Optimization of the multirevolutional non-coplanar low-thrust orbital transfers

Аuthors

Nikolichev I. A.

e-mail: ianikolichev@gmail.com

Abstract

Subject
The subject of this article examines the optimization of non-complanar multirevolutional low-thrust transfer.
Purpos
The purpose of this article is to demonstrate a number of new teaching techniques used in the formulation of the optimization problem, the definition of a mathematical model, as well as the approach for solution to the boundary problem using continuation method.
Methodology
The main methodological approach during the consideration of optimization problem is use of the Pontryagin maximum principle. Following the formalism of the maximum principle the optimization problem is reduced to the solution to boundary value problem. To solve the boundary value problem the continuation method is used. The quality criterion is the minimum engine time required to orbital transfer provided that the engine spacecraft is uncontrollable. In our mathematical model the equinoctial elements is used to describe the motion of the spacecraft. Right parts of the canonical system of optimal motion numerically determined in the process of integration. To determine the derivatives the complex step method is used. Also, the complex step method is used to determine the elements of the Jacobian matrix for the nonlinear system of the boundary value problem.
Results
The results of solution to the two types of problems have been analyzed. Two different orbital transfers from the original elliptical or circular orbit to GEO are submitted. The dependencies of determining the evolution of the trajectory and optimal control program for pitch and yaw channels are presented.
Practical implications
The results can be applied in the design of transfer vehicle with electric propulsion system. Also, these results can be applied in the guidance and control synthesis tasks for those systems.
Conclusions
The calculation of the right parts of the differential equations describing the optimal motion of the spacecraft in the integration process using the complex step method shown to be effective. This technique is possible to avoid the time-consuming procedures of analytical writing the right parts of the equations for the conjugate system.

Keywords:

spacecraft, electric propulsion system, equinoctial elements, boundary value problem, continuation method

References

1. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimalnykh protsessov (The mathematical theory of optimal processes), Moscow, Nauka, 1983, 392 p. 
2. Konstantinov M.S., Kamenkov E.F., Perelygin B.P., Bezverbyi V.K. Mekhanika kosmicheskogo poleta (Mechanics of the space flight), Moscow, Mashinostroenie, 1989, 408 p. 
3.  Sackett L.L., Malchow H.L., Edelbaum T.N. Solar Electric Geocentric Transfer with Attitude Constraints: Analysis, NASA CR-134927, 1975, 137 p.