Optimal Transfers in the Earth-Moon System through Polyhedral Representation of Invariant Manifolds

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth-Moon system and in alternative multibody environments. Several researches have been focused on this topic in the last decades, and some space missions have already taken advantage of the results of these studies. Recent efforts have been devoted to developing a suitable representation for the manifolds, which would be extremely useful for mission analysis and optimization. This work proposes and describes an intuitive polyhedral interpolative approach for each state component associated with manifold trajectories, both in two and in three dimensions. An adequate grid of data, coming from the numerical propagation of a finite number of manifold trajectories, is employed. Accuracy of this representation is evaluated with reference to the two invariant manifolds associated with a two-dimensional Lyapunov orbit and with a three-dimensional Halo orbit, and is proven to be satisfactory, with the exclusion of limited regions of the manifold. The polyhedral interpolation technique has several applications. Three of them are illustrated in this paper. First, the globally optimal two-impulse transfer between a specified low Earth orbit and a Lyapunov orbit (through its stable manifold) is determined. Second, the minimum-time low-thrust transfer from the same terminal orbits is found (using again the stable manifold) through satisfaction of the necessary conditions for optimal ity. Third, the totality of the intersections between the unstable manifold associated with a Halo orbit and two specified lunar periodic orbits are determined. This leads to establishing the globally optimal single-impulse transfer between the Halo orbit and each lunar periodic orbit. These three applications prove the effectiveness of the polyhedral interpolative technique and represent the premise for its application also to different problems involving invariant manifold dynamics. Copyright © 2014 by Mauro Pontani and Paolo Teofilatto.