1. Spinning Out, With Calculus J. Christian Gerdes Associate Professor Mechanical Engineering Department Stanford University

2. Dynamic Design LabStanford University- 2 Future Vehicles… Safe By-wire Vehicle Diagnostics Lanekeeping Assistance Rollover Avoidance Fun Handling Customization Variable Force Feedback Control at Handling Limits Clean Multi-Combustion-Mode Engines Control of HCCI with VVA Electric Vehicle Design

3. Dynamic Design LabStanford University- 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. Dynamic Design LabStanford University- 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 handwheel handwheel angle sensor handwheel feedback motor steering actuator shaft angle sensor power steering unit pinion steering rack

5. Dynamic Design LabStanford University- 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

6. Dynamic Design LabStanford University- 6 Lanekeeping on the Corvette

7. Dynamic Design LabStanford University- 7 Lanekeeping Assistance Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama…

8. Dynamic Design LabStanford University- 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

9. Dynamic Design LabStanford University- 9 When Do Cars Spin Out? Can we figure out when the car will spin and avoid it?

10. Dynamic Design LabStanford University- 10 Tires Let’s use your knowledge of Calculus to make a model of the tire…

11. Dynamic Design LabStanford University- 11 An Observation… A tire without lateral force moves in a straight line Tire without lateral force

12. Dynamic Design LabStanford University- 12 An Observation… A tire without lateral force moves in a straight line Tire without lateral force

13. Dynamic Design LabStanford University- 13 An Observation… A tire without lateral force moves in a straight line Tire without lateral force

14. Dynamic Design LabStanford University- 14 An Observation… A tire subjected to lateral force moves diagonally Tire with lateral force

15. Dynamic Design LabStanford University- 15 An Observation… A tire subjected to lateral force moves diagonally Tire with lateral force

16. Dynamic Design LabStanford University- 16 An Observation… A tire subjected to lateral force moves diagonally Tire with lateral force

17. Dynamic Design LabStanford University- 17 An Observation… A tire subjected to lateral force moves diagonally How is this possible? Shouldn’t the tire be stuck to the road?

18. Dynamic Design LabStanford University- 18 Tire Force Generation The contact patch does stick to the ground This means the tire deforms (triangularly)

19. Dynamic Design LabStanford University- 19 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 

20. Dynamic Design LabStanford University- 20 Saturation at Limits Eventually tire force saturates Friction limited Rear part of contact patch saturates first  FyFy 

21. Dynamic Design LabStanford University- 21 Simple Lateral Force Model Deflection initially triangular Defined by slip angle Force follows deflection Assume constant foundation stiffness c py q y (x) is force per unit length x = a x = -a  v(x) = (a-x) tan   q y (x) = c py (a-x) tan 

22. Dynamic Design LabStanford University- 22 Simple Lateral Force Model Calculate lateral force x = a x = -a  v(x) = (a-x) tan   q y (x) = c py (a-x) tan  Cornering stiffness

23. Dynamic Design LabStanford University- 23 Tire Forces with Saturation Tire force limited by friction Assume parabolic normal force distribution in contact patch q z (x)

24. Dynamic Design LabStanford University- 24 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 q y (x) <  q z (x)  s q z (x)  p q z (x)

25. Dynamic Design LabStanford University- 25 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 q y (x) <  q z (x) Result: the rear part of the contact patch is always sliding large slip small slip  s q z (x)  p q z (x)

26. Dynamic Design LabStanford University- 26 Calculate Lateral Force  s q z (x)  p q z (x) x sl

27. Dynamic Design LabStanford University- 27 Lateral Force Model The entire contact patch is sliding when  sl The lateral force model is therefore: Figures show shape of this relationship

28. Dynamic Design LabStanford University- 28 Lateral Force Behavior  s =1.0 and  p =1.0 Fiala model

29. Dynamic Design LabStanford University- 29 Coefficients of Friction Sliding (dynamic friction):  s = 0.8 Many force-slip plots have approximately this much friction after the peak, when the tire is sliding Seen in previous literature Adhesion (peak friction):  p = 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 F peak / F z = 1.0 Agrees with expectation

30. Dynamic Design LabStanford University- 30 Lateral Force with Peak and Slide Friction  s =0.8 and  p =1.6 Peak in curve Can we predict friction on road?

31. Dynamic Design LabStanford University- 31 Testing at Moffett Field

32. Dynamic Design LabStanford University- 32 linearnonlinear How Early Can We Estimate Friction? loss of control

33. Dynamic Design LabStanford University- 33 Ramp: Friction Estimates Friction estimated about halfway to the peak – very early! linearnonlinear loss of control

34. Dynamic Design LabStanford University- 34 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

35. Dynamic Design LabStanford University- 35 Bicycle Model Basic variables Speed V (constant) Yaw rate r – angular velocity of the car Sideslip angle  – Angle between velocity and heading Steering angle  – our input Model Get slip angles, then tire forces, then derivatives ff rr   V ba r

36. Dynamic Design LabStanford University- 36 Calculate Slip Angles ff rr   V ba r  f rr

37. Dynamic Design LabStanford University- 37 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,  and  Calculate front and rear forces from slip angles Calculate changes in r and 

38. Dynamic Design LabStanford University- 38 Making Sense of Yaw Rate and Sideslip What is happening with this car?

39. Dynamic Design LabStanford University- 39 For Normal Driving, Things Simplify Slip angles generate lateral forces Simple, linear tire model (no spin-outs possible) FyFy 

40. Dynamic Design LabStanford University- 40 Two Linear Ordinary Differential Equations

41. Dynamic Design LabStanford University- 41 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

42. Dynamic Design LabStanford University- 42 P1 Vehicle Parameters