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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

Stability at the Limits

Yung-Hsiang Judy Hsu

J. Christian Gerdes

Stanford University

November 10, 2005

did you know
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)


motion of a vehicle
motion of a vehicle


  • 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






tire curve
tire curve

maximum tire grip



Loss of control


vehicle response
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


loss of control
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


overall goals
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


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


vehicle model
vehicle model

Bicycle model

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


  • Small angles
  • Ux constant


equations of motion
equations of motion

Sum forces and moments:

Dugoff tire model:



tire estimation algorithm
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 


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


controller design
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


feedback linearization fbl
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:


fbl in action
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


fbl in action19
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


driver input saturation
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


overall control system
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


design considerations
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


  • 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


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


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


4 cases
4 cases

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

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


4 cases28
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)


new inputs
new inputs
  • Define new inputs v1 and v2

to represent system as


more general form of fbl
More general form of FBL

SISO algorithm:


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


tracking yaw rate
Tracking yaw rate
  • Choose new input

cr = 200

c = 50


estimating c f
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


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



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


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



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


tire estimation algorithm42
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