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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 By-wire Vehicle Diagnostics

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Vehicle Dynamics – It’s all about the Calculus…

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Vehicle dynamics it s all about the calculus

Vehicle Dynamics – It’s all about the Calculus…

J. Christian Gerdes

Associate Professor

Mechanical Engineering Department

Stanford University

Future vehicles

Future Vehicles…


Multi-Combustion-Mode Engines

Control of HCCI with VVA

Electric Vehicle Design


By-wire Vehicle Diagnostics

Lanekeeping Assistance

Rollover Avoidance


Handling Customization

Variable Force Feedback

Control at Handling Limits

Electric vehicle design

Electric Vehicle Design

  • How do we calculate the 0-60 time?

Basic dynamics

Basic Dynamics

  • Newton’s Second Law

  • With Calculus

  • If we know forces, we can figure out velocity

What are the forces

What are the Forces?

  • Forces from:

    • Engine

    • Aerodynamic Drag

    • Tire Rolling Resistance

Working in the motor characteristics

Working in the Motor Characteristics

Working in the motor characteristics1

Working in the Motor Characteristics

Some numbers for the tesla roadster

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

More numbers for the roadster

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:

    • 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

Motor issues

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:

Results of my simulation

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)

P1 steer by wire vehicle

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


Future systems

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

Steer by wire systems


handwheel angle sensor

handwheel feedback motor

shaft angle sensor

steering actuator

power steering unit


steering rack

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

Bicycle model









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

Vehicle model

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

Calculate slip angles

Calculate Slip Angles










d+ af

Lateral force behavior

Lateral Force Behavior

  • ms=1.0 and mp=1.0

    • Fiala model

When do cars spin out

When Do Cars Spin Out?

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

Comparing our model to reality

Comparing our Model to Reality

loss of control



Lanekeeping with potential fields

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

Lanekeeping on the corvette

Lanekeeping on the Corvette

Lanekeeping assistance

Lanekeeping Assistance

  • Energy predictions work!

  • Comfortable, guaranteed lanekeeping

  • Another example with more drama…

Handling limits

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?



  • 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





Example: Loss of rear tire traction

Lanekeeping at handling limits

Lanekeeping at Handling Limits

Video from dropped throttle tests

Video from Dropped Throttle Tests

Yaw stability from lanekeeping

Yaw Stability from Lanekeeping

Lanekeeping Active

Lanekeeping Deactivated

Controller countersteers to prevent spinout

A closer look

A Closer Look

Controller response to heading error prevents the vehicle from spinning



  • 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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