Gauss' Variational Equations for Low-Thrust Optimal Control Problems in Low-Energy Regimes

摘要

This paper proposes a new description of the equations of motion for low-thrust trajectory design in the presence of a third-body perturbation. The framework is formulated using Gauss' Variational Equations (GVE) with two distinct accelerations: the one produced by the electric engine and the disturbing term of the third-body effect. The latter is computed using the disturbing potential of the previously studied Keplerian Map, formulated from the Hamiltonian of the circular-restricted three-body problem. The presented GVE framework is advantageous in several respects: first, low-thrust sub-optimal control laws can be easily generated and explored to find a first guess solution near global optima. Second, bounds for the optimal control problem, as well as boundary values, can be easily defined, leading to a much faster convergence. This dynamical framework is accurate until very close to the sphere of influence of the perturbing body, and thus can be efficiently used to target low-energy hyperbolic invariant manifold structures associated with periodic orbits near it. This paper presents the methodology as well as a full retrieval trajectory for asteroid 2018 AV2, a small co-orbital asteroid that could be retrieved during its next Earth encounter in 2037.