Vehicle Dynamics – It’s all about the Calculus… J. Christian Gerdes Associate Professor...
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Transcript of Vehicle Dynamics – It’s all about the Calculus… J. Christian Gerdes Associate Professor...
Vehicle Dynamics – It’s all about the Calculus…
J. Christian Gerdes
Associate Professor
Mechanical Engineering Department
Stanford University
Dynamic Design LabStanford University - 2
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
Dynamic Design LabStanford University - 3
Electric Vehicle Design
How do we calculate the 0-60 time?
Dynamic Design LabStanford University - 4
Basic Dynamics
Newton’s Second Law
With Calculus
If we know forces, we can figure out velocity
2
2
dt
xdm
dt
dVmF
maF
Dynamic Design LabStanford University - 5
What are the Forces?
Forces from: Engine Aerodynamic Drag Tire Rolling Resistance wheel
gear
r
RV
2
2
1VACF
r
R
dt
dVm Drr
wheel
gearmotor
Dynamic Design LabStanford University - 6
Working in the Motor Characteristics
2
2
1VACF
r
R
dt
dVm Drr
wheel
gearmotor
plplslope
pl
motor
max
max
Dynamic Design LabStanford University - 7
Working in the Motor Characteristics
tf
t
Drrwheel
gearmotorf dtVACF
r
RVVm
0
20 2
1
plplslope
pl
motor
max
max
Dynamic Design LabStanford University - 8
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
Dynamic Design LabStanford University - 9
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
r
RV
Dynamic Design LabStanford University - 10
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 45045032.0375
rad/s 450375
Nm
Nmmotor
Dynamic Design LabStanford University - 11
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)
Dynamic Design LabStanford University - 12
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
Dynamic Design LabStanford University - 13
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
Dynamic Design LabStanford University - 14
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
Dynamic Design LabStanford University - 15
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 LabStanford University - 16
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 LabStanford University - 17
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 LabStanford University - 18
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 LabStanford University - 19
When Do Cars Spin Out?
Can we figure out when the car will spin and avoid it?
Dynamic Design LabStanford University - 20
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
Comparing our Model to Reality
loss of control
Dynamic Design LabStanford University - 21
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
Dynamic Design LabStanford University - 23
Lanekeeping Assistance
Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama…
Dynamic Design LabStanford University - 24
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?
Dynamic Design LabStanford University - 25
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
Dynamic Design LabStanford University - 28
Controller countersteers to prevent spinout
Lanekeeping Active Lanekeeping Deactivated
Yaw Stability from Lanekeeping
Dynamic Design LabStanford University - 29
Controller response to heading error prevents the vehicle from spinning
A Closer Look
Dynamic Design LabStanford University - 30
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