Linearization method in the problems of spacecraft with electric propulsion power plant transfer to geostationary orbit

DOI: 10.34759/trd-2020-115-09

Аuthors

Kasmerchuk P. V.^*, Vernigora L. V.^**

Lavochkin Research and Production Association, NPO Lavochkin, 24, Leningradskay str., Khimki, Moscow region, 141400, Russia

*e-mail: pavel.kazmerchuk@gmail.com
**e-mail: vlv@laspace.ru

Abstract

The work “Optimization of non-coplanar low-thrust flights by the linearization method” by P.V. Kazmerchuk and L.V. Vernigorа solved the problem of non-coplanar flight from a high elliptical orbit to a geostationary orbit (GSO) within the minimum time (the task of optimal response) employing modified linearization method. In this task, the thrust acts continuously, with a constant level, and control is performed by the thrust vector turn. Another important class of optimization problems of transferring the spacecraft (SC) with electric propulsion power plant to the GSO are the problems of the final mass maximization under the limitations on the transfer time (the problem with the fixed time). The flight time increase relative to the optimal response problem allows reducing fuel consumption and increase the final mass by disabling the propulsion system at the trajectory legs with less efficient control. The article considers the non-adjustable EPPs, which thrust can be either zero, or maximum.

Maximum principle is widely used to solve the tasks of the EPP SC flight optimization. Application of the Maximum principle allows reducing the original optimization task to solving a boundary value problem. When solving such boundary value problems, researchers face convergence problems, the initial approximation selection, stipulated by the locality of the Maximum principle, and the existence and uniqueness of solutions to the systems of nonlinear equations. To overcome the above said difficulties, the authors use various approaches, such as the continuation method, sequential refinement of motion models, combinations of numerical methods, and others. A specific feature of the fixed time task consists in the presence of a large number of local extrema, which further complicates the solution of the boundary value problem of the Maximum principle. This happens due to the fact that partial derivatives matrix of the discrepancy of a boundary value problem by its unknown parameters, one way or another used in all indirect methods, degenerates at the boundaries of the domains of attraction of local extrema.

Being a direct method operating in the controls space, the MLM belongs to the class of gradient methods (first order methods), which stipulates its large convergence domain. It turned out to be possible to use trivial initial approximations in the task of optimal response that allows expect its successful application for the tasks with a fixed time as well.

MLM also belongs to the class of local methods, but it is desirable to have an algorithm that allows getting solutions close to global ones. Looking ahead, we note that it was possible to parameterize the task with only two quantities: the initial value of the true longitude of the flight and the initial value of the EPS thrust. The resulting two-parameter family can be easily studied by direct enumeration to find the global minimum without the need to invoke global optimization methods.

The main problem when trying to use MLM for solving a task with a fixed time consists in determining the number of thrust switching points and their location on the trajectory, since there is no explicit information about the switching function. The solution algorithm should automatically determine the optimal values of these parameters along with the thrust vector direction optimizing at the active legs of trajectory. The article proposes an algorithm that allows regularly solving the tasks of EPP SC flight to the GSO with a fixed time.

Keywords:

linearization method, low thrust, nonlinear optimization, geostationary orbit, electrical propulsion power plant

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