Vehicle Dynamics – It’s all about the Calculus…

1. Vehicle Dynamics – It’s all about the 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. Electric Vehicle Design • How do we calculate the 0-60 time?

4. Basic Dynamics • Newton’s Second Law • With Calculus • If we know forces, we can figure out velocity

5. What are the Forces? • Forces from: • Engine • Aerodynamic Drag • Tire Rolling Resistance

6. Working in the Motor Characteristics

7. Working in the Motor Characteristics

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 1.13m • 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

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 • 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

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:

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)

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

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

14. handwheel handwheel angle sensor handwheel feedback motor shaft angle sensor steering actuator power steering unit pinion 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

15. a b b ar d V af r 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

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, d and b • Calculate front and rear forces from slip angles • Calculate changes in r and b

17. Calculate Slip Angles a b b ar d V af r ar d+ af

18. Lateral Force Behavior • ms=1.0 and mp=1.0 • Fiala model

19. When Do Cars Spin Out? • Can we figure out when the car will spin and avoid it?

20. Comparing our Model to Reality loss of control linear nonlinear

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

22. Lanekeeping on the Corvette

23. Lanekeeping Assistance • Energy predictions work! • Comfortable, guaranteed lanekeeping • Another example with more drama…

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 handling limits linear region Is the lanekeeping system safe at the limits?

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

26. Lanekeeping at Handling Limits

27. Video from Dropped Throttle Tests

28. Yaw Stability from Lanekeeping Lanekeeping Active Lanekeeping Deactivated Controller countersteers to prevent spinout

29. A Closer Look Controller response to heading error prevents the vehicle from spinning

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