Literature review

Minimum-Thrust Transfers to the Moon

1. V. Ivanyukhin, V. G. Petukhov, Sung Wook Yoon

Moscow

Abstract—This paper considers the problem of optimizing low-thrust transfers to lunar orbits, to the libration points of the Earth–Moon system, and to the halo orbits around them. The goal of optimization is to minimize the thrust value. To solve the minimum-thrust problem, an indirect method based on the use of the maximum principle and the continuation method is used. The optimization problem for the considered type of trajectories gave rise to the necessity to overcome a number of problems associated with computational instability, limited size of the domain of existence of the solution and the necessity to choose the correct relation between the angular distance and the transfer duration in the geocentric and the selenocentric segments of trajectory. To overcome these difficulties, we propose to use a sequential solution of the problem of optimizing the limited power trajectories and the minimum-thrust problem in the formulation with the fixed angular distance and free transfer duration. The accuracy of calculating the derivatives that necessary to solve the boundary value problem of the maximum principle is ensured by the application of the automatic differentiation using complex dual numbers. The numerical examples of optimal direct and low-energy trajectories including the segment of motion along a stable manifold are given.

CONCLUSIONS

A numerical method to solve the minimum-thrust problem for a multi-revolution low-thrust transfer to the Moon has been developed based on the maximum principle, the continuation method, the automatic differentiation using complex dual numbers, and the use of auxiliary longitude as an independent variable and formulation of problem with a fixed angular distance and free transfer duration. The numerical results of optimizing trajectories from the elliptical EO to the circular and the elliptical LO are presented, as well as trajectories from the circular EO to the halo orbits around libration points EML1 and EML2. The obtained results are compared with trajectories with suboptimal feedback control. In the considered examples, optimal control provides a reduction in thrust value by 6–8% compared to suboptimal feedback control at the same angular distance. The developed method for calculating the minimum-thrust trajectories to the Moon can be used as a tool for diagnosing the existence of a solution in the problem of optimizing trajectories with limited thrust: with the given angular distance of transfer, the trajectory with constant exhaust velocity and limited thrust exists only if the given value of thrust is not lower than the minimum value.

End-to-End Optimization of Power-Limited Earth–Moon Trajectories

Viacheslav Petukhov and Sung Wook Yoon

Moscow, Russia

Abstract: The aim of this study is to analyze lunar trajectories with the optimal junction point of geocentric and selenocentric segments. The major motivation of this research is to answer two questions: (1) how much of the junction of the trajectory segments at the liberation point between the Earth and the Moon is non-optimal? and (2) how much can the trajectory be improved by optimizing the junction point of the two segments? The formulation of the end-to-end optimization problem of power-limited trajectories to the Moon and a description of the method of its solution are given. The proposed method is based on the application of the maximum principle and continuation method. Canonical transformation is used to transform the costate variables between geocentric and selenocentric coordinate systems. For the initial guess, a collinear libration point between the Earth and the Moon is used as a junction point, and the transformation to the optimal junction of these segments is carried out using the continuation method. The developed approach does not require any user-supplied initial guesses. It provides the computation of the optimal transfer duration for trajectories with a given angular distance and facilitates the incorporation of the perturbing accelerations in the mathematical model. Numerical examples of low-thrust trajectories from an elliptical Earth orbit to a circular lunar orbit considering a four-body ephemeris model are given, and a comparison is made between the trajectories with an optimal june