Orthogonal Approximation of Invariant Manifolds in the Circular Restricted Three-Body Problem

Abstract

Methods to parameterize and approximate the hyperbolic invariant manifolds of particular solutions in the circular restricted three-body problem (CR3BP) are presented in this paper. Analytical representations obtained from these manifold approximations are instrumental in the synthesis of optimal trajectories for cislunar transport. A multivariate Chebyshev series is used to approximate the surfaces, thereby serving as tractable parametric representations of the complex properties of motion. It is demonstrated that the continuum of ballistic capture trajectories and their associated sensitivities on the manifold can be realized in functional form using simple algebraic operations. Two applications making use of the Chebyshev manifold approximations as a terminal constraint surface are presented. The first is a low-thrust trajectory optimization problem formulated such that the optimal free final state lying on the manifold is determined as an additional set of design parameters. The second is a guidance law designed to target the manifold in the vicinity of the nominal patch point. Each of these methods takes advantage of the Chebyshev approximations to provide additional flexibility for mission design in multibody dynamic environments. These applications offer tremendous optimism about the utility of function approximation methods in arriving at a formal representation for the invariant manifolds in the three-body problem for efficient generation of optimal trajectories.