Estimation of Road Friction Coefficient Based on the Brush Model

Road friction coefficients are highly effective for advanced vehicle control technologies, although the estimation at four individual tires has not been practically used for ordinary vehicles. This study describes the essential relation between the tire forces and the aligning torque that can be rearranged as an estimation equation for the grip margin. The grip margin is readily convertible into the friction coefficient. The brush model is reanalyzed, beginning from the conventional simple physical model, and intrinsic expressions are derived. The grip margin, which is defined as the residual tire force normalized by the radius of friction circle, was estimated using three components of the tire forces and the aligning torque. A simple cubic equation is obtained as a grip margin equation for an isotropic brush model. Previous studies assumed an anisotropic brush model and obtained an imperfect quintic equation. In the present study, a new term is added to the algebraic equation, which was shown to be consistent with the isotropic model. The solutions to the equations are approximated by Chebyshev polynomials. The estimation methods are tested by numerical simulations using CarSim, which is a popular vehicle simulation software application. The estimated friction coefficients agree well with the values that are set during each run of the simulations, especially for the cases of smaller grip margins and lower friction conditions.