Bayesian Inference in Terrain Navigation

Abstract: Terrain navigation is a concept for autonomous aircraft navigation. If measurements of the terrain height over mean sea-level are collected along the aircraft flight path, an estimate of the aircraft position can be formed by matching these measurements with a digital reference terrain map. This matching is a recursive nonlinear estimation problem. Due to the unstructured nonlinear reference map, local approximation schemes, like the extended Kalman Filter, fail in this application.In this work, the optimal Bayesian approach to the recursive inference of the measurement sources is taken. In the Bayesian approach the uncertainty about the aircraft position is condensed in the conditional probability density function. The analytical expression for the recursive propagation of this function is derived.Due to the unstructured nonlinear terrain reference map, the propagation of the conditional density is impossible to perform in practice. To circumvent this problem, an approximation of the conditional density is introduced. The recursionis expressed as an update of a set of point-mass weights distributed over an adaptive grid. An efficient implementation of this update is developed.In realistic simulations, using a commercial map, the proposed point-mass implementation gives reliable estimates with small estimation errors. The most important feature of the algorithm is the ability to track several position hypotheses in the terrain. A Monte Carlo analysis shows that the filter meets the Cramér-Rao lower bound as the grid resolution increases.The Cramér-Rao bound is also utilized to derive an information counterpart of the reference map. This information map reveals the areas where high accuracy navigation can be performed.Furthermore, extensions of the point-mass implementation that account for bias errors in the geographic altitude are proposed.

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