Application of dual numbers for solving the problems of interorbital flight optimization

Аuthors

Nikolichev I. A.

e-mail: ianikolichev@gmail.com

Abstract

Subject

The subject of this article is to analyze two aspects of application of the dual numbers for solving optimization problems of the multi revolution interorbital flight of the spacecraft with electric propulsion system.

Purpose

The purpose of this article is to demonstrate the possibility of using dual numbers in solving complicated optimization problems of the interorbital flight.

Methodology

The paper analyzed two aspects of application of the dual numbers to calculate the required derivatives during the solving optimization problems of the interorbital flight of the spacecraft with electric propulsion system. The use of dual numbers allows to determine the values of the derivatives with relative accuracy equals to the precision of function computation. The first aspect corresponds to the use of dual numbers with a single dual part together with the continuation method for calculating the elements of the sensitivity matrix of the system of nonlinear equations corresponding to the boundary value problem of the Pontryagin maximum principle. The second aspect is the use of dual numbers with vector dual part to calculate the right-hand sides of the system of differential equations, which describes the optimal process. By virtue of the canonical formalism of the maximum principle it is necessary to calculate the optimal Hamiltonian in the dual representation. This approach is used for solving the optimization problem of the interorbital flight when the model of the spacecraft motion takes into account various disturbances.

Results

The results of two types optimization problems of the interorbital flight of the spacecraft with electric propulsion system between the initial elliptical orbit and geostationary orbit are presented in this paper. The first type of problem corresponds to the motion model of the spacecraft in the central gravitational field under the influence of the reactive acceleration. For the second type the model takes into account the effect of the Moon and Sun attraction, and disturbance caused by the Earths gravitational field noncentrality. The functional that corresponds to the minimum mass of required fuel quantity is considered for both problems. Solution of the first type was obtained by using the continuation method and approximation of elements of the sensitivity matrix using dual numbers with a single dual part. For the problem of the second type, dual numbers with vector dual part is often used when we determining the right- hand sides of differential equations of optimal motion of the spacecraft. The nonlinear boundary value problem in this case was solved by a hybrid Powell algorithm.

Practical implications

Overall, the methodological approach of the using dual numbers for automatic differentiation outlined in this paper can be used for solving any optimization problems, where model of the object motion is described in a complicated manner.

Conclusions

This paper shows, that the using of the device of the dual numbers for computing required derivatives can efficiently solve complicated optimization problems of the interorbital flight of the spacecraft with electric propulsion system.

Keywords:

spacecraft, electric propulsion system, dual numbers, interorbital flight

References

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