Fast Computing on Vehicle Dynamics Using Chebyshev Series Expansions

This paper focuses on faster computation techniques to integrate mechanical models in electronic advanced active safety applications. It shows the different techniques of approximation in series of functions and differential equations applied to vehicle dynamics. This allows the achievement of approximate polynomial and rational solutions with a very fast and efficient computation. First, the whole theoretical basic principles related to the techniques used are presented: orthogonality of functions, function expansion in Chebyshev and Jacobi series, approximation through rational functions, the Minimax-Remez algorithm, orthogonal rational functions, and the perturbation of dynamic systems theory, that reduces the degree of the expansion polynomials used. As an application, it is shown the obtaining of approximate solutions to the longitudinal dynamics, vertical dynamics, steering geometry, and a tyre model, all obtained through development in series of orthogonal functions with a computation much faster than those of its equivalents in the classic vehicle theory. These polynomial partially symbolic solutions present very low errors and very favorable analytical properties due to their simplicity, becoming ideal for real-time computation as those required for the simulation of evasive manoeuvres prior to a crash. This set of techniques had never been applied to vehicle dynamics before.