The most popular semi-empirical models for predicting different aspects of the pneumatic tires performance under steady-state conditions are the Friction Ellipse Model (FEM) and the Magic Formula Model (MFM). The Friction Ellipse Model calculates the longitudinal and the lateral forces in the tire contact patch based on the slip ratio, the slip angle, the normal forces at the tire, and the friction coefficients between the tire and the road surface. The Magic Formula Model describes the cornering forces, the braking forces, and the aligning moment as functions of the slip ratio, the slip angle, and the normal forces at the tire. In the real operational environment, key parameters at the interface of the vehicle with the road, such as the slip ratio, the slip angle, the friction coefficients, and the normal force do not have constant values, but always change in time; thus, it is not possible to capture the effect of such uncertainties on the tire behavior (resultant force and moments) using a deterministic model. In addition, current measuring techniques have certain limitations and sometime non-negligible measurement errors could be a source of relatively rough approximations in estimating some important parameters involved in vehicle dynamics simulations and control algorithms.
In this study we treat the uncertainty in key parameters associated with the tire-road interface using a polynomial chaos approach. The approach has been proved to be more computationally efficient than traditional stochastic methods such as Monte Carlo (MC), while it can nicely accommodate nonlinear systems with large uncertainties. In this paper, FEM and the MFM have been extended from deterministic to stochastic models, to account for the uncertainties in the tire-road friction coefficient, the slip ratio, the slip angle, and the normal forces in the contact patch. Although a uniform distribution has been considered for each of the stochastic parameters of interest, the approach presented is not limited to this type of distribution. In addition to the analysis of the impact that the uncertainty in one parameter has on the dynamics of the tire for the specific tire model considered, we also studied the tire response under the scenario where multiple parameters behave in a stochastic manner simultaneously. The modeling approach presented in this paper is able to capture the stochastic nature of parameters of interest and to predict the response of the system under those uncertainties, in an effort to provide a better understanding and a more realistic prediction of the tire-road interaction than a deterministic formulation. This is possible since the stochastic models give the response as a range of possible values and the analysis can further benefit from the corresponding probability density function.