Spinning Out, With Calculus J. Christian Gerdes Associate Professor Mechanical Engineering...

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


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


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 2



keFvirtual 2


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


Lanekeeping on the Corvette


Lanekeeping Assistance

Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama…


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


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…


Dynamic Design Lab

An Observation…

A tire without lateral force moves in a straight line

Tire without lateral force


Dynamic Design Lab

An Observation…




Dynamic Design Lab

An Observation…

A tire subjected to lateral force moves diagonally

Tire with lateral force


Dynamic Design Lab

An Observation…




Dynamic Design Lab

An Observation…


How is this possible?Shouldn’t the tire be stuck to the road?


Dynamic Design Lab

Tire Force Generation

The contact patch does stick to the ground This means the tire deforms (triangularly)


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


Dynamic Design Lab

Saturation at Limits

Eventually tire force saturates Friction limited Rear part of contact

patch saturates first

Fy


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


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


Dynamic Design Lab

Tire Forces with Saturation

Tire force limited by friction Assume parabolic normal force

distribution in contact patch

qz(x)


Dynamic Design Lab



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)


Dynamic Design Lab



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)


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


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


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


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


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


Dynamic Design Lab

Testing at Moffett Field


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


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


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


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


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


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

)(


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


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


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

)(


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


Dynamic Design Lab

P1 Vehicle Parameters

21100

1724

13800015.1

9000035.1

mkgI

kgmrad

NCmb

rad

NCma

z

r

f

Spinning Out, With Calculus J. Christian Gerdes Associate Professor Mechanical Engineering...

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