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Spinning Out, With Calculus

Spinning Out, With 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 Lanekeeping Assistance

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Spinning Out, With Calculus

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  1. 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. 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

  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. Lanekeeping on the Corvette

  7. 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

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

  10. Tires • Let’s use your knowledge of Calculus to make a model of the tire…

  11. An Observation… • A tire without lateral force moves in a straight line Tire without lateral force

  12. An Observation… • A tire without lateral force moves in a straight line Tire without lateral force

  13. An Observation… • A tire without lateral force moves in a straight line Tire without lateral force

  14. An Observation… • A tire subjected to lateral force moves diagonally Tire with lateral force

  15. An Observation… • A tire subjected to lateral force moves diagonally Tire with lateral force

  16. An Observation… • A tire subjected to lateral force moves diagonally Tire with lateral force

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

  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 a

  20. Saturation at Limits • Eventually tire force saturates • Friction limited • Rear part of contact patch saturates first a Fy a

  21. Simple Lateral Force Model • Deflection initially triangular • Defined by slip angle • Force follows deflection • Assume constant foundation stiffness cpy • qy(x) is force per unit length x = a x = -a a v(x) = (a-x) tana a qy(x) = cpy(a-x) tana

  22. 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

  23. Tire Forces with Saturation • Tire force limited by friction • Assume parabolic normal force distribution in contact patch qz(x)

  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 • qy(x) <mqz(x) msqz(x) mpqz(x)

  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 • qy(x) <mqz(x) • Result: the rear part of the contact patch is always sliding large slip small slip msqz(x) mpqz(x)

  26. Calculate Lateral Force xsl msqz(x) mpqz(x)

  27. Lateral Force Model • The entire contact patch is sliding when a = asl • The lateral force model is therefore: • Figures show shape of this relationship

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

  29. 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

  30. Lateral Force with Peak and Slide Friction • ms=0.8 and mp=1.6 • Peak in curve • Can we predict friction on road?

  31. Testing at Moffett Field

  32. How Early Can We Estimate Friction? loss of control linear nonlinear

  33. Friction estimated about halfway to the peak – very early! Ramp: Friction Estimates linear nonlinear loss of control

  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. 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

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

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

  38. Making Sense of Yaw Rate and Sideslip • What is happening with this car?

  39. For Normal Driving, Things Simplify • Slip angles generate lateral forces • Simple, linear tire model (no spin-outs possible) a Fy

  40. Two Linear Ordinary Differential Equations

  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. P1 Vehicle Parameters

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