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TECHNICAL PAPERS

# Dynamic Friction Model-Based Tire-Road Friction Estimation and Emergency Braking Control

[+] Author and Article Information
Luis Alvarez1

Professoralvar@pumas.iingen.unam.mx Instituto de Ingeniería Universidad Nacional Autónoma de México, 04510 Coyoacán DF, México

Jingang Yi2

Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740jgyi@me.berkeley.edu

Roberto Horowitz

Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740horowitz@me.berkeley.edu

Luis Olmos

Instituto de Ingeniería Universidad Nacional Autónoma de México, 04510 Coyoacán DF, Méxicoluiso10@yahoo.com

This simplified model is common in the tire-road friction literature.

This is a reasonable assumption as most modern vehicles already have wheel angular velocity measurements, and solid-state accelerometers are cheap and easy to install.

Angular acceleration can be calculated from wheel angular velocity measurements, while the master cylinder braking pressure can be obtained from the brake actuator.

It should be noted that the function $f(vr)$ is not differentiable with respect to $vr$ when $vr=0$. It is assumed, however, that during emergency braking the sign of $vr$ does not change.

The braking force is not maximum because initially the controller does not apply full brakes, that is the optimal value. Instead, it tries to track asymptotically the desired slip given by $λ̂m$.

1

Corresponding author.

2

Currently with Lam Research Corporation.

J. Dyn. Sys., Meas., Control 127(1), 22-32 (Jun 21, 2004) (11 pages) doi:10.1115/1.1870036 History: Received January 17, 2003; Revised June 21, 2004

## Abstract

An adaptive control scheme for emergency braking of vehicles is designed based on a LuGre dynamic model for the tire-road friction. The wheel angular speed and longitudinal vehicle acceleration information are used to design a fast convergence observer to estimate the vehicle velocity and the internal state of the friction model. The unknown parameters of the dynamic friction model are estimated through a parameter adaptation law. A Lyapunov-based state estimator and a stabilizing braking controller are designed to achieve near to maximum braking capability of the vehicle. Underestimation of the maximum friction coefficient, a very desirable feature from the perspective of safety, is guaranteed by a proper choice of adaptation gains and initial values of the estimated friction parameters.

###### FIGURES IN THIS ARTICLE
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Copyright © 2005 by American Society of Mechanical Engineers
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## Figures

Figure 1

Comparison between the psedostatic solution of the LuGre dynamic model and the “magic formula” for a braking case

Figure 2

Dynamic surface s̃

Figure 3

Estimated velocity (v̂) and relative velocity (v̂r)

Figure 4

Friction coefficient μ and braking pressure Pb(KPa)

Figure 5

Braking acceleration

Figure 6

Internal state (z, dashed), estimated internal state (ẑ, solid), and estimation error (z̃)

Figure 7

Adapted parameters; underestimation case, reference value dashed

Figure 8

Adapted parameters; non-underestimation case, reference value dashed

Figure 9

Estimated of velocity (v̂) and relative velocity (v̂r); non-underestimation case

Figure 10

Dynamic surface s̃ with mass M changed 30% and Kb changed 10%

## Errata

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