Vehicle Dynamics Its all about the Calculus

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Vehicle Dynamics Its all about the Calculus

• J. Christian Gerdes
• Associate Professor
• Mechanical Engineering Department
• Stanford University

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Future Vehicles
Clean Multi-Combustion-Mode Engines Control of
HCCI with VVA Electric Vehicle Design
Safe By-wire Vehicle Diagnostics Lanekeeping
Assistance Rollover Avoidance
Fun Handling Customization Variable Force
Feedback Control at Handling Limits
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Electric Vehicle Design

• How do we calculate the 0-60 time?

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Basic Dynamics

• Newtons Second Law
• With Calculus
• If we know forces, we can figure out velocity

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What are the Forces?

• Forces from
• Engine
• Aerodynamic Drag
• Tire Rolling Resistance

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Working in the Motor Characteristics
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Working in the Motor Characteristics
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Some numbers for the Tesla Roadster

• From Teslas web site
• m mass 1238 kg
• Rgear final drive gear ratio 8.28
• A Frontal area Heightwidth
• 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

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More numbers for the roadster

• From other sources
• rwheel wheel radius 0.33m (a reasonable
  value)
• Frr rolling resistance 0.01mg
• For reference, see
• http//www.greenseal.org/resources/reports/CGR_tir
  e_rollingresistance.pdf
• r 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

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Motor issues

• The website lists a motor peak torque of 375 Nm
  up to 4500RPM. This doesnt 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

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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

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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

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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

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Calculate Slip Angles
b
a
ar
b
d
V
af
r
ar
d af
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Lateral Force Behavior

• ms1.0 and mp1.0
• Fiala model

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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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Comparing our Model to Reality
linear
nonlinear
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

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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Yaw Stability from Lanekeeping
Lanekeeping Active
Lanekeeping Deactivated
Controller countersteers to prevent spinout
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A Closer Look
Controller response to heading error prevents the
vehicle from spinning
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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