Stability at the Limits

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# Stability at the Limits - PowerPoint PPT Presentation

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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Presentation Transcript

### Stability at the Limits

Yung-Hsiang Judy Hsu

J. Christian Gerdes

Stanford University

November 10, 2005

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

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

tire curve

maximum tire grip

Linear

Saturation

Loss of control

4

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

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

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

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

8

vehicle model

Bicycle model

• 2 states: β and r
• Nonlinear tire model (Dugoff)
• Steer-by-wire

Assume

• Small angles
• Ux constant

9

equations of motion

Sum forces and moments:

Dugoff tire model:

-C

10

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

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

15

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

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

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

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

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

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

design considerations
• Relative importance of  vs. r
• Which produces a more predictable response?
• differential drive
• rear steering

22

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

controller validation
• Simulate control system on more complete vehicle model

25

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

26

4 cases

Case 1: Both tires are linear (f¸ 1 and r¸ 1)

Case 2: Both tires saturating (f < 1 and r < 1)

27

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

new inputs
• Define new inputs v1 and v2

to represent system as

29

More general form of FBL

SISO algorithm:

30

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

32

Tracking yaw rate
• Choose new input

cr = 200

c = 50

33

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

Experimental Tire Curve
• P1: Ramp steer from 0 to 9° in 24 s at V = 31 mph

35

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

37

steer-by-wire

• electronically actuate steering system separately from driver’s commands
• decouple underlying dynamics from driver force feedback

Conventional steering

Steer-by-wire

38

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

41

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

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