# Restricted Quadratic Optimal Control of a Spacecraft Turning in a Fixed Time Period

• M. V. Levskii
CONTROL SYSTEMS OF MOVING OBJECTS

### Abstract

A dynamic problem of a spacecraft (SC) turning from an arbitrary initial position to the required terminal angular position with restricted control is considered and solved. The termination time of the maneuver is known. The control program is optimized using the quadratic criterion of quality; the minimized functional characterizes the energy consumed in the turn. The construction of the optimal control of the turn is based on quaternion variables and Pontryagin’s maximum principle. It is shown that during the optimal turn, the moment of force is parallel to the straight line fixed in the inertial space, and the direction of the angular momentum in the process of the SC’s rotation is constant relative to the inertial coordinate system. The formalized equations and computational expressions for determining the optimal turning program are obtained. A special mode of control is studied in detail, and the conditions of the impossibility of occurrence of this mode are formulated. The control algorithms allow performing turns over a fixed time period with the minimum angular kinetic energy. A comprehensive solution to the control problem is given for a dynamically symmetric SC: the dependences as explicit functions of time for the control variables and relations for calculating the key parameters of the turn maneuver’s control law are obtained. A numerical example and the results of the mathematical modeling of the SC’s motion with the optimal control are given, which demonstrate the practical feasibility of the method for controlling the attitude of the SC.

## REFERENCES

1. 1.
V. N. Branets and I. P. Shmyglevskii, Application of Quaternions in Problems of Attitude Control of a Rigid Body (Nauka, Moscow, 1973) [in Russian].
2. 2.
K. B. Alekseev and G. G. Bebenin, Spacecraft Control (Mashinostroenie, Moscow, 1974) [in Russian].Google Scholar
3. 3.
N. E. Zubov, “Optimal control of the terminal reorientation of the spacecraft based on an algorithm with a predictive model,” Kosm. Issled. 29, 340–351 (1991).Google Scholar
4. 4.
A. I. Van’kov, “Adaptive robust control of spacecraft angular motion using predictive models,” Kosm. Issled. 32 (4–5), 13–21 (1994).Google Scholar
5. 5.
M. A. Velishchanskii, A. P. Krishchenko, and S. B. Tkachev, “Synthesis of spacecraft reorientation algorithms using the concept of the inverse dynamic problem,” J. Comput. Syst. Sci. Int. 42, 811 (2003).
6. 6.
A. V. Molodenkov and Ya. G. Sapunkov, “Special control regime in optimal turn problem of spherically symmetric spacecraft,” J. Comput. Syst. Sci. Int. 48, 891 (2009).
7. 7.
M. V. Levskii, “About method for solving the optimal control problems of spacecraft spatial orientation,” Probl. Nonlin. Anal. Eng. Syst. 21, 61–75 (2015).Google Scholar
8. 8.
B. V. Raushenbakh and E. N. Tokar’, Orientation Control of Spacecraft (Nauka, Moscow, 1974) [in Russian].Google Scholar
9. 9.
S. Liu and T. Singh, “Fuel/time optimal control of spacecraft maneuvers,” Guidance 20, 394–397 (1996).
10. 10.
S. Scrivener and R. Thompson, “Survey of time-optimal attitude maneuvers,” AIAA Guidance. Control Dyn. 17, 225–233 (1994).
11. 11.
H. Shen and P. Tsiotras, “Time-optimal control of axi-symmetric rigid spacecraft with two controls,” AIAA Guidance, Control Dyn. 22, 682–694 (1999).
12. 12.
A. V. Molodenkov and Ya. G. Sapunkov, “A solution of the optimal turn problem of an axially symmetric spacecraft with bounded and pulse control under arbitrary boundary conditions,” J. Comput. Syst. Sci. Int. 46, 310 (2007).
13. 13.
V. N. Branets, M. B. Chertok, and Yu. V. Kaznacheev, “Optimal rotation of a rigid body with one symmetry axis,” Kosm. Issled. 22, 352–360 (1984).Google Scholar
14. 14.
M. V. Levskii, “The use of universal variables in problems of optimal control concerning spacecrafts orientation,” Mekhatron., Avtomatiz., Upravl., No. 1, 53–59 (2014).Google Scholar
15. 15.
M. V. Levskii, “Optimal spacecraft terminal attitude control synthesis by the quaternion method,” Mech. Solids 44, 169 (2009).
16. 16.
M. V. Levskii, “Spacecraft spatial reversal control system,” RF Patent No. 2006431 (1994).Google Scholar
17. 17.
M. V. Levskii, “Quadratic optimal control in reorienting a spacecraft in a fixed time period in a dynamic problem statement,” J. Comput. Syst. Sci. Int. 57, 131 (2018).
18. 18.
M. V. Levskii, “Pontryagin’s maximum principle in optimal control problems of orientation of a spacecraft,” J. Comput. Syst. Sci. Int. 47, 974 (2008).
19. 19.
M. V. Levskii, “A special case of spacecraft optimal attitude control,” J. Comput. Syst. Sci. Int. 51, 587 (2012).
20. 20.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, et al., The Mathematical Theory of Optimal Processes (Nauka, Moscow, 1983; Wiley, New York, London, 1962).Google Scholar
21. 21.
M. V. Levskii, “Device for forming parameters of regular precession of a solid body,” RF Patent No. 2146638 (2000).Google Scholar
22. 22.
M. V. Levskii, “On optimal spacecraft damping,” J. Comput. Syst. Sci. Int. 50, 144 (2011).