Genetic Algorithm of Energy Consumption Optimization for Reorientation of the Spacecraft Orbital Plane

The paper is dedicated to the problem of finding optimal spacecraft trajectories. The equations of spacecraft motion are written in quaternion form. The spacecraft moves on its orbit under acceleration from the limited in magnitude jet thrust. It is necessary to minimize the energy costs for the process of reorientation of the spacecraft orbital plane. The equations of spacecraft motion are written in orbital coordinate system. It is assumed that spacecraft orbit is circular and control has constant value on each part of active spacecraft motion. In this case the lengths of the sections of the spacecraft motion are unknown. We need to find the length of each section, their quantity and value of control on each section. The equations of the problem were written in dimensionless form. It simplifies the numerical investigation of the obtained problem. There is a characteristic dimensionless parameter in the phase equations of the problem. This parameter is a combination of dimension variables describing the spacecraft and its orbit. Usually the problems of spaceflight mechanic are solved with the maximum principle. And we have to solve boundary value problems with some kind of shooting method (Newton’s method, gradient descent method etc.) Each shooting method requires initial values of conjugate variables, but we have no analytical formulas to find them. In this paper spacecraft flight trajectories were found with new genetic algorithm. Each gene contains additional parameter which equals to " True" , if the gene forms the control and equals to "False" otherwise. It helps us determine the quantity of spacecraft active motion parts. The input of proposed algorithm does not contain information about conjugate variables. It is well-known that the differential equations of the problem have a partial solution when the spacecraft orbit is circular and control is constant. The genetic algorithm involves this partial solution and its speed is increased. Numerical examples were constructed for two cases: when the difference between angular variables for start and final orientations of the spacecraft orbital plane equals to a few (or tens of) degrees. Final orientation of the spacecraft plane of orbit coincides with GLONASS orbital plane. The graphs of components of the quaternion of orientation of the orbital coordinate system, the longitude of the ascending node, the orbit inclination and optimal control are drawn. Tables were constructed showing the dependence of the value of the quality functional and the time spent on the reorientation of the orbital plane on the maximum length of the active section of motion.

1. Kirpichnikov S. N., Bobkova A. N., Os’kina Yu. V. Minimum-time impulse transfers between coplanar circular orbits, Kosmicheskie Issledovaniia, 1991, vol. 29, no. 3, pp. 367—374 (in Russian).

2. Grigoriev K. G., Grigoriev I. S., Petrikova Yu. D. The fastest maneuvers of a spacecraft with a jet engine of a large limited thrust in a gravitational field in a vacuum, Cosmic Research, 2000, vol. 38, no. 2, pp. 160—181.

3. Kiforenko B. M., Pasechnik Z. V., Kyrychenko S. B., Vasiliev I. Yu. Minimum time transfers of a low-thrust rocket in strong gravity fields, Acta Astronautica, 2003, vol. 52, no. 8, pp. 601—611.

4. Fazelzadeh S. A., Varzandian G. A. Minimum-time earthmoon and moon-earth orbital maneuevers using time-domain finite element method, Acta Astronautica, 2010, vol. 66, no. 3—4, pp. 528—538.

5. Grigoriev K. G., Fedyna A. V. Optimal flights of a spacecraft with jet engine large limited thrust between coplanar circular orbits, Kosmicheskie Issledovaniia, 1995, vol. 33, no. 4, pp. 403—416 (in Russian).

6. Ryzhov S. Y., Grigoriev I. S. On solving the problems of optimization of trajectories of many-revolution orbit transfers of spacecraft, Cosmic Research, 2006, vol. 44, no. 3, pp. 258—267.

7. Grigoriev I. S., Grigoriev K. G. The use of solutions to problems of spacecraft trajectory optimization in impulse formulation when solving the problems of optimal control of trajectories of a spacecraft with limited thrust engine: I, Cosmic Research, 2007, vol. 45, no. 4, pp. 339—347.

8. Grigoriev I. S., Grigoriev K. G. The use of solutions to problems of spacecraft trajectory optimization in impulse formulation when solving the problems of optimal control of trajectories of a spacecraft with limited thrust engine: II, Cosmic Research, 2007, vol. 45, no. 6, pp. 523—534.

9. Kirpichnikov S. N., Bobkova A. N. Optimal impulse interorbital flights with aerodynamic maneuvers, Kosmicheskie Issledovaniia, 1992, vol. 30, no. 6, pp. 800—809 (in Russian).

10. Kirpichnikov S. N., Kuleshova L. A., Kostina Yu. L. A qualitative analysis of impulsive minimum-fuel flight paths between coplanar circular orbits with a given launch time, Cosmic Research, 1996, vol. 34, no. 2, pp. 156—164.

11. Ishkov S. A., Romanenko V. A. Forming and correction of a high-elliptical orbit of an earth satellite with low-thrust engine, Cosmic Research, 1997, vol. 35, no. 3, pp. 268—277.

12. Kamel O. M., Soliman A. S. On the optimization of the generalized coplanar Hohmann impulsive transfer adopting energy change concept, Acta Astronautica, 2005, vol. 56, no. 4, pp. 431—438.

13. Mabsout B. E., Kamel O. M., Soliman A. S. The optimization of the orbital Hohmann transfer, Acta Astronautica, 2009, vol. 65, no. 7—8, pp. 1094—1097.

14. Miele A., Wang T. Optimal transfers from an Earth orbit to a Mars orbit, Acta Astronautica, 1999, vol. 45, no. 3, pp. 119—133.

15. Branets V. N., Shmyglevskii I. P. Use of quaternions in the problems of orientation of solid bodies, Moscow, Nauka, 1973, 320 p. (in Russian).

16. Chelnokov Yu. N. The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a newtonian gravitational field: I, Cosmic Research. 2001, vol. 39, no. 5, pp. 470—484.

17. Abalakin V. K., Aksenov E. P., Grebennikov E. A., Demin V. G., Riabov Iu. A. Reference guide on celestial mechanics and astrodynamics, Moscow, Nauka, 1976, 864 p. (in Russian).

18. Duboshin G. N. Celestial mechanics. Main tasks and methods, Moscow, Nauka, 1968, 799 p. (in Russian).

19. Pankratov I. A. Genetic algorithm for minimizing the energy costs for the reorientation of the plane of the spacecraft orbit, Nauchno-tekhnicheskiy vestnik Bryanskogo gosudarstvennogo universiteta. 2017, no. 3, pp. 353—360 (in Russian)

20. Pankratov I. A. Calculating of the fastest spacecraft flights between circular orbits, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika. 2017, vol. 17, no. 3, pp. 344—352 (in Russian).

21. Chelnokov Yu. N., Pankratov I. A., Sapunkov Ya. G. Optimal reorientation of spacecraft orbit, Archives of Control Sciences. 2014, vol. 24, no. 2, pp. 119—128.

22. Kozlov E. A., Pankratov I. A., Chelnokov Yu. N. Investigation of the problem of optimal correction of angular elements of the spacecraft orbit using quaternion differential equation of orbit orientation, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika. 2016, vol. 16, no. 3, pp. 336—344 (in Russian).

23. Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. Numerical Recipes 3rd Edition: The Art of Scietific Computing, Cambridge University Press, 2007, 1235 p.

24. Goldberg D. E. Genetic Algorithms in Search, Optimization and Machine Learning, Reading, Massachusets, AddisonWesley P ublishing Company, Inc., 1989, 412 p.

25. Zebulum R., Pacheco M., Velasco M. Evolutionary Electronics: Automatic Design of Electronic Circuits and Systems by Genetic Algorithms, Boca Raton, London, New York, Washington, CRC Press, 2001, 320 p.

26. Coverstone-Carrol V., Hartmann J. W., Mason W. J. Optimal multi-objective low-thrust spacecraft trajectories, Computer methods in applied mechanics and engineering, 2000, vol. 186, no. 2—4, pp. 387-402.

27. Dachwald B. Optimization of very-low-thrust trajectories using evolutionary neurocontrol, Acta Astronautica, 2005, vol. 57, no. 2—8, pp. 175—185.

28. Eiben A. E., Smith J. E. Introduction to Evolutionary Computing, Berlin, Springer-Verlag, 2015, 294 p.

29. Chelnokov Yu. N. On determining vehicle orientation in the Rodrigues-Hamilton parameters from its angular velocity, Mechanic of Solids. 1977, vol. 37, no. 3, pp. 8—16.

30. Bordovitsyna T. V. Modern numerical methods in problems of celestial mechanics, Moscow, Nauka, 1984, 136 p. (in Russian).