Abstract
This work is devoted to a survey and generalization of results obtained in the theory of optimal motion control for a material point in the central Newtonian gravitational field using the Pontryagin’s maximum principle and quaternion models of orbital motion. This theory is very important in space flight mechanics, being the background of the solution of optimal control problems of the motion of the center of mass of a space vehicle. In the first part of this work, a survey of quaternion models of the motion of a material point in a central Newtonian gravitational field is given, their advantages and disadvantages are analyzed. The formulation of the optimal control problem of the motion of a material point in the central Newtonian gravitational field and its correlation with the optimal control problem of the motion of the center of mass of a space vehicle is considered. The main problems arising in the solution of optimal control problems of the motion of a material point using the maximum principle, including instability in the Lyapunov’s sense of solutions to adjoint equations, are studied. It is shown that efficiency of analytical investigation and numerical solution of the corresponding boundary-value problems can be increased by the application of quaternion models of orbital motion.
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References
A. I. Lur’e, Analytical Mechanics (Fizmatgiz, Moscow, 1961) [in Russian].
V. N. Branets and I. P. Shmyglevskii, Application of Quaternions in the Problems of Orientation of a Rigid Body (Nauka, Moscow, 1973) [in Russian].
Yu. N. Chelnokov, Quaternion and Bi-quaternion Models in Mechanics of Solid Body and Their Applications: Geometry and Motion Kinematics (Fizmatlit, Moscow, 2005) [in Russian].
A. Deprit, “Ideal Frames for Perturbed Keplerian Motions,” Celestial Mechanics 13(2), 253–263 (1976).
V. A. Brumberg, Analytical Algorithms of Celestial Mechanics (Nauka, Moscow, 1980) [in Russian].
A. F. Bragazin, V. N. Branets, and I. P. Shmyglevskii, “A Description of Optimal Motion with the Use of Quaternions and Velocity Parameters,” in Abstracts of Papers of the 6th All-Union Conference of Theoretical and Applied Mechanics (Fan, Tashkent, 1986), p. 133 [in Russian].
V. N. Branets and I. P. Shmyglevskii, Introduction to Strapdown Interital Navigation Systems (Nauka, Moscow, 1992) [in Russian].
Yu. N. Chelnokov, “On Regularization of the Equations of Spatial Problem of Two Bodies,” Izv. Akad. Nauk SSSR, Mekhanika Tverdogo Tela, No. 6, 12–21 (1981).
Yu. N. Chelnokov, “On Regularization of the Equations of Spatial Problem of Two Bodies,” Izv. Akad. Nauk SSSR, Mekhanika Tverdogo Tela, No. 1, 151–158 (1984).
Yu. N. Chelnokov, “A Quaternion Regularization and Stabilization of Perturbed Central Motion, I and II,” Izv. Ros. Akad. Nauk. Mekhanika Tv. Tela, Nos. 1 and 2, 20–30 and 3–11 (1993).
Yu. N. Chelnokov, “Application of Quaternions in the Theory of Orbital Motion of an Artificial Satellite, I, II,” Kosm. Issl. 30(9), 759–770 (1993 and 31 (3), 3–15 (1993).
Yu. N. Chelnokov, “Construction of Optimal Controls and Motion Trajectories of a Space Vehicle that Uses a Quaternion Description of the Spatial Orbit Orientation,” Kosm. Issl. 35(5), 534–542 (1997).
E. L. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics (Springer, Berlin, 1971; Nauka, Moscow, 1975).
Yu. N. Chelnokov and Ya. G. Sapunkov, “Construction of Optimal Controls and Trajectories of a Spacecraft Based on Regular Quaternion Equations of the Problem of Two Bodies,” Kosm. Issl. 34(2), 150–158 (1996).
Yu. N. Chelnokov and V. A. Yurko, “Quaternion Construction of Optimal Controls and Motion Trajectories of a Spacecraft in the Newton Gravitational Field,” Izv. Ross. Akad. Nauk: Mekh. Tverd. Tela, No. 6, 3–13 (1996).
Ya. G. Sapunkov, “Application of KS-variables to the Problem of Optimal Control of a Spacecraft,” Kosm. Issl. 34(4), 428–433 (1996).
Ya. G. Sapunkov, “Quaternion Elements of the Orbit in the Problem of Optimal Control of a Rendezvous of Two Spacecraft,” in Mathematics, Mechanics: Collection of Papers (Sarat. Gos. Univ., Saratov, 2002), No. 4, pp. 210–213 [in Russian].
Ya. G. Sapunkov, “Optimal Trajectories and Controls in the Problem of a Rendezvous of Two Spacecraft,” in Mathematics, Mechanics: Collection of Papers (Sarat. Gos. Univ., Saratov, 2003), No. 5, pp. 171–174 [in Russian].
T. V. Bordovitsyna, Modern Numerical Methods in Problems of Celestial Mechanics (Nauka, Moscow, 1984) [in Russian].
Yu. N. Chelnokov, “Optimal Motion Control of a Spacecraft in the Newton Gravitational Field: Application of Quaternions for a Description of the Orbit Orientation,” Kosm. Issl. 37(4), 433–440 (1999).
Yu. N. Chelnokov, “Application of Quaternions in Problems of Optimal Control of the Motion of the Center of Mass of a Spacecraft in the Newton Gravitational Field. I–III,” Kosm. Issl. 39(5), 502–517 (2003), 41 (1), 92–107 (2003), and 41 (5), 502–517 (2003).
V. K. Abalakin, E. P. Aksenov, E. A. Grebenikov, et al., Handbook and Celestial Mechanics and Astrodynamics (Nauka, Moscow, 1976) [in Russian].
V. A. Il’in and G. E. Kuzmak, Optimal Flightd of Spacecraft (Nauka, Moscow, 1976) [in Russian].
G. N. Duboshin, Celestial Mechanics. The Main Problems and Methods (Nauka, Moscow, 1968) [in Russian].
L. Euler, “De Motu Rectilineo Trium Corporum Se Mutuo Attrahentium,” Nov. Comm. Petrop, No. 11, 144 (1765).
T. Levi-Civita, “Sur La Resolution Qualitative Du Probleme Restreint Des Trois Corps,” Opere mathematiche, No. 2, 411–417 (1956).
P. Kustaanheimo, “Spinor Regularization of the Kepler Motion,” Ann. Univ. Turku. Ser. Al, No. 73 (1964).
P. Kustaanheimo and E. Stiefel, “Perturbation Theory of Kepler Motion Based on Spinor Regularization,” J. Reine Angew. Math, No. 218, 204–219 (1965).
R. Bellman, Introduction to Matrix Analysis (Nauka, Moscow, 1969; McGraw-Hill, New York, 1960).
P. K. Plotnikov and Yu. N. Chelnokov, “Application of Quaternion Matrices in the Theory of a Finite Turn of a Rigid Body,” in Collection of Papers on Theoretical Mechanics (Vysshaya Shkola, Moscow, 1981), No. 11, pp. 122–129 [in Russian].
Yu. N. Chelnokov, “Quaternion Algorithms of Systems of Spatial Inertial Navigation,” Izv. Akad. Nauk SSSR, Mekhanika Tverdogo Tela, No. 6, 14–21 (1983).
G. L. Grodzovskii, D. E. Okhotsimskii, V. V. Beletskii, et al., “Mechanics of Space Flight,” in Mechanics in USSR for 50 years, Vol. 1. General and Applied Mechanics: Collection of Papers (Nauka, Moscow, 1968), pp. 265–319 [in Russian].
G. L. Grodzovskii, Yu. N. Ivanov, and V. V. Tokarev, Mechanics of Space Flight. Optimization Problems (Nauka, Moscow, 1975) [in Russian].
G. E. Kuzmak and A. Z. Braude, “Approximative Construction of Optimal Flights in a Small Neighborhood of a Circular Orbit,” Kosm. Issl. 7(3), 323–338 (1969).
V. V. Beletskii and V. A. Egorov, “Interplanetary Flights with Engines of Constant Power,” Kosm. Issl. 2(3) (1964).
S. Pines, “Motion Constants for Optimal Active Trajectories in a Central Force Field,” Raketn. Tekhn. Kosmon. 2(11) (1964).
N. N. Krasovskii, “The Theory of Optimal Control Systems,” in Mechanics in USSR for 50 years, Vol. 1. General and Applied Mechanics: Collection of Papers (Nauka, Moscow, 1968), pp. 179–244 [in Russian].
A. I. Lur’e, “A Free Fall of a Material Point in the Satellite Cockpit,” Prikl. Math. Mekh. 27(1) (1963).
V. D. Andreev, The Theory of Inertial Navigation. Autonomous Systems (Nauka, Moscow, 1966) [in Russian].
E. A. Devyanin and N. A. Parusnikov, “On the Motion Stability of a Point in the Field of Forces of an Attracting Center,” Izv. Akad. Nauk SSSR, Mekhanika Tverdogo Tela, No. 4 (1969).
D. F. Lawden, “Fundamentals of Space Navigation,” J. Brit. Interplanet. Soc. 13(2) (1954).
A. M. Lyapunov, The General Problem of Motion Stability (Gostekhteorizd, Moscow, 1950) [in Russian].
V. V. Rumyantsev, On Stability of Stationary Motions of Satellites (Vych. Ts. Akad. Nauk SSSR, Moscow, 1967) [in Russian].
Yu. N. Chelnokov and S. V. Nenakhov, “A Quaternion Solution of the Problem of Optimal Control of the Orientation of the Orbit of a Spacecraft,” in Proceedings of International Conference on Onboard Integrated Complexes and Modern Control Problems, Moscow, 1998 (Mosk. Aviats. Inst., Ioscow, 1998), pp. 59–60 [in Russian].
D. A. Sergeev and Yu. N. Chelnokov, “Optimal Control of Orbit Orientation of a Spacecraft,” in Collection of Papers on Mathematics and Mechanics (Sarat. Gos. Univ., Saratov, 2001), No. 3, pp. 185–188 [in Russian].
D. A. Sergeev and Yu. N. Chelnokov, “Optimal Control of the Orientation of an Orbit of a Spacecraft,” in Proceedings of IPTMU RAN “Problems of Precision Mechanics and Control” (Sarat. Gos. Tekh. Univ., Saratov, 2002), pp. 64–75 [in Russian].
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Original Russian Text © Yu.N. Chelnokov, 2007, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2007, No. 5, pp. 18–44.
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Chelnokov, Y.N. Analysis of optimal motion control for a material point in a central field with application of quaternions. J. Comput. Syst. Sci. Int. 46, 688–713 (2007). https://doi.org/10.1134/S1064230707050036
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DOI: https://doi.org/10.1134/S1064230707050036