Transcript Stability at the Limits

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
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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
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To understand how loss of control occurs, need
to know what determines vehicle motion
1
2
National Highway Traffic Safety Administration. Traffic safety facts (2002)
USA Today. Study of deadly crashes involving 16-19 year old drivers (2003)
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motion of a vehicle
SIDE VIEW
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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
Contact Patch
Ground
BOTTOM VIEW
a
Fy
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tire curve
maximum tire grip
Linear
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Saturation
Loss of control
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vehicle response
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Normally, we operate in linear region
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Predictable vehicle response
But during slick road conditions,
emergency maneuvers, or
aggressive/performance driving
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Enter nonlinear tire region
Response unanticipated by driver
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loss of control
Imagine making an aggressive turn
 If front tires lose grip first, plow out of turn
(limit understeer)
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If rear tires saturate, rear end kicks out (limit
oversteer)
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may go into oscillatory response
driver loses ability to influence vehicle motion
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
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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

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Vehicle/Tire models
Tire parameter estimation
2. Produce stable, predictable response
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Feedback linearizing controller
Driver input saturation
Simulation results
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vehicle model
Bicycle model
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2 states: β and r
Nonlinear tire model
(Dugoff)
Steer-by-wire
Assume
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Small angles
Ux constant
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equations of motion
Sum forces and
moments:
Dugoff tire model:
-Ca

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tire estimation algorithm
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Find af: use GPS/INS
Find Fyf: SBW motor
give steering torque
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Estimate Ca f and 
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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
steering angle
 (deg)
0
-10
-20
26
28
30
32
34
36
38
40
42
36
38
40
42
36
38
40
42
a f (deg)
front slip angle
15
10
5
0
26
28
30
32
34
F
yf
(N)
front lateral force
0
-2000
-4000
-6000
-8000
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28
30
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time (s)
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getting the data
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estimation technique
9000
8000
7000
side force -F
yf
(N)
6000
5000
4000
3000
2000
1000
0
-1000
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0
2
4
6
8
10
slip angle a f (deg)
14
12
14
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parameter estimates
Begin estimating  after entering NL region
Ca f estimate is steady
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7
Incremental Fit Error
x 10
2.5
1.06
MSE (N2)
 estimate
1.08
1.04
1.02
els
2
enls
1.5
1
0.5
1
26
28
30
32
34
36
38
40
0
42
26
28
30
32
34
36
38
40
42
38
40
42
4
x 10
7
MSE difference (N2)
9.5
9
8.5
26
Incremental Fit Error Difference
x 10
C
af
estimate
10
28
30
32
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time (s)
36
38
40
2
1
0
42
26
15
28
30
32
34
time (s)
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controller design
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Desired vehicle response
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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
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Could compromise between the two
Or, track one state exactly
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feedback linearization (FBL)
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Nonlinear control technique
Applicable to systems that look like:
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Use input to cancel system nonlinearities.
In our case,
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Apply linear control theory to track desired
trajectory:
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FBL in action
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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
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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
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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?
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Prevents vehicle from needing more side
force than is available
Keeps vehicle in linearizable handling region
Saturation algorithm
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If a < ath, driver commands are OK
If a ¸ ath, gradually saturate driver’s steering
capability
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overall control system
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Ramp steer from 0 to 6° at 20 m/s (45 mph) in 1 s
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Tracks linear model yaw rate, then saturates input
Reduced sideslip
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design considerations
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Relative importance of  vs. r
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Which produces a more predictable
response?
Could add additional input to track 
and r
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differential drive
rear steering
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conclusions
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Overall approach
1.
Sense tire saturation and actively compensate
for them with SBW inputs
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2.
Make vehicle response more predictable
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Algorithm can characterize tires (Ca, ) using GPSbased af and estimates of Fyf,
Up to capabilities of tires, controller tracks linear yaw
rate trajectory
Reduces sideslip
Current work
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Estimate Ca,  on board in real-time
Implement overall controller on research vehicle
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controller validation
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Simulate control system on more complete
vehicle model
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validation results II
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input: ramp steer from 0 to 5° at 45 mph in 0.5 s
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4 cases
Case 1: Both tires are linear (f ¸ 1 and r ¸ 1)
  Caf  Car

   
mV

  
C ar b  C a f a
 r 

Iz

 C ar b  Ca f a  
 Caf

 1  

   
m V2


  Cm Va


2
2
 C a f a  C ar b   r   af

 I z
I zV

Car 

m V   f 
Car b    r 
I z 
Case 2: Both tires saturating (f < 1 and r < 1)
2
2


F
f nf
   f Fnf Car

Car b




r

r
 4C m V
2
    m V
m
V
m
V
 af
2 
    a F
 a 2f Fnf2
bC
C
b
f nf

r
a
r
a
r


 



r


Iz
Iz
I zV   4Caf I z

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Car 

m V   v1 
 
Car b   r 


I z 
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4 cases
Case 3: front is nonlinear, rear is linear (f ¸ 1 and r < 1)
Caf a
   r Fnr Caf
  Caf



r

r
  mV
2
    m V
m
V
m
V

2
    b F
aC
C
a
af
af
  Caf a
r nr
 r  



r
 I
  I
Iz
I zV
z

  z
 r2 Fnr2 

4Car m V   f 
b r2 Fnr2   v 2 

4Car I z 
Case 4: front is linear, rear is nonlinear (f ¸ 1 and r < 1)
2
2


F
f
nf




r



 
mV
 4Caf m V




 
2
2
a

F

b

F

a

F
f
nf
r
nr

r

f nf
  



Iz
 4Caf I z
  f Fnf   r Fnr
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 r2 Fnr2 

4Car m V   v1 
 
b r2 Fnr2  v 2 


4Car I z 
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Dynamic Design Laboratory
new inputs

Define new inputs v1 and v2


1
v1  
   ra  

f
V











1

v2  
rb
     
r
V


to represent system as
x  f ( x)  g ( x)  u
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More general form of FBL
y 
SISO algorithm:
x  f ( x)  g ( x)u
y  h( x )
h
h
f ( x) 
g ( x)u

x 

x

 

L f h( x)
Lg h ( x )
if L g h( x)   ,
u

1
 L f h( x )  w
L g h( x )

if L g h( x)  0,
y 
L f h
L g h
f ( x) 
g ( x)u

x

x
 
L2f h ( x )
Lg L f h ( x )
if L g L f h( x)   ,
u
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
1
 L2f h( x)  w
L g L f h( x )
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
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driver saturation algorithm
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Front steering only approach
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Model Fyf as:
Substitute into system equations:
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Tracking yaw rate
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Choose new input
cr = 200
c = 50
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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
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Using least squares
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Experimental Tire Curve
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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
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Motivation
Background
Controller design

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Feedback linearization
Driver input saturation
Validation on complex model
Conclusions
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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
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Steer-by-wire
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Linear tire model
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Nonlinear tire model
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comparing vehicle responses
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Ramp steer to from 0 to 4o at 45 mph in 0.5 s
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tire estimation algorithm

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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
NLS fit to Dugoff
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Compare fit errors to tell us
if in nonlinear region
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