Stochastic modelling of 1-D and 2-D terrain profiles using a polynomial chaos approach

Abstract

One fundamental difficulty in understanding the physics of vehicular off-road traction and in predicting vehicle performance is the variability of the terrain profile. These operating conditions are uniquely defined at a given spatial location and a given time. It is not practically feasible to measure them at a sufficiently large number of points to be able to accurately represent the terrain in models, or to use all the data collected to recreate the terrain profile. This renders traditional analysis tools insufficient when dealing with rough terrain. In this study, mathematical tools to quantify the impact of uncertainties in the terrain profile on vehicle mobility are developed. A polynomial chaos approach is used to reconstruct one-dimensional (along longitudinal direction) stationary and non-stationary terrain profiles. Also, an efficient mathematical method based on the Karhunen-Loeve expansion and the approach for 1-D stochastic terrain profile is developed to reconstruct two-dimensional (along longitudinal and lateral directions) terrain profiles. The proposed mathematical methods calculate the autocorrelation of terrain profiles, solve eigenvalues and eigenvectors of the autocorrelation function, and obtain the corresponding orthogonal random variables directly. The original terrain profile is reconstructed by Karhunen-Loeve expansions, requesting a small, limited computational effort, without the need to verify the terrain data for Gaussianity, stationary, and linearity, and without the need to choose the order of the expansion and the corresponding fitting coefficient artificially. Promising simulation results based on experimental data are obtained using the proposed methods. The schemes to choose the number of eigenvalues and eigenvectors are discussed. The proposed mathematical methods can be used to simulate the terrain profile for on-road and off-road vehicle dynamics or robotic applications.