Vehicle dynamics it s all about the calculus
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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…

J. Christian Gerdes

Associate Professor

Mechanical Engineering Department

Stanford University

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

  • How do we calculate the 0-60 time?

Basic Dynamics

  • Newton’s Second Law

  • With Calculus

  • If we know forces, we can figure out velocity

What are the Forces?

  • Forces from:

    • Engine

    • Aerodynamic Drag

    • Tire Rolling Resistance

Working in the Motor Characteristics

Working in the Motor Characteristics

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

  • 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

  • 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

  • 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

    • 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

  • 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


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

  • 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

  • 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










d+ af

Lateral Force Behavior

  • ms=1.0 and mp=1.0

    • Fiala model

When Do Cars Spin Out?

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

Comparing our Model to Reality

loss of control



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 Assistance

  • Energy predictions work!

  • Comfortable, guaranteed lanekeeping

  • Another example with more drama…

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

Video from Dropped Throttle Tests

Yaw Stability from Lanekeeping

Lanekeeping Active

Lanekeeping Deactivated

Controller countersteers to prevent spinout

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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