Bivariate Chebyshev Expansion of the Pacejka's Tyre Model

Abstract

Pacejka's tyre model is widely used and well‐known by the community of automotive engineers. The magic formula basically describes the brake force, side force and self‐aligning torque in terms of the longitudinal slip and slip angle, with corrections due to the variation of the load force and camber angle. Obtaining continuous approximate solutions in Chebyshev series expansions of full vehicle dynamics can help in the real time solving of vehicle equations, for collision avoidance purposes. We contribute to solve the specific problem of the tyre's model expansion and its integration with the longitudinal, lateral and vertical behaviour of the car. The present work describes the approximation of the magic formula with Chebyshev's series development of rational polynomials, maintaining a moderate error of the model respect to the original formula, with a triple objective: firstly to obtain a very fast processing of the formula, secondly to allow the inclusion of the formula in DAE systems of vehicular dynamic modelling solved continuously, not numerically, by means of the expansion of the complete system in Chebyshev's series, and thirdly, the final expressions can be evaluated, integrated and derived easily.