Download State and Parameter Estimation for Vehicle Dynamics Control Using

Spinning Out, With Calculus
J. Christian Gerdes
Associate Professor
Mechanical Engineering Department
Stanford University
Future Vehicles…
Clean
Multi-Combustion-Mode Engines
Control of HCCI with VVA
Electric Vehicle Design
Safe
Fun
By-wire Vehicle Diagnostics
Lanekeeping Assistance
Rollover Avoidance
Handling Customization
Variable Force Feedback
Control at Handling Limits
Stanford University
-2
Dynamic Design Lab
Future Systems


Change your handling… … in
software
Customize real cars like those in
a video game
Stanford University


Use GPS/vision to assist the
driver with lanekeeping
Nudge the vehicle back to the
lane center
-3
Dynamic Design Lab
Steer-by-Wire Systems

Like fly-by-wire aircraft
 Motor for road wheels
 Motor for steering wheel
 Electronic link
handwheel
handwheel angle sensor

handwheel feedback motor
Like throttle and brakes
shaft angle sensor

What about safety?
 Diagnosis
 Look at aircraft
steering actuator
power steering unit
pinion
steering rack
Stanford University
-4
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
-5
Dynamic Design Lab
Lanekeeping on the Corvette
Stanford University
-6
Dynamic Design Lab
Lanekeeping Assistance



Stanford University
Energy predictions work!
Comfortable, guaranteed lanekeeping
Another example with more drama…
-7
Dynamic Design Lab
P1 Steer-by-wire Vehicle

“P1” Steer-by-wire vehicle



Independent front steering
Independent rear drive
Manual brakes
steering
motors

handwheel
Entirely built by students

5 students, 15 months from start to first driving tests
Stanford University
-8
Dynamic Design Lab
When Do Cars Spin Out?

Can we figure out when the car will spin and avoid it?
Stanford University
-9
Dynamic Design Lab
Tires

Let’s use your knowledge of Calculus to make a
model of the tire…
Stanford University
- 10
Dynamic Design Lab
An Observation…

A tire without lateral force moves in a straight line
Tire without lateral force
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 subjected to lateral force moves diagonally
Tire with 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
How is this possible?
Shouldn’t the tire be stuck to the road?
Stanford University
- 17
Dynamic Design Lab
Tire Force Generation


The contact patch does stick to the ground
This means the tire deforms (triangularly)
Stanford University
- 18
Dynamic Design Lab
Tire Force Generation


a
Fy  Caa

Stanford University
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
- 19
Dynamic Design Lab
Saturation at Limits

a
Eventually tire force
saturates
 Friction limited
 Rear part of contact
patch saturates first
Fy
a
Stanford University
- 20
Dynamic Design Lab
Simple Lateral Force Model
x=a
x = -a
a

Deflection initially triangular

Defined by slip angle
v(x) = (a-x) tana
a

Force follows deflection

qy(x) = cpy(a-x) tana
Stanford University

Assume constant foundation
stiffness cpy
qy(x) is force per unit length
- 21
Dynamic Design Lab
Simple Lateral Force Model

x=a
x = -a
a
Calculate lateral force
a
Fy   q y ( x)dx
a
a
v(x) = (a-x) tana
a
qy(x) = cpy(a-x) tana
Stanford University
   c py (a  x) tan adx
a
 2c py a 2 tan a  Ca tan a
Cornering stiffness
- 22
Dynamic Design Lab
Tire Forces with Saturation

Tire force limited by friction

Assume parabolic normal force
distribution in contact patch
qz(x)
Stanford University
- 23
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
msqz(x)
mpqz(x)
Lateral force and deflection are friction limited

qy(x) <mqz(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
msqz(x)
mpqz(x)
Lateral force and deflection are friction limited


qy(x) <mqz(x)
Result: the rear part of the contact patch is always sliding
large slip
Stanford University
small slip
- 25
Dynamic Design Lab
Calculate Lateral Force
Fy 
q
y
adhesion
( x)dx 
q
y
( x)dx
sliding
a
xsl
xsl
a
   c py (a  x) tan adx   m s q z ( x)dx
xsl
3Fz  a 2  x 2 


q z ( x) 
2
4a  a 
msqz(x)
q y ( xsl )  m p q z ( xsl )
mpqz(x)
Stanford University
- 26
Dynamic Design Lab
Lateral Force Model
The entire contact patch is sliding when a  asl
3m p Fz
tan a sl 
Ca
 The lateral force model is therefore:

2

Ca
 Ca tan a 
Fy (a )  
3m p Fz



3




m
C
 2  s  tan a tan a  a 1  2m s  tan 3 a
2 2 


m
9
m
Fz  3m p 
p
p


 m s Fz sgn a
a  a sl
a  a sl
Figures show shape of this relationship
Stanford University
- 27
Dynamic Design Lab
Lateral Force Behavior

ms=1.0 and mp=1.0

Fiala model
1
F/Fz
0.9
tp/t p0
0.8
0.6
0.5
F/F
z
and t /t
p p0
0.7
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
 C qtan a
2
2.5
3
a
m p Fz
Stanford University
- 28
Dynamic Design Lab
Coefficients of Friction
Sliding (dynamic friction): ms = 0.8


Adhesion (peak friction): mp = 1.6



Many force-slip plots have
approximately this much friction after
the peak, when the tire is sliding
Seen in previous literature
1
0.8
y

F

0.6
0.4
0.2
0
0
5
10
15
20
25
a
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
Stanford University
- 29
Dynamic Design Lab
Lateral Force with Peak and Slide Friction

ms=0.8 and mp=1.6

1
Peak in curve
0.8
0.6
p p0
0.2
and t /t
z
tp/t p0
0.4
F/F
F/Fz
0
-0.2
-0.4
0
0.5
1
1.5
 C qtan a
2
2.5
a

Can we predict friction on road?
Stanford University
m p Fz
- 30
Dynamic Design Lab
3
Testing at Moffett Field
Stanford University
- 31
Dynamic Design Lab
How Early Can We Estimate Friction?
Tire Curve
Front slip angle
8000
GPS
NL Observer
0.2
0.1
0
0
2
4
6
8
10
12
14
16
ar (rad)
Rear slip angle
0.1
0.05
-Lateral Front Tire Force Fyf (N)
af (rad)
0.3
7000
6000
5000
4000
3000
2000
1000
0
0
0
2
4
6
8
10
12
14
16
Time (s)
linear
Stanford University
0
0.05
0.1
0.15
0.2
0.25
0.3
Slip angle af (rad)
nonlinear loss of
control
- 32
Dynamic Design Lab
Ramp: Friction Estimates
Friction estimated about halfway to the peak – very early!

Friction coefficient
Steering Angle
2
1.8
1.6
0
2
4
6
8
10
12
14
16
af (rad)
Front Slip Angle
0.3
0.2
0.1
0
2
4
6
8
10
12
14
16
1.2
1
0.8
0.6
Lateral Acceleration
ay (g)
1.4
Estimated m
 (rad)
0
-0.1
-0.2
-0.3
0
0.4
-0.5
0.2
-1
0
0
2
4
6
8
10
12
14
16
0
2
4
8
10
12
14
16
Time (s)
Time (s)
linear
Stanford University
6
nonlinear
- 33
loss of
control
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
- 34
Dynamic Design Lab
Bicycle Model

Basic variables





Speed V (constant)
Yaw rate r – angular velocity of the car
Sideslip angle b – Angle between velocity and heading
Steering angle  – our input
Model

Get slip angles, then tire forces, then derivatives

af
Stanford University
a
b
b
ar
V r
- 35
Dynamic Design Lab
Calculate Slip Angles

af
a
b
b
ar
V r
V cos b
V sin b  ar
 af
V cos b
V sin b  br
V sin b  ar
V cos b
a
a f  b  r 
V
tan a f    
Stanford University
ar
V sin b  br
V cos b
b
ar  b  r
V
tan a r 
- 36
Dynamic Design Lab
Vehicle Model

Get forces from slip angles (we already did this)
 Vehicle Dynamics

Fy  ma y
Fyf  Fyr  mV ( b  r )
 z  I z r
aFyf  bFyr  I z r
This is a pair of first order differential equations



Calculate slip angles from V, r,  and b
Calculate front and rear forces from slip angles
Calculate changes in r and b
Stanford University
- 37
Dynamic Design Lab
Making Sense of Yaw Rate and Sideslip
r / rad/s
0.4
0.2
0
0
2
4
t/s
6
8
4
t/s
6
8
b / rad
0
-0.1
-0.2
-0.3
0

actual
desired
2
What is happening with this car?
Stanford University
- 38
Dynamic Design Lab
For Normal Driving, Things Simplify

Slip angles generate lateral forces
a
Fy

Simple, linear tire model (no spin-outs possible)
Fyf  Caf a f
Fyr  Cara r
Stanford University
a


Fyf  Caf  b  r   
V


b 

Fyr  Car  b  r 
V 

- 39
Dynamic Design Lab
Two Linear Ordinary Differential Equations
Fyf  Fyr  mV ( b  r )
aFyf  bFyr  I z r
 Caf  Car


b  
mV
 r    aC  bC
ar
   af

Iz


Stanford University
a


Fyf  Caf  b  r   
V


b 

Fyr  Car  b  r 
V 


 Caf 

 1


2
b


mV
mV 




2
2


a Caf  b Car   r   Caf 




I
I zV
 z 

aCaf  bCar
- 40
Dynamic Design Lab
Conclusions

Engineers really can change the world


Many of these changes start with Calculus




In our case, change how cars work
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
- 41
Dynamic Design Lab
P1 Vehicle Parameters
a  1.35 m
b  1.15 m
m  1724 kg
N
Caf  90000
rad
N
Car  138000
rad
I z  1100 kg  m 2
Stanford University
- 42
Dynamic Design Lab