Vehicle Dynamics – It’s all about the 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
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Dynamic Design Lab
Electric Vehicle Design

How do we calculate the 0-60 time?
Stanford University
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Dynamic Design Lab
Basic Dynamics

Newton’s Second Law
F  ma

With Calculus
2
dV
d x
F m
m 2
dt
dt

If we know forces, we can figure out velocity
Stanford University
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Dynamic Design Lab
What are the Forces?

Forces from:



Engine
Aerodynamic Drag
Tire Rolling Resistance

V  Rgear
rwheel
dV  motor Rgear
1
m

 Frr  ACDV 2
dt
rwheel
2
Stanford University
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Dynamic Design Lab
Working in the Motor Characteristics
 max
   pl

 motor  
 max   slope     pl     pl
dV  motor Rgear
1
2
m

 Frr  ACDV
dt
rwheel
2
Stanford University
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Dynamic Design Lab
Working in the Motor Characteristics
 max
   pl

 motor  
 max   slope     pl     pl
  motor Rgear
1
2
mV f  V0    
 Frr  ACDV dt
rwheel
2

t 0 
tf
Stanford University
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Dynamic Design Lab
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 1.13m
 Overall width is 1.85m
 This 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
Stanford University
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Dynamic Design Lab
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
V  Rgear
The Tesla data is in RPM

rwheel
1 rad/s = .1047 RPM
 (or 0.1 for back of the envelope calculations)
1mph = 0.44704 m/s
Stanford University
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Dynamic Design Lab
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:
375 Nm
  450 rad/s

 motor  
375 Nm  0.32    450   450 rad/s
Stanford University
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Dynamic Design Lab
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)
Stanford University
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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
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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
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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
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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 d – our input
Model

Get slip angles, then tire forces, then derivatives
d
af
Stanford University
a
b
b
ar
V r
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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, d and b
Calculate front and rear forces from slip angles
Calculate changes in r and b
Stanford University
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Dynamic Design Lab
Calculate Slip Angles
d
af
a
b
b
ar
V r
V cos b
V sin b  ar
d af
V cos b
V sin b  br
V sin b  ar
V cos b
a
a f  b  r d
V
tan a f  d  
Stanford University
ar
V sin b  br
V cos b
b
ar  b  r
V
tan a r 
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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
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Dynamic Design Lab
When Do Cars Spin Out?

Can we figure out when the car will spin and avoid it?
Stanford University
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Dynamic Design Lab
Comparing our Model to Reality
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
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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
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Dynamic Design Lab
Lanekeeping on the Corvette
Stanford University
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Dynamic Design Lab
Lanekeeping Assistance



Stanford University
Energy predictions work!
Comfortable, guaranteed lanekeeping
Another example with more drama…
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Dynamic Design Lab
Handling Limits

What happens when tire forces saturate?
 Front tire
6000


Reduces “spring” force
Loss of control input
Rear tire


Vehicle will tend to spin
Loss of stability
5000
4000
-Fy (N)

handling limits
3000
linear region
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
alpha (rad)
0.3
0.35
0.4
Is the lanekeeping system safe at the limits?
Stanford University
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Dynamic Design Lab
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
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Dynamic Design Lab
Lanekeeping at Handling Limits
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Dynamic Design Lab
Video from Dropped Throttle Tests
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Dynamic Design Lab
Yaw Stability from Lanekeeping
Lanekeeping Active
Lanekeeping Deactivated
Controller countersteers to prevent spinout
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Dynamic Design Lab
A Closer Look
Controller response to heading error
prevents the vehicle from spinning
Stanford University
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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
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Dynamic Design Lab