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
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Electric Vehicle Design
How do we calculate the 0-60 time?
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Basic Dynamics
Newton’s Second Law
With Calculus
If we know forces, we can figure out velocity
2
2
dtxdm
dtdVmF
maF
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What are the Forces?
Forces from: Engine Aerodynamic Drag Tire Rolling Resistance wheel
gear
rRV
2
21 VACF
rR
dtdVm Drr
wheel
gearmotor
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Working in the Motor Characteristics
2
21 VACF
rR
dtdVm Drr
wheel
gearmotor
plplslope
plmotor
max
max
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Working in the Motor Characteristics
tf
tDrr
wheel
gearmotorf dtVACF
rR
VVm0
20 2
1
plplslope
plmotor
max
max
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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When Do Cars Spin Out?
Can we figure out when the car will spin and avoid it?
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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
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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
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Lanekeeping on the Corvette
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Lanekeeping Assistance
Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama…
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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?
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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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Lanekeeping at Handling Limits
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Video from Dropped Throttle Tests
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Controller countersteers to prevent spinout
Lanekeeping Active Lanekeeping Deactivated
Yaw Stability from Lanekeeping
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Controller response to heading error prevents the vehicle from spinning
A Closer Look
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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
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