# Analytical Solution of the Minimum Time Slew Maneuver Problem for an Axially Symmetric Spacecraft in the Class of Conical Motions

- 16 Downloads

## Abstract

The conventional problem of the time-optimal slew of a spacecraft considered as a solid body with a single symmetry axis subject to arbitrary boundary conditions for the attitude and angular velocity is considered in the quaternion statement. By making certain changes of variables, the original dynamic Euler equations are simplified, and the problem turns into the optimal slew problem for a solid body with a spherical distribution of mass containing one additional scalar differential equation. For this problem, a new analytical solution in the class of conical motions is found; in this solution, the initial and terminal attitudes of the space vehicle belong to the same cone realized under a bounded control. A modification of the optimal slew problem in the class of generalized conical motions is made that makes it possible to obtain its analytical solution under arbitrary boundary conditions for the attitude and angular velocity of the spacecraft. A numerical example of a spacecraft’s conical motion and examples demonstrating the proximity of the solutions of the conventional and modified optimal slew problems of an axially symmetric spacecraft are discussed.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.V. N. Branets and I. P. Shmyglevskii,
*Application of Quaternions in Problems of Solid Body Orientation*(Nauka, Moscow, 1973) [in Russian].zbMATHGoogle Scholar - 2.S. L. Scrivener and R. C. Thompson, “Survey of time-optimal attitude maneuvers,” J. Guidance, Control, Dyn.
**17**(2) (1994).Google Scholar - 3.B. N. Petrov, V. A. Bodner, and K. B. Alekseev, “An analytical solution of the problem control of the spatial turn maneuver,” Dokl. Akad. Nauk SSSR
**192**(6) (1970).Google Scholar - 4.V. N. Branets, M. B. Chertok, and Yu. V. Kaznacheev, “Optimal turn of a solid body with one symmetry axis,” Kosm. Issled.
**22**(3) (1984).Google Scholar - 5.A. N. Sirotin, “Optimal control of reorientation of symmetric solid body from a rest position to a rest position,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 1 (1989).Google Scholar
- 6.A. N. Sirotin, “On time-optimal spatial reorientation in a rest position of a rotating spherically symmetric body,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 3 (1997).Google Scholar
- 7.M. V. Levskii, “Pontryagin’s maximum principle in optimal control problems of orientation of a spacecraft,” J. Comput. Syst. Sci. Int.
**47**, 974 (2008).MathSciNetCrossRefzbMATHGoogle Scholar - 8.M. V. Levskii, “Optimal spacecraft terminal attitude control synthesis by the quaternion method,” Mech. Solids
**44**, 169 (2009).CrossRefGoogle Scholar - 9.A. V. Molodenkov and Ya. G. Sapunkov, “A new class of analytic solutions in the optimal turn problem for a spherically symmetric body,” Mech. Solids
**47**, 167 (2012).CrossRefzbMATHGoogle Scholar - 10.A. V. Molodenkov, “A solution to the problem of optimal U-turn of a spherically symmetric spacecraft for a partial case,” in
*Proceedings of 6th International Conference of Systems Analysis and Control of Spacecraft, Evpatoriya, Ukraine, 2001*(MAI, Moscow, 2001).Google Scholar - 11.A. V. Molodenkov and Ya. G. Sapunkov, “Analytical solution of the time-optimal slew problem of a spherically symmetric spacecraft in the class of conical motions,” J. Comput. Syst. Sci. Int.
**53**, 159 (2014).MathSciNetCrossRefzbMATHGoogle Scholar - 12.A. V. Molodenkov and Ya. G. Sapunkov, “Analytical solution of the optimal slew problem for an axisymmetric spacecraft in the class of conical motions,” J. Comput. Syst. Sci. Int.
**55**, 969 (2016).MathSciNetCrossRefzbMATHGoogle Scholar - 13.A. V. Molodenkov and Ya. G. Sapunkov, “Analytical approximate solution of the problem of a spacecraft’s optimal turn with arbitrary boundary conditions,” J. Comput. Syst. Sci. Int.
**54**, 458 (2015).MathSciNetCrossRefzbMATHGoogle Scholar - 14.L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko,
*The Mathematical Theory of Optimal Processes*(Nauka, Moscow, 1961; Gordon and Breach, New York, 1986).Google Scholar - 15.A. V. Molodenkov, “On the solution of the Darboux problem,” Mech. Solids
**42**, 167 (2007).CrossRefGoogle Scholar - 16.P. M. Bainum, “Numerical approach for solving rigid spacecraft minimum time attitude maneuvers,” J. Guidance, Contr., Dyn.
**13**(1) (1990).Google Scholar - 17.Ya. G. Sapunkov and A. V. Molodenkov, “The investigation of characteristics of distant sounding system of the earth with the help of cosmic device,” Mekhatron., Avtomatiz., Upravl., No. 6, 10–15 (2008).Google Scholar