Optimal Path Planning in a Threat Environment

Abstract

Analytical and discrete optimization approaches for routing an aircraft in a threat environment have been developed. The model considers an aircraft’s trajectory in three dimensional (3D) space and presents the aircraft by a symmetrical ellipsoid with the axis of symmetry orienting the trajectory direction. The threat is associated with the risk of aircraft detection by radars, sensors or surface air missiles. Using the analytical and discrete optimization approaches, the deterministic problem of finding an aircraft’s optimal risk trajectory subject to a constraint on the trajectory length has efficiently been solved. Through techniques of Calculus of Variations, the analytical approach reduces the original risk optimization problem to a vectorial nonlinear differential equation. In the case of a single detecting installation, the solution to this equation is expressed by a quadrature. The discrete optimization approach approximates the original problem by the Constrained Shortest Path Problem (CSPP) for a 3D network with a flexible structure. The CSPP has been solved for various ellipsoid shapes and different length constraints in the cases of several radars. The impact of ellipsoid shape on the geometry of an optimal trajectory as well as impact of variable RCS on performance of the discrete optimization approach have been analyzed and illustrated with several numerical examples.