Synthesis of control of the angular motion of a spacecraft on the basis of generalization of the direct Lyapunov method

1Volosov, VV, 1Shevchenko, VM
1Space Research Institute of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine, Kyiv, Ukraine
Space Sci.&Technol. 2018, 24 ;(4):03-13
https://doi.org/10.15407/knit2018.04.003
Publication Language: Russian
Abstract: 
The work deals with solving the problems of attitude control synthesis for an orbital spacecraft based on the known Zubov-Krasovskii generalizations of the direct Lyapunov method for studying the stability of individual solutions of differential equations to analyze the stability of closed bounded sets in their phase space. The kinematic equations of the angular motion of a spacecraft are represented by differential equations in the vector-matrix form with a state vector in the real Euclidean space. The elements of this space are the vectors of Rodrigues-Hamilton parameters. The given equations are characterized by the following generally known peculiarities. One of them lies in the fact that the same specified physical orientation of a spacecraft corresponds to two state vectors whose components differ only in sign. The second peculiarity of the differential equations is the presence of the geometrical motion integral – conservation of the norm of the vector of orientation parameters. It is noted that, as a consequence, solutions of the equations cannot be asymptotically stable by Lyapunov. They can be characterized only by conditional stability.
           With regard for these peculiarities, the corresponding problem of control synthesis for spacecraft orientation is for the first time formulated as a problem of the conditional asymptotic stability of the solutions of the differential equations or stability on manifolds. More precisely, it is a problem of stability of two-point sets in the phase space. A new non-smooth piecewise-quadratic Lyapunov function is proposed to solve the problem. The obtained solutions of the problems of control synthesis provide the achievement and stabilization of the given orientations of a spacecraft in the inertial and orbital coordinate systems. The effectiveness of the proposed algorithms is illustrated by computer simulation.
Keywords: control synthesis, Lyapunov function, orientation, spacecraft
References: 
1. Abalakin V. K., Aksenov Ye. P., Grebennikov Ye. A., et al. Reference Guide on celestial mechanics and astrodynamics. Ed. 2. M.: Nauka, 1971, 584 p. [in Russian].
2. Beletsky V. V. Satellite motion relative to the center of mass in a gravitational field. M.: Lomonosov MGU Press, 1975, 308 p. [in Russian].
3. Branets V. N., Shmyglevsky I. P. IThe use of quaternions in problems of solid-state orientation. M.: Nauka, 1973, 320 p. [in Russian].
4. Butenin N. V., Lunts Yu. Ya., Merkin D. R. Course of theoretical mechanics. M.: Nauka, 1979. Vol. 2, 544 p. [in Russian].
5. Bucholtz N. N. Basic course of theoretical mechanics. M.: Nauka,1969. P. II, 332 p. [in Russian].
6. Vi B., Ueys Kh., Erepostasis E. Control of the rotation of the spacecraft around its own axis with feedback on the components of the quaternion. Aerokosmicheskaya tekhnika. 1990, N 3, P. 3—11 [in Russian].
7. Wittenburg Y. Dynamics of systems of rigid bodies. M.: Mir, 1990, 292 p. [in Russian].
8. Voyevodin V. V., Kuznetsov Yu. A. Matrices and calculations. M.: Nauka, 1984, 320 p. [in Russian].
9. Volosov V. V. Attitude control of a spacecraft in the orbital coordinate system using ellipsoidal estimates of its state vector. J. Automation and Information Sciences, 1999, 31(4-5), P. 24—32.
10. Volosov V. V., Kutsenko I. A., Popadinets V. I. Mathematical models of rotational motion of spacecrafts with superfluous systems of gyrodins and flywheels and problems of control of their attitude. Ph. I, II. J. Automation and Information Sciences, 2003, Vol. 35, N 2, P.101— 116, Vol. 35, N 6, P. 109—116.
11. Volosov V. V., Kutsenko I. A., Selivanov Yu. A. Development and investigation of the robust algorithms of ellipsoidal estimation of the inertia characteristics of a spacecraft controlled by powered gyroscopes. J. Automation and Information Sciences, 2005, Vol. 37, N 4, P. 44—57.
12. Volosov V. V., Tyutyunnik L. I. Synthesis of spacecraft attitude control algorithms using quaternions. Kosm. nauka tehnol., 1999, 5(4), P. 61—69 [in Russian].
https://doi.org/10.15407/knit1999.04.061
13. Volosov V. V., Khlebnikov M. V., Shevchenko V. N. Algorithm of precision control of spacecraft orientation under action of uncontrollable disturbance. J. Automation and Information Sciences, 2011, Vol. 43, N 3. P. 59—66.
14. Gantmacher F. R. Matrix theory. M.: Nauka, 1967. 575 p. [in Russian].
15. Golubev Yu. F. Quaternion algebra in rigid body kinematics. M. V. Keldysh. Institute of Applied Mathematics Press, 2013, N 39. 23 p. [in Russian].
16. Gordeyev V. N. Quaternions and three-dimensional geometry. Kiev, 2012, 60 p. [in Russian].
17. Demidovich B. P. Lectures on the mathematical stability theory. M.: Nauka, 1967, 472 p. [in Russian].
18. Efimov N. V., Rozendorn Ye. A. Linear algebra and multidimensional geometry. M.: Nauka, 1970,  528 p. [in Russian].
19. Zubov V. I. Motion stability. Vysshaya shkola, 1973, 272 p. [in Russian].
20. Kirichenko N. F., Matviyenko V. T. Algorithms of asymptotic, terminal and adaptive stabilization of rotary motions of a rigid body. J. Automation and Information Sciences, 2003, 35(1), P. 1—9.
21. Krasovsky N. N. Generalization of the theorems of the Lyapunov second method. In Malkin I. G. Theory of motion stability. M.: Nauka, 1966. P. 463—467 [in Russian]
 22. Kryzhevich S. G. The generalization of the Lyapunov theorem on conditional stability foe non-analytic case. Differential equations and control processes, 1998, N 3. P. 43—55. 
 http://www.neva.ru/journal [in Russian].
23. Kuzovkov N. T., Salychev O. S. Inertial navigation and optimal filtrations. M.: Mashinostroyeniye, 1982, 216 p. [in Russian].
24. La Salle J., Lefschetz S., Stability by Liapunov›s direct method. M.: Mir, 1964, 168 p. [in Russian].
25. Lebedev D. V., Tkachenko A. I., Shtepa Yu. P. Magnetic system for controlling the angular motion of a micro-satellite. Kosm. nauka tehnol., 1996, 2(5-6). P. 17—25 [in Russian].
https://doi.org/10.15407/knit1996.05.017
26. Lebedev D. V. Attitude control of a rigid body utilizing Rodrigues-Hamilton parameters. Avtomatika, 1974, N 4. P. 29—32 [in Russian].
27. Onishchenko S. M. Optimal stabilization of the Earth artificial satellite with redundant system of fly-wheels. J. Automation and Information Sciences, 2016, 48(12). P. 1—12.
 28. Raushenbakh B. V., Tokar Ye. N. Spacecraft attitude control. M.: Nauka, 1974, 600 p. [in Russian].
29. Ryabova A. V., Tertychny-Dauri V. Yu. Elements of stability theory. ITMO S.-Pb, 2015, 208 p. [in Russian].
30. Sarychev V. A., Belyaev M. Yu., Zaykov S. G., Sazonov V. V., Teslenko B. P. Mathematical modeling of Eulerian turns of the Mir orbital complex by using gyrodines. Space Research, 1991, 29(4). P. 458—468.
31. Filimonov A. B., Filimonov N. B. Non-smooth analyses and synthesis of regulation systems based on the direct Lyapunov method. Priborostroyeniye, 1994, 37(7—8). P. 5—15 [in Russian].
32. Filippov A. F. Differential equations with discontinuous right-hand side. M.: Nauka, 1985, 224 p. [in Russian].
33. Filippov Yu. I. Effective algorithm of transformation of a quaternion to the system of Euler-Krylov angles. Polet, 2009, N 6. P. 32—35 [in Russian].
34. Chelnokov Yu. N. Quaternion models and methods in dynamics, navigation, and motion control. M.: Fizmatlit, 2011, 560 p. [in Russian].