Using a Galerkin Approach to Define Terrain Surfaces

Terrain is the principal source of vertical excitation to the vehicle and must be accurately represented in order to correctly predict the vehicle response. Ideally, an efficient terrain surface definition could be developed that maintains the high-fidelity information required to accurately excite vehicle models. It is also desirable to minimize the effect of the choice of measurement system used to sample the terrain surface. Nondeformable, anisotropic (path-specific) terrain surfaces are defined as a sequence of vectors, where each vector comprises terrain heights at locations oriented perpendicular to the direction of travel. A vector space is formed by the span of these vectors and a corresponding set of empirical basis vectors is developed. A set of analytic basis vectors is formed from Gegenbauer polynomials, parameterized to approximate the empirical basis vectors. A weighted inner product is defined to form a Hilbert space and the terrain surface vectors are projected onto the set of analytic basis vectors. The weighting matrix is developed such that these projections are insensitive to the number and placement of the discrete transverse locations at which the terrain heights are defined. This method is successfully demonstrated on sets of paved road surfaces to show that a high-fidelity but compact definition of terrain surfaces is developed. This representation is applied to an example of off-road terrain to demonstrate the representation’s scope of applicability. Future work will investigate generating stochastic terrain surfaces with these vectors.