusions
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Dynamic Design Laboratory
steer-by-wire
Removes 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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Dynamic Design Laboratory
Linear tire model
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Dynamic Design Laboratory
Nonlinear tire model
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Dynamic Design Laboratory
comparing vehicle responses
Ramp steer to from 0 to 4o at 45 mph in 0.5 s
2005 ASME IMECE 41 Stanford University
Dynamic Design Laboratory
tire estimation algorithm
Find af: GPS/INS measures , r, V
Find Fyf: SBW motor give steering
torque
Estimate Ca f and from (Fyf, af) data
LS fit to line Compare fit errors to tell us
if in nonlinear region
NLS fit to Dugoff
2005 ASME IMECE 42 Stanford University
Dynamic Design Laboratory