Performance Guarantees for Hazard Based Lateral Vehicle

Control

Eric Rossetter

J. Christian Gerdes

Stanford University

Department of Mechanical

Engineering

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 2

Outline• Motivation for Lanekeeping Assistance Systems• Potential Field Approach

– ‘Artificial’ potential functions in control– Implementing control forces derived from the potential– Video

• Recap Linear Stability Results• Linear Lyapunov Bounding Function• Simulation• Conclusions/Future Work

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 3

Motivation and Goals• A large percentage of vehicle fatalities are caused

by lane departures.

• Active lanekeeping assistance systems can help the driver remain in the lane

• What performance

guarantees can be placed

on these systems?

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 4

Potential Field Approach

• ‘Artificial’ potential energy is analogous to hazard level

• Control forces are derived from artificial potential functions

• Control inputs are added on top of driver commands

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 5

Control Law

),(),()( controldriver uqguqgqfqM

InertialForces

UncontrolledForces (rear tire)

Driver Controlled Forces(steering, braking)

Final System:T

driver q

w

w

wVuqgqfqM

)(

),()(

Replace withControl Forces

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 6

Creating a Virtual Force• Steering & Differential Braking

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 7

Creating a Virtual Force• Steering only: Moment applied is dependent on

the side force

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 8

Potential Field Video

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 9

Bounding Lateral Motion

• By design, the potential field controller is not a tracking controller

• Desirable to bound the lateral motion of the vehicle

• Question: Given a set of initial conditions, how should the potential function be scaled to avoid a collision or lane departure?

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 10

Lyapunov Approach• Create an energy-like function of the states

)(xL positive definite

)(xL negative semi-definite

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 11

Controlled Vehicle DynamicsF

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 12

Recap of Linear Stability Results1. Control force must be

applied in front of the neutral steer point

2. Control force must come from a projection into the potential function

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 13

Creating a Candidate Lyapunov Function

Generalized Damping Forces

Are these derivablefrom a potential?

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 14

Candidate Lyapunov Function

• All our forces are derivable from the potential or are generalized damping forces

• Candidate Lyapunov Fcn:

'V

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 15

Lyapunov Function Conditions

• Two conditions must be checked

• Condition 1:

• Using Sylvester’s Criterion:

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 16

Lyapunov Conditions

• Condition 2:

• Using Sylvester’s Criterion:

• Therefore,

is a Lyapunov function

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 17

Using the Lyapunov Fcn. To Bound Lateral Motion

• How do we use this Lyapunov function to avoid collisions?

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 18

SimulationPotential Function Gain Chosen to Avoid Obstacle 0.75maway with initial conditions: Ux=40 m/s, Degrees

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 19

Conclusions• The presented Lyapunov function gives

an analytic expression that yields useful bounds for lateral vehicle motion

• The structure of this function is useful in finding bounds for certain non-linearities– Large heading angles

– Higher order potentials

Dynamic Design LabStanford University

Performance Guarantees for Hazard Based Lateral Vehicle Control

Slide 20

Future Work

• Add time varying external disturbances to the bounding function– Road Curvature– Driver Inputs

• Experimentally verify the results on the steer-by-wire test vehicle