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Stability at the Limits

Stability at the Limits. Yung-Hsiang Judy Hsu J. Christian Gerdes Stanford University. did you know…. Every day in the US, 10 teenagers are killed in teen-driven vehicles in crashes 1 Loss of control accounts for 30% of these deaths

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Stability at the Limits

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  1. Stability at the Limits Yung-Hsiang Judy Hsu J. Christian Gerdes Stanford University November 10, 2005

  2. did you know… • Every day in the US, 10 teenagers are killed in teen-driven vehicles in crashes1 • Loss of control accounts for 30% of these deaths • Inexperienced drivers make more driving errors, exceed speed limits & run off roads at higher rates • In 2002, motor vehicle traffic crashes were the leading cause of death for ages 3-33.2 To understand how loss of control occurs, need to know what determines vehicle motion 1 National Highway Traffic Safety Administration. Traffic safety facts (2002) 2 USA Today. Study of deadly crashes involving 16-19 year old drivers (2003) 2

  3. motion of a vehicle SIDE VIEW • Motion of a vehicle is governed by tire forces • Tire forces result from deformation in contact patch • Lateral tire force is a function of tire slip Contact Patch Ground BOTTOM VIEW a Fy 3

  4. tire curve maximum tire grip Linear Saturation Loss of control 4

  5. vehicle response • Normally, we operate in linear region • Predictable vehicle response • But during slick road conditions, emergency maneuvers, or aggressive/performance driving • Enter nonlinear tire region • Response unanticipated by driver 5

  6. loss of control Imagine making an aggressive turn • If front tires lose grip first, plow out of turn (limit understeer) • may go into oscillatory response • driver loses ability to influence vehicle motion • If rear tires saturate, rear end kicks out (limit oversteer) • may go into a unstable spin • driver loses control • Both can result in loss of control 6

  7. overall goals We’d like to design a control system to • Stabilize vehicle in nonlinear handling region • Make vehicle response consistent and predictable for drivers • Communicate to driver when limits of handling are approaching 7

  8. Outline • Identify tire operating region • Vehicle/Tire models • Tire parameter estimation • Produce stable, predictable response • Feedback linearizing controller • Driver input saturation • Simulation results 8

  9. vehicle model Bicycle model • 2 states: β and r • Nonlinear tire model (Dugoff) • Steer-by-wire Assume • Small angles • Ux constant 9

  10. equations of motion Sum forces and moments: Dugoff tire model: -C  10

  11. tire estimation algorithm • Find f: use GPS/INS • Find Fyf: SBW motor give steering torque • Estimate C f and  • LS fit to linear tire model • NLS fit to Dugoff model • Compare residual of fits to tell us if we’re in the nonlinear region  estimate  11

  12. tire parameter estimation 12

  13. getting the data 13

  14. estimation technique 14

  15. parameter estimates • Begin estimating  after entering NL region • C f estimate is steady 15

  16. controller design • Desired vehicle response • Track response of bicycle model with linear tires • Be consistent with what driver expects • When tires saturate, compensate for decreasing forces with steer-by-wire input • One input f; two states ,r • Could compromise between the two • Or, track one state exactly 16

  17. feedback linearization (FBL) • Nonlinear control technique Applicable to systems that look like: • Use input to cancel system nonlinearities. In our case, • Apply linear control theory to track desired trajectory: 17

  18. FBL in action • Ramp steer from 0 to 4o at 20 m/s (45 mph) in 1 s • Controller results in exact tracking of linear tire model yaw rate trajectory 18

  19. FBL in action • Ramp steer from 0 to 6o at 20 m/s (45 mph) in 1 s • FBL works well up to physical capabilities of tires 19

  20. driver input saturation • Road naturally saturates driver’s steering capability often unexpectedly • Here, we safely limit steering capability in a predictable, safe manner • Why do we need it? • Prevents vehicle from needing more side force than is available • Keeps vehicle in linearizable handling region • Saturation algorithm • If  < th, driver commands are OK • If ¸th, gradually saturate driver’s steering capability 20

  21. overall control system • Ramp steer from 0 to 6° at 20 m/s (45 mph) in 1 s • Tracks linear model yaw rate, then saturates input • Reduced sideslip 21

  22. design considerations • Relative importance of  vs. r • Which produces a more predictable response? • Could add additional input to track and r • differential drive • rear steering 22

  23. conclusions • Overall approach • Sense tire saturation and actively compensate for them with SBW inputs • Algorithm can characterize tires (C, ) using GPS-based f and estimates of Fyf, • Make vehicle response more predictable • Up to capabilities of tires, controller tracks linear yaw rate trajectory • Reduces sideslip • Current work • Estimate C,  on board in real-time • Implement overall controller on research vehicle 23

  24. 24

  25. controller validation • Simulate control system on more complete vehicle model 25

  26. validation results II • input: ramp steer from 0 to 5° at 45 mph in 0.5 s 26

  27. 4 cases Case 1: Both tires are linear (f¸ 1 and r¸ 1) Case 2: Both tires saturating (f < 1 and r < 1) 27

  28. 4 cases Case 3: front is nonlinear, rear is linear (f¸ 1 and r < 1) Case 4: front is linear, rear is nonlinear (f¸ 1 and r < 1) 28

  29. new inputs • Define new inputs v1 and v2 to represent system as 29

  30. More general form of FBL SISO algorithm: 30

  31. driver saturation algorithm 31

  32. Front steering only approach • Model Fyf as: • Substitute into system equations: 32

  33. Tracking yaw rate • Choose new input cr = 200 c = 50 33

  34. Estimating Cf • Find f:Use GPS/INS to measure r and f and estimate  • Find Fyf: Estimate tm from steering geometry, model tp as and use disturbance torque estimate from SBW system to find Fyf • Estimate : • Using least squares 34

  35. Experimental Tire Curve • P1: Ramp steer from 0 to 9° in 24 s at V = 31 mph shad_2004-12-11_l.mat 35

  36. questions? 36

  37. overview • Motivation • Background • Controller design • Feedback linearization • Driver input saturation • Validation on complex model • Conclusions 37

  38. steer-by-wire Removes mechanical linkage between steering wheel and road wheels • electronically actuate steering system separately from driver’s commands • decouple underlying dynamics from driver force feedback Conventional steering Steer-by-wire 38

  39. Lineartire model 39

  40. Nonlineartire model 40

  41. comparing vehicle responses • Ramp steer to from 0 to 4o at 45 mph in 0.5 s 41

  42. tire estimation algorithm • Find f: GPS/INS measures , r, V • Find Fyf: SBW motor give steering torque  • Estimate C f and  from (Fyf, f) data • LS fit to line • NLS fit to Dugoff Compare fit errors to tell us if in nonlinear region 42

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