Spinning Out, With Calculus

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Spinning Out, With Calculus

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

2
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
3
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

4
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

5
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

8
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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When Do Cars Spin Out?

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

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Tires

• Lets use your knowledge of Calculus to make a
  model of the tire

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An Observation

• A tire without lateral force moves in a straight
  line

Tire without lateral force
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An Observation

• A tire without lateral force moves in a straight
  line

Tire without lateral force
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An Observation

• A tire without lateral force moves in a straight
  line

Tire without lateral force
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An Observation

• A tire subjected to lateral force moves
  diagonally

Tire with lateral force
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An Observation

• A tire subjected to lateral force moves
  diagonally

Tire with lateral force
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An Observation

• A tire subjected to lateral force moves
  diagonally

Tire with lateral force
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An Observation

• A tire subjected to lateral force moves
  diagonally

How is this possible? Shouldnt the tire be stuck
to the road?
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Tire Force Generation

• The contact patch does stick to the ground
• This means the tire deforms (triangularly)

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Tire Force Generation

• Force distribution is triangular
• More force at rear
• Force proportional to slip angle initially
• Cornering stiffness
• Force is in opposite direction as velocity
• Side forces dissipative

a
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Saturation at Limits

• Eventually tire force saturates
• Friction limited
• Rear part of contact patch saturates first

a
Fy
a
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Simple Lateral Force Model
x a
x -a
a

• Deflection initially triangular
• Defined by slip angle
• Force follows deflection
• Assume constant foundation stiffness cpy
• qy(x) is force per unit length

v(x) (a-x) tana
a
qy(x) cpy(a-x) tana
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Simple Lateral Force Model

• Calculate lateral force

x a
x -a
a
v(x) (a-x) tana
a
qy(x) cpy(a-x) tana
Cornering stiffness
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Tire Forces with Saturation

• Tire force limited by friction
• Assume parabolic normal force
  distribution in contact patch

qz(x)
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Tire Forces with Saturation

• Tire force limited by friction
• Assume parabolic normal force
  distribution in contact patch
• Rubber has two friction coefficients adhesion
  and sliding
• Lateral force and deflection are friction limited
  
• qy(x)

msqz(x)
mpqz(x)
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Tire Forces with Saturation

• Tire force limited by friction
• Assume parabolic normal force
  distribution in contact patch
• Rubber has two friction coefficients adhesion
  and sliding
• Lateral force and deflection are friction limited
  
• qy(x)
• Result the rear part of the contact patch is
  always sliding
• large slip
  small
  slip

msqz(x)
mpqz(x)
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Calculate Lateral Force
xsl
msqz(x)
mpqz(x)
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Lateral Force Model

• The entire contact patch is sliding when a asl
• The lateral force model is therefore
• Figures show shape of this relationship

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Lateral Force Behavior

• ms1.0 and mp1.0
• Fiala model

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Coefficients of Friction

• Sliding (dynamic friction) ms 0.8
• Many force-slip plots haveapproximately this
  much friction after the peak, when the tire is
  sliding
• Seen in previous literature
• Adhesion (peak friction) mp 1.6
• Tire/road friction, tested in stationary
  conditions, has been demonstrated to be
  approximately this much
• Seen in previous literature
• Model predicts that these values give Fpeak / Fz
  1.0
• Agrees with expectation

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Lateral Force with Peak and Slide Friction

• ms0.8 and mp1.6
• Peak in curve
• Can we predict friction on road?

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Testing at Moffett Field
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How Early Can We Estimate Friction?
linear
nonlinear
loss of control
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Ramp Friction Estimates

• Friction estimated about halfway to the peak
  very early!

linear
nonlinear
loss of control
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Bicycle Model

• Outline model
• How does the vehicle move when I turn the
  steering wheel?
• Use the simplest model possible
• Same ideas in video games and car design just
  with more complexity
• Assumptions
• Constant forward speed
• Two motions to figure out turning and lateral
  movement

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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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Calculate Slip Angles
b
a
ar
b
d
V
af
r
ar
d af
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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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Making Sense of Yaw Rate and Sideslip

• What is happening with this car?

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For Normal Driving, Things Simplify

• Slip angles generate lateral forces
• Simple, linear tire model (no spin-outs possible)

a
Fy
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Two Linear Ordinary Differential Equations
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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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P1 Vehicle Parameters