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Alexander S. Samokhin
(Lomonosov Moscow State University, V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences)

Methodology for Pontryagin's extremals constructing in the problems of end-to-end trajectory optimization of interplanetary flights, taking into account planet-centric sections


The research considers a three-dimensional end-to-end optimization problem for the trajectory of an interplanetary flight to Mars of the spacecraft with a one whole functional, a detailed consideration of planetocentric sections without the use of zero-dimensional gravispheres. The mission is considered in three different formulations: 1) with a direct hit to Phobos; 2) with a three-pulse approach to Phobos; 3) with a many-revolution no-pulse approach to Phobos.

In the first part of the mission, the spacecraft launches from an artificial Earth satellite orbit, corresponding to the launch from Baikonur, and arrives to Mars. Then the spacecraft conducts research on Phobos, including the extraction of soil samples, lasting at least 30 days. In the second part of the mission, the spacecraft flies back from Phobos to the Earth. The gravitational fields of the Sun, the Earth, and Mars are considered central Newtonian. The total duration of the expedition is limited. The positions of the Earth, Mars, and Phobos in the first two schemes of the mission correspond to DE424 and MAR097 ephemeris, the third scheme is considered in the circular formulation. The research assumes that the spacecraft and Phobos are non-attractive material points, their coordinates, and velocities at the final moment of the first part and the initial moment of the second part coincide. The start and finish moments of the spacecraft are optimized. The spacecraft is equipped with a combined thrust propulsion system and is controlled by the magnitude and direction of the jet thrust vector. The start and finish times, the angular position of the spacecraft on the initial orbit of an artificial Earth satellite, the moments of switching on and off the thrust are optimized. One whole functional is considered: in the first two mission schemes the mass costs are minimized, in the third the time-optimal problem is considered.

A methodology for solving the described multi-extremes problems of interplanetary trajectory optimization with the return to Earth, taking into account ephemeris, with rigid phasing, combined large and small piecewise continuous limited thrust, including the solution of a series of auxiliary problems in a simplified formulation and continuation of the solution by parameter, is proposed. Numerical methods have been developed to solve the corresponding boundary value problems of the maximum principle arising in the control of a set of dynamical systems, taking into account the effect of accuracy losses and trajectory structure restructuring when the number of active parts varies during the continuation of the solution by parameter. A technique based on the optimization of combinations of the Lambert problem by direct methods and solutions of problems in the pulse formulation has been developed for constructing an initial approximation for the parameters of the firing method needed to find the area of finding the globally optimal solution and to obtain values of conjugate variables required for the convergence of the modified Newton method.

On the basis of the proposed techniques and numerical methods, the program complex in C language was realized using the NASA SPICE package for the ephemeris accounting. The 9-point boundary value problems of order 70 were solved numerically. As a result, specific extrema were constructed for all three different variants of the spacecraft's expedition to Mars and its satellite Phobos. On the basis of the analysis of the constructed Pontryagin extremals, the benefit from the use of low thrust in the delivery of soil samples from Phobos is estimated, allowing to conclude the expediency of equipping the spacecraft with such engine. The comparison of various schemes of the expedition is made.

The report is based on the thesis for the degree of Candidate of Sciences in Physics and Mathematics (supervisors - Ilya S. Grigoriev, Maxim P. Zapletin). The text of the dissertation is available on the website of the Keldysh IPM RAS Council D 002.024.01

Full text of the Dissertation (in Russian) is also available here: