Vehicle Dynamics – It’s all about the Calculus…

Vehicle Dynamics – It’s all about the Calculus…

J. Christian GerdesAssociate Professor

Mechanical Engineering DepartmentStanford University

Dynamic Design LabStanford University - 2

Future Vehicles…

SafeBy-wire Vehicle Diagnostics

Lanekeeping AssistanceRollover Avoidance

Fun Handling CustomizationVariable Force FeedbackControl at Handling Limits

CleanMulti-Combustion-Mode Engines

Control of HCCI with VVAElectric Vehicle Design


Electric Vehicle Design

How do we calculate the 0-60 time?


Basic Dynamics

Newton’s Second Law

With Calculus

If we know forces, we can figure out velocity

2

2

dtxdm

dtdVmF

maF


What are the Forces?

Forces from: Engine Aerodynamic Drag Tire Rolling Resistance wheel

gear

rRV

2

21 VACF

rR

dtdVm Drr

wheel

gearmotor


Working in the Motor Characteristics

2

21 VACF

rR

dtdVm Drr

wheel

gearmotor

plplslope

plmotor

max

max


Working in the Motor Characteristics

tf

tDrr

wheel

gearmotorf dtVACF

rR

VVm0

20 2

1

plplslope

plmotor

max

max


Some numbers for the Tesla Roadster

From Tesla’s web site: m = mass = 1238 kg Rgear = final drive gear ratio = 8.28 A = Frontal area = Height*width

Overall height is 1.13mOverall width is 1.85mThis gives A = 2.1m2 but the car is not a box. Taking

into account the overall shape, I think A = 1.8 m2 is a better value to use.

CD = drag coefficient = 0.365 This comes from the message board but seems

reasonable


More numbers for the roadster From other sources

rwheel = wheel radius = 0.33m (a reasonable value) Frr = rolling resistance = 0.01*m*g

For reference, see:http://www.greenseal.org/resources/reports/CGR_tire_rollingresistance.pdf

= air density = 1.2 kg/m3

Density of dry air at 20 degrees C and 1 atm To keep in mind:

Engine speed w is in radians/sec The Tesla data is in RPM 1 rad/s = .1047 RPM

(or 0.1 for back of the envelope calculations) 1mph = 0.44704 m/s

wheel

gear

rRV

http://www.greenseal.org/resources/reports/CGR_tire_rollingresistance.pdf


Motor issues

The website lists a motor peak torque of 375 Nm up to 4500RPM. This doesn’t match the graph.

They made changes to the motor when they chose to go with a single speed transmission. I think the specs are from the new motor and the graph from the old one.

Here is something that works well with the new specs:

rad/s 45045032.0375rad/s 450375

Nm

Nmmotor


Results of my simulation

Pretty cool – it gives a 0-60 time of about 3.8s Tesla says “under 4 seconds” Top speed is 128 mph (they electronically limit to 125)


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


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

handwheel angle sensor

handwheel feedback motor

steering actuatorshaft angle sensor

power steering unitpinion

steering rack


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 d – our input

Model Get slip angles, then tire forces, then derivatives

af

ard bV

ba

r


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, d and b Calculate front and rear forces from slip angles Calculate changes in r and b

rI

maF

zz

yy

rIbFaF

rmVFF

zyryf

yryf

)(b


Calculate Slip Angles

rVbr

Va

VbrV

VarV

rf

rf

badba

bba

bbda

cossintan

cossintan

af

ard bV

ba

r

d af

bcosV

arV bsinar

bcosV

brV bsin


Lateral Force Behavior

ms=1.0 and mp=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 a

nd t

p/t p0

F/Fz

tp/tp0

zpFCm

aa tan


When Do Cars Spin Out?

Can we figure out when the car will spin and avoid it?


0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

Front slip angle

a f (rad

)

GPSNL Observer

0 2 4 6 8 10 12 14 16

0

0.05

0.1

Rear slip angle

Time (s)

a r (rad

)

0 0.05 0.1 0.15 0.2 0.25 0.30

1000

2000

3000

4000

5000

6000

7000

8000Tire Curve

-Lat

eral

Fro

nt T

ire F

orce

Fyf

(N)

Slip angle af (rad)

linear nonlinear

Comparing our Model to Reality

loss of control


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…


Handling Limits

What happens when tire forces saturate? Front tire

Reduces “spring” force Loss of control input

Rear tire Vehicle will tend to spin Loss of stability

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

1000

2000

3000

4000

5000

6000

alpha (rad)

-Fy

(N) handling limits

linear region

Is the lanekeeping system safe at the limits?


Countersteering Simple lanekeeping algorithm will countersteer

Lookahead includes heading error Large heading error will change direction of steering

Lanekeeping system also turns out of a skid

Lateral error

Projected error

Example: Loss of rear tire traction


Lanekeeping at Handling Limits


Video from Dropped Throttle Tests


Controller countersteers to prevent spinout

Lanekeeping Active Lanekeeping Deactivated

Yaw Stability from Lanekeeping


Controller response to heading error prevents the vehicle from spinning

A Closer Look


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

Documents

Vehicle Dynamics – It’s all about the Calculus…