# Calculating of the Fastest Spacecraft Flights between Circular Orbits

The problem of optimal reorientation of spacecraft orbit is considered in quaternion formulation. Control (jet thrust vector orthogonal to the plane of the orbit) is limited in magnitude. It is necessary to minimize the duration of the process of reorientation of the spacecraft orbit. To describe the motion of the spacecraft center of mass quaternion differential equations of the orientation of the orbital coordinate system was used. The actual special case of the problem, when the spacecraft’s orbit is circular and control equals to its maximum (in modulus) value on adjacent parts of active spacecraft motion, was considered. Original genetic algorithm for finding the trajectories of spacecraft fastest flights is built. In this case the lengths of the active sections of the spacecraft motion are unknown. This method does not require any information about the unknown initial values of conjugate variables. The high speed of operation of the proposed genetic algorithm is achieved through the use of existing, in this case, a known analytical solution of equations of the problem. Examples of numerical solution of the problem for the case when the difference between the initial and final orientations of the spacecraft’s orbit equals to a few degrees in angular measure, are given. The final orientation of the spacecraft’sorbitcor respondsto one of the satellites of Russian GLONASS orbital grouping. The graphs of components of the quaternion of orientation of the orbital coordinate system, the deviation of the current position of the spacecraft’s orbit to the required and optimal control are drawn. Specific features and regularities of the process of optimum reorientation of the spacecraft’s orbit are given.

1. 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, vol. 39, iss. 5, 2001, pp. 470–484. DOI: 10.1023/A:1012345213745.

2. Pontriagin L. S., Boltianskii V. G., Gamkrelidze R. V., Mishchenko E. F. Matematicheskaia teoriia optimal’nykh protsessov [The mathematical theory of optimal processes]. Moscow, Nauka, 1983. 393 p. (in Russian).

3. Kirpichnikov S. N., Bobkova A. N., Os’kina Yu. V. Minimal’nye po vremeni impul’snye perelety mezhdu krugovymi komplanarnymi orbitami [Minimum-time impulse transfers between coplanar circular orbits]. Kosmicheskie issledovaniia [Cosmic Research], vol. 29, no 3, 1991, pp. 367—374 (in Russian).

4. 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, iss. 2, pp. 160–181.

5. 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. DOI: 10.1016/S0094-5765(02)00130-3.

6. Fazelzadeh S. A., Varzandian G. A. Minimum-time earth-moon and moon-earth orbital maneuevers using time-domain finite element method. Acta Astronautica, 2010, vol. 66, no. 3–4, pp. 528–538. DOI: 10.1016/j.actaastro.2009.07.021.

7. Pankratov I. A., Sapunkov Ya. G., Chelnokov Yu. N. Solution of a Problem of Spacecraft’s Orbit Optimal Reorientation Using Quaternion Equations of Orbital System of Coordinates Orientation. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 1, pt. 1, pp. 84–92 (in Russian).

8. Pankratov I. A., Sapunkov Ya. G., Chelnokov Yu. N. Chislennoe issledovanie zadachi pereorientatsii orbity kosmicheskogo apparata s ispol’zovaniem orbital’noi sistemy koordinat [Numerical study of the problem of reorientation of the spacecraft’s orbit using the orbital coordinate system]. Matematika. Mehanika [Mathematics. Mechanics]. Saratov, Saratov Univ. Press, 2012, iss. 14, pp. 132–136 (in Russian).

9. Moiseev N. N. Chislennye metody v teorii optimal’nykh sistem [Numerical methods in the theory of optimal systems]. Moscow, Nauka, 1971. 424 p. (in Russian).

10. Dachwald B. Optimization of very-low-thrust trajectories using evolutionary neurocontrol. Acta Astronautica, 2005, vol. 57, no. 2–8, pp. 175–185. DOI: 10.1016/j.actaastro.2005.03.004.

11. 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.

12. Panchenko T. V. Geneticheskie algoritmy [Genetic algorithms]. Astrakhan, Publ. House „Astrakhan University“, 2007. 87 p. (in Russian).

13. Pankratov I. A., Chelnokov Yu. N. Analytical Solution of Differential Equations of Circular Spacecraft’s Orbit Orientation. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2011, vol. 11, iss. 1, pp. 84–89 (in Russian).

14. 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: III. Cosmic Research, 2003, vol. 41, iss. 5, pp. 460–477. DOI: 10.1023/A:1026098216710