Stability at the limits
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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 l.jpg

Stability at the Limits

Yung-Hsiang Judy Hsu

J. Christian Gerdes

Stanford University

November 10, 2005

Did you know l.jpg
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 l.jpg
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 l.jpg
tire curve

maximum tire grip



Loss of control


Vehicle response l.jpg
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 l.jpg
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 l.jpg
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


Outline l.jpg

  • Identify tire operating region

    • Vehicle/Tire models

    • Tire parameter estimation

  • Produce stable, predictable response

    • Feedback linearizing controller

    • Driver input saturation

    • Simulation results


Vehicle model l.jpg
vehicle model

Bicycle model

  • 2 states: β and r

  • Nonlinear tire model (Dugoff)

  • Steer-by-wire


  • Small angles

  • Ux constant


Equations of motion l.jpg
equations of motion

Sum forces and moments:

Dugoff tire model:



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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 we’re in the nonlinear region  estimate 


Parameter estimates l.jpg
parameter estimates

  • Begin estimating  after entering NL region

  • C f estimate is steady


Controller design l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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


Conclusions l.jpg

  • 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 l.jpg
controller validation

  • Simulate control system on more complete vehicle model


Validation results ii l.jpg
validation results II

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


4 cases l.jpg
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 l.jpg
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 l.jpg
new inputs

  • Define new inputs v1 and v2

    to represent system as


More general form of fbl l.jpg
More general form of FBL

SISO algorithm:


Front steering only approach l.jpg
Front steering only approach

  • Model Fyf as:

  • Substitute into system equations:


Tracking yaw rate l.jpg
Tracking yaw rate

  • Choose new input

cr = 200

c = 50


Estimating c f l.jpg
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 l.jpg
Experimental Tire Curve

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



Overview l.jpg

  • Motivation

  • Background

  • Controller design

    • Feedback linearization

    • Driver input saturation

  • Validation on complex model

  • Conclusions


Steer by wire l.jpg

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



Slide39 l.jpg

Lineartire model


Slide40 l.jpg

Nonlineartire model


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comparing vehicle responses

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


Tire estimation algorithm42 l.jpg
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