2005 ASME IMECE

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Stability at the Limits

• Yung-Hsiang Judy Hsu
• J. Christian Gerdes
• Stanford University

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

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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

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overall goals

• Wed 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

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Outline

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

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vehicle model

• Bicycle model
• 2 states ß and r
• Nonlinear tire model (Dugoff)
• Steer-by-wire
• Assume
• Small angles
• Ux constant

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equations of motion

• Sum forces and moments
• Dugoff tire model

-C?
?
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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 were in
  the nonlinear region ? estimate ?

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tire parameter estimation
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getting the data
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estimation technique
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parameter estimates

• Begin estimating ? after entering NL region
• C? f estimate is steady

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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

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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

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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

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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

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driver input saturation

• Road naturally saturates drivers 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 ?
• If ? ?th, gradually saturate drivers steering
  capability

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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

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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

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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

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controller validation

• Simulate control system on more complete vehicle
  model

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validation results II

• input ramp steer from 0 to 5 at 45 mph in 0.5 s

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4 cases

• Case 1 Both tires are linear (?f 1 and ?r
  1)
• Case 2 Both tires saturating (?f

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4 cases

• Case 3 front is nonlinear, rear is linear (?f
  1 and ?r
• Case 4 front is linear, rear is nonlinear (?f
  1 and ?r

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new inputs

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

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More general form of FBL

• SISO algorithm

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driver saturation algorithm
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Front steering only approach

• Model Fyf as
• Substitute into system equations

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Tracking yaw rate

• Choose new input

cr 200 c? 50
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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

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Experimental Tire Curve

• P1 Ramp steer from 0 to 9 in 24 s at V 31 mph

shad_2004-12-11_l.mat
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questions?
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overview

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

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steer-by-wire

• Removes mechanical linkage between steering wheel
  and road wheels
• electronically actuate steering system separately
  from drivers commands
• decouple underlying dynamics from driver force
  feedback

Conventional steering
Steer-by-wire
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Linear tire model
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Nonlinear tire model
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comparing vehicle responses

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

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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