Geodesic
Analytical algorithm of energy and time quasioptimal turn of~a~spacecraft under arbitrary boundary conditions
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 213-226.

Voir la notice de l'article provenant de la source Math-Net.Ru

The optimal attitude maneuver control problem without control constraints is studied in the quaternion statement for an axially symmetric spacecraft as a rigid body under arbitrary boundary conditions on angular position and angular velocity of a spacecraft. The performance criterion is given by a functional combining the time and energy used for the attitude maneuver. Using substitutions of variables, the original problem is simplified (in terms of dynamic Euler equations) to the optimal slew problem for a rigid body with a spherical mass distribution. The simplified problem contains one additional scalar differential equation. In the class of generalized conical motions, the traditional optimal slew problem is modified to obtain analytical solutions for motion equations. The solutions contain arbitrary constants and two arbitrary scalar functions (generalized conical motion parameters). The proposed approach fits well with the Poinsot concept that any arbitrary angular motion of a rigid body around a fixed point can be considered as some generalized conical motion of a rigid body. Moreover, for the cases of analytic solvability of the classical problem of optimal reversal spherically symmetric spacecraft, when restrictions are imposed on the edge conditions of the problem (plane Euler turn, conical motion) solutions of the classical and modified tasks are completely the same. An optimization problem is formulated and solved with respect to these functions, the second derivatives of which serve as controls in the optimization problem. The resulting analytical solution of the modified problem can be treated as an approximate (quasioptimal) solution of the traditional optimal slew problem under arbitrary boundary conditions. The quasioptimal algorithm of the optimal turn of a spacecraft is given. Numerical example showing the closeness of the solutions of the traditional and modified optimal slew problems for an axially symmetric spacecraft is given. 
@article{ISU_2021_21_2_a7,
     author = {Ya. G. Sapunkov and A. V. Molodenkov},
     title = {Analytical algorithm of energy and time quasioptimal turn of~a~spacecraft under arbitrary boundary conditions},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {213--226},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2021_21_2_a7/}
}
TY  - JOUR
AU  - Ya. G. Sapunkov
AU  - A. V. Molodenkov
TI  - Analytical algorithm of energy and time quasioptimal turn of~a~spacecraft under arbitrary boundary conditions
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2021
SP  - 213
EP  - 226
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2021_21_2_a7/
LA  - ru
ID  - ISU_2021_21_2_a7
ER  - 
%0 Journal Article
%A Ya. G. Sapunkov
%A A. V. Molodenkov
%T Analytical algorithm of energy and time quasioptimal turn of~a~spacecraft under arbitrary boundary conditions
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2021
%P 213-226
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2021_21_2_a7/
%G ru
%F ISU_2021_21_2_a7
Ya. G. Sapunkov; A. V. Molodenkov. Analytical algorithm of energy and time quasioptimal turn of~a~spacecraft under arbitrary boundary conditions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 213-226. http://geodesic.mathdoc.fr/item/ISU_2021_21_2_a7/

[1] Branets V. N., Shmyglevskij I. P., The Use of Quaternions in Problems of Orientation of Solid Bodies, Nauka, M., 1973, 320 pp. (in Russian) | MR

[2] Scrivener S. L., Thompson R. C., “Survey of time-optimal attitude maneuvers”, Journal of Guidance, Control, and Dynamics, 17:2 (1994), 225–233 | DOI

[3] Petrov B. N., Bodner V. A., Alekseev K. B., “Analytical solution of the spatial slew manuever”, Reports of the USSR Academy of Sciences, 192:6 (1970), 1235–1238 (in Russian) | Zbl

[4] Branets V. N., Chertok M. B., Kaznacheev Yu. V., “Optimal slew of a solid body with a single symmetry axis”, Space Research, 22:3 (1984), 352–360 (in Russian)

[5] Sirotin A. N., “Optimal reorientation of a symmetric solid body from a state of rest to another state of rest”, News of the USSR Academy of Sciences. Mechanics of Solids, 1989, no. 1, 36–47 (in Russian)

[6] Chelnokov Yu. N., “Quaternion solution of kinematic problems in rigid body orientation control – equations of motion, problem statement, programmed motion, and control”, News of the Academy of Sciences. Mechanics of solids, 1993, no. 4, 7–14 (in Russian)

[7] Levskij M. V., “Pontryagin's maximum principle in optimal control problems of orientation of a spacecraft”, Journal of Computer and System Sciences International, 47:6 (2008), 974–986 | DOI | MR | Zbl

[8] Molodenkov A. V., Sapunkov Ya. G., “Analytical solution of the time-optimal slew problem of a spherically symmetric spacecraft in the class of conical motion”, Journal of Computer and Systems Sciences International, 53:2 (2014), 159–171 | DOI | DOI | MR | Zbl

[9] Pontriagin L. S., Boltianskii V. G., Gamkrelidze R. V., Mishchenko E. F., The Mathematical Theory of Optimal Processes, Nauka, M., 1961, 391 pp. (in Russian) | MR

[10] Li F., Bainum P. M., “Numerical approach for solving rigid spacecraft minimum time attitude maneuvers”, Journal of Guidance, Control, and Dynamics, 13:1 (1990), 38–45 | DOI | MR

[11] Molodenkov A. V., Sapunkov Ya. G., “Analytical approximate solution of the problem of a spacecraft's turn with arbitrary boundary conditions”, Journal of Computer and Systems Sciences International, 54:3 (2015), 458–468 | DOI | DOI | MR | Zbl