This paper explores the optimum two-impulse transfer problem between a low Earth orbit and sample cycler orbits in the framework of the circular restricted three-body framework, emphasizing the optimization strategy. Cyclers are those type of periodic orbits that encounter both the Earth and the Moon periodically. Cyclers have gained recent interest as baseline orbits for several Earth-Moon mission concepts. In this paper we show that a direct application of Lambert initial guess may not be adequate for these problems, and a two-step process is investigated to improve upon the range of boundary conditions where convergence is reached. The first step consists of finding feasible trajectories with a given transfer time. Here two methods are investigated: the use of a shooting method from a Lambert initial guess, or smooth deformation of the dynamics from a Lambert solution using continuation methods.The second step optimizes the impulse over transfer time, which thus results in the minimum impulse transfer for fixed end points. Contour maps of optimal impulses in the phase space of departure and arrival points are then computed to summarize the results and show the limitation of the method. In particular, the direct optimization fails to converge for most boundary conditions, while the continuation from Lambert initial guess do not capture most Moon gravity assists transfers. Lambert solutions, however, are seen as providing a good approximation to the transfer cost, albeit not the velocity directions.
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