#### For citation:

Pankratov I. A., Sapunkov Y. G., Chelnokov Y. N. Quaternion Models and Algorithms for Solving the General Problem of Optimal Reorientation of Spacecraft Orbit. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2020, vol. 20, iss. 1, pp. 93-104. DOI: 10.18500/1816-9791-2020-20-1-93-104

# Quaternion Models and Algorithms for Solving the General Problem of Optimal Reorientation of Spacecraft Orbit

The problem of optimal reorientation of the spacecraft orbit is considered in quaternion formulation. Control (vector of the acceleration of the jet thrust) is limited in magnitude. To solve the problem it is required to determine the optimal orientation of this vector in space. It is necessary to minimize the duration of the process of reorientation of the spacecraft orbit. To describe the motion of the center of mass of the spacecraft we used quaternion differential equation of the orientation of the spacecraft orbit. The problem was solved using the maximum principle of L. S. Pontryagin. Accounting the known particular solution of the equation for the variable conjugated to a true anomaly, we allowed to simplify the equations of the problem. The problem of optimal reorientation of the spacecraft orbit is reduced to a boundary value problem with a moving right end trajectory described by a system of nonlinear differential equations of the fifteenth order. For the numericalsolution of the obtained boundary value problem the transition to dimensionless variables was carried out. At the same time a characteristic dimensionless parameter of the problem appeared in the phase and conjugate equations. The original numerical algorithm for finding unknown initial values of conjugate variables, which is a combination of the 4th order Runge –Kutta method, modified Newton method and gradient descent method is constructed. The use of two methods for solving boundary value problems has improved the accuracy of the boundary value problem solution of optimal control. Examples of numerical solution of the problem are given for the case when the difference between the initial and final orientations of the spacecraft orbit equals to a few (or tens of) degrees in angular measure. Graphs of component changes of the spacecraft orbit orientation quaternion; variables characterizing the shape and dimensions of the spacecraft orbit; optimal control are plotted. The analysis of the obtained solutions is given. The features and regularities of the optimal reorientation of the spacecraft orbit are established.

- 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: II. Cosmic Research, 2003, vol. 41, no. 1, pp. 85–99. DOI: https://doi.org/10.1023/A:1022359831200
- Abalakin V. K., Aksenov E. P., Grebennikov E. A., Demin V. G., Riabov Yu. A. Spravochnoe rukovodstvo po nebesnoi mekhanike i astrodinamike [Reference guide on celestial mechanics and astrodynamics]. Moscow, Nauka, 1976. 864 p. (in Russian).
- Duboshin G. N. Nebesnaia mekhanika. Osnovnye zadachi i metody [Celestial mechanics. Main tasks and methods]. Moscow, Nauka, 1968. 799 p. (in Russian).
- Pankratov I. A., Sapunkov Ya. G., Chelnokov Yu. N. About a Problem of Spacecraft’s Orbit Optimal Reorientation. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2012, vol. 12, iss. 3, pp. 87–95 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2012-12-3-87-95
- 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). DOI: https://doi.org/10.18500/1816-9791-2013-13-1-1-84-92
- Kirpichnikov S. N., Bobkova A. N., Os’kina Yu. V. Minimum-time impulse transfers between coplanar circular orbits. Kosmicheskie issledovaniia [Cosmic Research], 1991, vol. 29, no. 3, pp. 367–374 (in Russian).
- 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.
- 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, iss. 8, pp. 601–611. DOI: https://doi.org/10.1016/S0094-5765(02)00130-3
- 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, iss. 3–4, pp. 528–538. DOI: https://doi.org/10.1016/j.actaastro.2009.07.021
- Grigorev K. G., Fedyna A. V. Optimal flights of a spacecraft with jet engine large limited thrust between coplanar circular orbits. Kosmicheskie issledovaniia [Cosmic Research], 1995, vol. 33, no. 4, pp. 403–416 (in Russian).
- Ryzhov S. Yu., Grigoriev I. S. On solving the problems of optimization of trajectories of many-revolution orbit transfers of spacecraft. Cosmic Research, 2006, vol. 44, iss. 3, pp. 258–267. DOI: https://doi.org/10.1134/S0010952506030105
- 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. DOI: https://doi.org/10.1134/S0010952507040077
- 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. DOI: https://doi.org/10.1134/S0010952507060093
- Kirpichnikov S. N., Bobkova A. N. Optimal impulse interorbital flights with aerodynamic maneuvers. Kosmicheskie issledovaniia [Cosmic Research], 1992, vol. 30, no. 6, pp. 800– 809 (in Russian).
- 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.
- Condurache D., Martinusi V. Quaternionic Exact Solution to the Relative Orbital Motion Problem. Journal of Guidance, Control, and Dynamics, 2010, vol. 33, no. 4, pp. 1035– 1047. DOI: https://doi.org/10.2514/1.47782
- Condurache D., Burlacu A. Onboard Exact Solution to the Full-Body Relative Orbital Motion Problem. Journal of Guidance, Control, and Dynamics, 2016, vol. 39, no. 12, pp. 2638–2648. DOI: https://doi.org/10.2514/1.G000316
- 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.
- Salmin V. V., Sokolov V. O. Approximate calculation of the formation manoeuvres of the satellite orbit the Earth with a small engine thrusts. Kosmicheskie issledovaniia [Cosmic Research], 1991, vol. 29, no. 6, pp. 872–888 (in Russian).
- Afanas’eva Yu. V., Chelnokov Yu. N. The problem of optimal control of the orientation of an orbit of a spacecraft as a deformable figure. Journal of Computer and Systems Sciences International, 2008, vol. 47, no. 4, pp. 621–634. DOI: https://doi.org/10.1134/S106423070804014X
- 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).
- 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).
- 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, no. 5, pp. 460–477. DOI: https://doi.org/10.1023/A:1026098216710
- Bordovitsyna T. V. Sovremennye chislennye metody v zadachakh nebesnoi mekhaniki [Modern numerical methods in problems of celestial mechanics]. Moscow, Nauka, 1984. 136 p. (in Russian).

- 800 reads