Post on 26-Dec-2015
Spinning Out, With Calculus
J. Christian Gerdes
Associate Professor
Mechanical Engineering Department
Stanford University
Stanford University- 2 Dynamic Design Lab
Future Vehicles…
Safe
By-wire Vehicle DiagnosticsLanekeeping Assistance
Rollover Avoidance
Fun
Handling CustomizationVariable Force FeedbackControl at Handling Limits
Clean
Multi-Combustion-Mode EnginesControl of HCCI with VVA
Electric Vehicle Design
Stanford University- 3 Dynamic Design Lab
Future Systems
Change your handling… … in software
Customize real cars like those in a video game
Use GPS/vision to assist the driver with lanekeeping
Nudge the vehicle back to the lane center
Stanford University- 4 Dynamic Design Lab
Steer-by-Wire Systems
Like fly-by-wire aircraft Motor for road wheels Motor for steering wheel Electronic link
Like throttle and brakes
What about safety? Diagnosis Look at aircraft
handwheel
2)( keeV
handwheel angle sensor
handwheel feedback motor
steering actuatorshaft angle sensor
power steering unitpinion
steering rack
keFvirtual 2keFvirtual 2 keFvirtual 2
keFvirtual 2keFvirtual 2 keFvirtual 2
keFvirtual 2
keFvirtual 2keFvirtual 2 keFvirtual 2
keFvirtual 2keFvirtual 2 keFvirtual 2
keFvirtual 2
Stanford University- 5 Dynamic Design Lab
Lanekeeping with Potential Fields
Interpret lane boundaries as a potential field
Gradient (slope) of potential defines an additional force
Add this force to existing dynamics to assist
Additional steer angle/braking
System redefines dynamics of driving but driver controls
Stanford University- 6 Dynamic Design Lab
Lanekeeping on the Corvette
Stanford University- 7 Dynamic Design Lab
Lanekeeping Assistance
Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama…
Stanford University- 8 Dynamic Design Lab
P1 Steer-by-wire Vehicle
“P1” Steer-by-wire vehicle Independent front steering Independent rear drive Manual brakes
Entirely built by students 5 students, 15 months from start to first driving tests
steering motors
handwheel
Stanford University- 9 Dynamic Design Lab
When Do Cars Spin Out?
Can we figure out when the car will spin and avoid it?
Stanford University- 10
Dynamic Design Lab
Tires
Let’s use your knowledge of Calculus to make a model of the tire…
Stanford University- 11
Dynamic Design Lab
An Observation…
A tire without lateral force moves in a straight line
Tire without lateral force
Stanford University- 12
Dynamic Design Lab
An Observation…
A tire without lateral force moves in a straight line
Tire without lateral force
Stanford University- 13
Dynamic Design Lab
An Observation…
A tire without lateral force moves in a straight line
Tire without lateral force
Stanford University- 14
Dynamic Design Lab
An Observation…
A tire subjected to lateral force moves diagonally
Tire with lateral force
Stanford University- 15
Dynamic Design Lab
An Observation…
A tire subjected to lateral force moves diagonally
Tire with lateral force
Stanford University- 16
Dynamic Design Lab
An Observation…
A tire subjected to lateral force moves diagonally
Tire with lateral force
Stanford University- 17
Dynamic Design Lab
An Observation…
A tire subjected to lateral force moves diagonally
How is this possible?Shouldn’t the tire be stuck to the road?
Stanford University- 18
Dynamic Design Lab
Tire Force Generation
The contact patch does stick to the ground This means the tire deforms (triangularly)
Stanford University- 19
Dynamic Design Lab
Tire Force Generation
Force distribution is triangular
More force at rear Force proportional to slip
angle initially Cornering stiffness
Force is in opposite direction as velocity
Side forces dissipative
CFy
Stanford University- 20
Dynamic Design Lab
Saturation at Limits
Eventually tire force saturates Friction limited Rear part of contact
patch saturates first
Fy
Stanford University- 21
Dynamic Design Lab
Simple Lateral Force Model
Deflection initially triangular Defined by slip angle
Force follows deflection Assume constant foundation
stiffness cpy
qy(x) is force per unit length
x = ax = -a
v(x) = (a-x) tan
qy(x) = cpy(a-x) tan
Stanford University- 22
Dynamic Design Lab
Simple Lateral Force Model
Calculate lateral forcex = ax = -a
v(x) = (a-x) tan
qy(x) = cpy(a-x) tan
tantan2
tan)(
)(
2 Cac
dxxac
dxxqF
py
a
a
py
a
a
yy
Cornering stiffness
Stanford University- 23
Dynamic Design Lab
Tire Forces with Saturation
Tire force limited by friction Assume parabolic normal force
distribution in contact patch
qz(x)
Stanford University- 24
Dynamic Design Lab
Tire Forces with Saturation
Tire force limited by friction Assume parabolic normal force
distribution in contact patch Rubber has two friction
coefficients: adhesion and sliding
Lateral force and deflection are friction limited qy(x) <qz(x)
sqz(x)
pqz(x)
Stanford University- 25
Dynamic Design Lab
Tire Forces with Saturation
Tire force limited by friction Assume parabolic normal force
distribution in contact patch Rubber has two friction
coefficients: adhesion and sliding
Lateral force and deflection are friction limited qy(x) <qz(x) Result: the rear part of the contact patch is always sliding
large slip small slip
sqz(x)
pqz(x)
Stanford University- 26
Dynamic Design Lab
Calculate Lateral Force
dxxqdxxac
dxxqdxxqF
sl
sl
x
a
zs
a
x
py
sliding
y
adhesion
yy
)(tan)(
)()(
2
22
4
3)(
a
xa
a
Fxq z
z
sqz(x)
pqz(x)
xsl
)()( slzpsly xqxq
Stanford University- 27
Dynamic Design Lab
Lateral Force Model
The entire contact patch is sliding when sl
The lateral force model is therefore:
Figures show shape of this relationship
slzs
slp
s
zpp
s
zpy
F
F
C
F
CC
F
sgn
tan3
21
9tantan2
3tan
)(3
22
32
C
Fzpsl
3tan
Stanford University- 28
Dynamic Design Lab
Lateral Force Behavior
s=1.0 and p=1.0 Fiala model
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
q
F/F
z an
d t
p/tp0
F/Fz
tp/t
p0
zpF
C
tan
Stanford University- 29
Dynamic Design Lab
Coefficients of Friction
Sliding (dynamic friction): s = 0.8 Many force-slip plots have
approximately this much friction after the peak, when the tire is sliding
Seen in previous literature
Adhesion (peak friction): p = 1.6 Tire/road friction, tested in stationary conditions, has been
demonstrated to be approximately this much Seen in previous literature
Model predicts that these values give Fpeak / Fz = 1.0 Agrees with expectation
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Fy
Stanford University- 30
Dynamic Design Lab
Lateral Force with Peak and Slide Friction
s=0.8 and p=1.6 Peak in curve
Can we predict friction on road?
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
q
F/F
z an
d t
p/tp0
F/Fz
tp/t
p0
zpF
C
tan
Stanford University- 31
Dynamic Design Lab
Testing at Moffett Field
Stanford University- 32
Dynamic Design Lab
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
Front slip angle
f (ra
d)
GPS
NL Observer
0 2 4 6 8 10 12 14 16
0
0.05
0.1
Rear slip angle
Time (s)
r (ra
d)
0 0.05 0.1 0.15 0.2 0.25 0.30
1000
2000
3000
4000
5000
6000
7000
8000Tire Curve
-La
tera
l Fro
nt T
ire F
orc
e F
yf (
N)
Slip angle f (rad)
linear nonlinear
How Early Can We Estimate Friction?
loss of control
Stanford University- 33
Dynamic Design Lab
Ramp: Friction Estimates
Friction estimated about halfway to the peak – very early!
0 2 4 6 8 10 12 14 16
-0.3
-0.2-0.1
0Steering Angle
(r
ad
)
0 2 4 6 8 10 12 14 16
0.1
0.2
0.3
Front Slip Angle
f (ra
d)
0 2 4 6 8 10 12 14 16-1
-0.5
0
Lateral Acceleration
a y (g
)
Time (s)
0 2 4 6 8 10 12 14 160
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Friction coefficient
Est
ima
ted
Time (s)
linear nonlinear loss of control
Stanford University- 34
Dynamic Design Lab
Bicycle Model
Outline model How does the vehicle move when I turn the steering
wheel? Use the simplest model possible Same ideas in video games and car design just with more
complexity
Assumptions Constant forward speed Two motions to figure out – turning and lateral movement
Stanford University- 35
Dynamic Design Lab
Bicycle Model
Basic variables Speed V (constant) Yaw rate r – angular velocity of the car Sideslip angle – Angle between velocity and heading Steering angle – our input
Model Get slip angles, then tire forces, then derivatives
f
r V
ba
r
Stanford University- 36
Dynamic Design Lab
Calculate Slip Angles
rV
br
V
aV
brV
V
arV
rf
rf
cos
sintan
cos
sintan
f
r V
ba
r
f
cosV
arV sinr
cosV
brV sin
Stanford University- 37
Dynamic Design Lab
Vehicle Model
Get forces from slip angles (we already did this) Vehicle Dynamics
This is a pair of first order differential equations Calculate slip angles from V, r, and Calculate front and rear forces from slip angles Calculate changes in r and
rI
maF
zz
yy
rIbFaF
rmVFF
zyryf
yryf
)(
Stanford University- 38
Dynamic Design Lab
Making Sense of Yaw Rate and Sideslip
What is happening with this car?
0 2 4 6 80
0.2
0.4
t / s
r / r
ad
/s
0 2 4 6 8
-0.3
-0.2
-0.1
0
t / s
/ r
ad
actualdesired
Stanford University- 39
Dynamic Design Lab
For Normal Driving, Things Simplify
Slip angles generate lateral forces
Simple, linear tire model (no spin-outs possible)
rryr
ffyf
CF
CF
Fy
rV
bCF
rV
aCF
ryr
fyf
Stanford University- 40
Dynamic Design Lab
Two Linear Ordinary Differential Equations
z
f
f
z
rf
z
rf
rfrf
I
CmV
C
r
VI
CbCa
I
bCaCmV
bCaC
mV
CC
r 22
21
rV
bCF
rV
aCF
ryr
fyf
rIbFaF
rmVFF
zyryf
yryf
)(
Stanford University- 41
Dynamic Design Lab
Conclusions
Engineers really can change the world In our case, change how cars work
Many of these changes start with Calculus Modeling a tire Figuring out how things move Also electric vehicle dynamics, combustion…
Working with hardware is also very important This is also fun, particularly when your models work! The best engineers combine Calculus and hardware
Stanford University- 42
Dynamic Design Lab
P1 Vehicle Parameters
21100
1724
13800015.1
9000035.1
mkgI
kgmrad
NCmb
rad
NCma
z
r
f