×

Hamiltonian structures of dynamics of a gyrostat in a gravitational field. (English) Zbl 1267.37059

Summary: The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, \(\mathrm{SO}(2)\) and \(\mathrm{SO}(3)\) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.

MSC:

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
37N05 Dynamical systems in classical and celestial mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barkin, Y.V.: ’Oblique’ regular motions of a satellite and some small effects in the motions of the Moon and Phobos. Cosm. Res. 23, 20–30 (1985)
[2] Beck, J.A., Hall, C.D.: Relative equilibria of a rigid satellite in a circular Keplerian orbit. J. Astronaut. Sci. 40(3), 215–247 (1998)
[3] Bloch, A.M.: Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics, vol. 24. Springer, New York (2003) · Zbl 1045.70001
[4] Bloch, A.M., Marsden, J.E.: Stabilization of rigid body dynamics by the Energy-Casimir method. Syst. Control Lett. 14, 341–346 (1990) · Zbl 0701.93083 · doi:10.1016/0167-6911(90)90055-Y
[5] Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Alvarez, G.S.: Stabilization of rigid body dynamics by internal and external torques. Automatica 28(4), 745–756 (1992) · Zbl 0781.70020 · doi:10.1016/0005-1098(92)90034-D
[6] Bloch, A.M., Hussein, I.I., Leok, M., Sanyal, A.K.: Geometric structure-preserving optimal control of a rigid body. J. Dyn. Control Syst. 15(3), 307–330 (2009) · Zbl 1203.70042 · doi:10.1007/s10883-009-9071-2
[7] Cheng, Y., Huang, K., Lu, Q.: Hamiltonian structure for rigid body with flexible attachments in a circular orbit. Acta Mech. Sin. 9(1), 72–79 (1993) · Zbl 0776.70008 · doi:10.1007/BF02489164
[8] Cheng, Y., Huang, K., Lu, Q.: Stability of a class of coupled rigid-elastic systems with symmetry breaking. Sci. China Ser. A 37(9), 1062–1069 (1994) · Zbl 0807.70018
[9] Cheng, Y., Lu, Q.: Planar motion and stability for a rigid body with a beam in a field of central gravitational force. Sci. China Ser. A 45(11), 1479–1486 (2002) · Zbl 1145.70310 · doi:10.1007/BF02880043
[10] Crouch, P.E.: Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models. IEEE Trans. Autom. Control 29(4), 321–331 (1984) · Zbl 0536.93029 · doi:10.1109/TAC.1984.1103519
[11] Duboshin, G.N.: Differential equations of the translational-rotational motion of mutually attracting bodies. Astron. Zh. 35, 265–276 (1958)
[12] Fahnestock, E.G., Lee, T., Leok, M., McClamroch, N.H., Scheeres, D.J.: Polyhedral potential and variational integrator computation of the full two body problem (2006). http://arxiv.org/abs/math/0608695v1
[13] Guirao, J.L.G., Vera, J.A.: Dynamics of a gyrostat on cylindrical and inclined Eulerian equilibria in the three-body problem. Acta Astronaut. 66, 595–604 (2010) · doi:10.1016/j.actaastro.2009.07.024
[14] Guirao, J.L.G., Vera, J.A.: Lagrangian relative equilibria for a gyrostat in the three-body problem: bifurcations and stability. J. Phys. A, Math. Theor. 43, 195203 (2010) · Zbl 1250.70010
[15] Hall, C.D.: Attitude dynamics of orbiting gyrostats. In: Prêtka-Ziomek, H., Wnuk, E., Seidelmann, P.K., Richardson, D. (eds.) Dynamics of Natural and Artificial Celestial Bodies, pp. 177–186. Kluwer Academic, Dordrecht (2001)
[16] Hall, C.D., Beck, J.A.: Hamiltonian mechanics and relative equilibria of orbiting gyrostats. J. Astronaut. Sci. 55(1), 53–65 (2007) · doi:10.1007/BF03256514
[17] Koon, W.-S., Marsden, J.E., Ross, S.D., Lo, M., Scheeres, D.J.: Geometric mechanics and the dynamics of asteroid pairs. Ann. N.Y. Acad. Sci. 1017, 11–38 (2004) · doi:10.1196/annals.1311.002
[18] Krishnaprasad, P.S.: Lie-Poisson structures and dual-spin spacecraft. In: Proceedings of the 22nd IEEE Conference on Decision and Control, San Antonio, Texas, pp. 814–824 (1983)
[19] Krishnaprasad, P.S., Marsden, J.E.: Hamiltonian structure and stability for rigid bodies with flexible attachments. Arch. Ration. Mech. Anal. 98, 137–158 (1987) · Zbl 0624.58010
[20] Krishnaprasad, P.S., Marsden, J.E., Posbergh, T.A.: Stability analysis of a rigid body with a flexible attachment using the Energy-Casimir method. Contemp. Math. 68, 253–273 (1987) · Zbl 0645.70011 · doi:10.1090/conm/068/924816
[21] Krupa, M., Schagerl, M., Steindl, A., Szmolyan, P., Troger, H.: Relative equilibria of tethered satellite systems and their stability for very stiff tethers. Dyn. Syst. 16, 253–278 (2001) · Zbl 0998.70020
[22] Lee, T.: Computational geometric mechanics, control, and estimation of rigid bodies (2008). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.175.6057&rep=rep1&type=pdf
[23] Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem. Comput. Methods Appl. Mech. Eng. 196, 2907–2924 (2007) · Zbl 1120.70004 · doi:10.1016/j.cma.2007.01.017
[24] Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem in orbital mechanics. Celest. Mech. Dyn. Astron. 98, 121–144 (2007) · Zbl 1330.70003 · doi:10.1007/s10569-007-9073-x
[25] Lee, T., Leok, M., McClamroch, N.H.: Optimal attitude control of a rigid body using geometrically exact computations on SO(3). J. Dyn. Control Syst. 14(4), 465–487 (2008) · Zbl 1203.70044 · doi:10.1007/s10883-008-9047-7
[26] Lee, T., McClamroch, N.H., Leok, M.: A lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum. In: Proceeding of the 2005 IEEE Conference on Control Applications, Toronto, Canada, 28–31 August 2005, pp. 962–967 (2005)
[27] Maciejewski, A.J.: Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astron. 63, 1–28 (1995) · Zbl 0883.70007 · doi:10.1007/BF00691912
[28] Maciejewski, A.J.: A simple model of the rotational motion of a rigid satellite around an oblate planet. Acta Astron. 47, 387–398 (1997)
[29] Marsden, J.E.: Lectures on Mechanics. London Math. Society, Lecture Note Series, vol. 174. Cambridge University Press, Cambridge (1992) · Zbl 0744.70004
[30] Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. TAM Series, vol. 17, pp. 349–352. Springer, New York (1999) · Zbl 0933.70003
[31] Molina, R., Mondéjar, F.: Equilibria and stability for a gyrostat satellite in circular orbit. Acta Astronaut. 54, 77–82 (2003) · doi:10.1016/S0094-5765(02)00282-5
[32] Mondéjar, F., Vigueras, A.: The Hamiltonian dynamics of the two gyrostats problem. Celest. Mech. Dyn. Astron. 73, 303–312 (1999) · Zbl 0965.70013 · doi:10.1023/A:1008375820146
[33] Mondéjar, F., Vigueras, A., Ferrer, S.: Symmetries, reduction and relative equilibria for a gyrostat in the three-body problem. Celest. Mech. Dyn. Astron. 81, 45–50 (2001) · Zbl 0996.70008 · doi:10.1023/A:1013303002722
[34] Scheeres, D.J.: Spacecraft at small NEO (2006). arXiv:physics/0608158v1
[35] Vera, J.A.: Eulerian equilibria of a triaxial gyrostat in the three body problem: rotational Poisson dynamics in Eulerian equilibria. Nonlinear Dyn. 55, 191–201 (2009) · Zbl 1200.70007 · doi:10.1007/s11071-008-9355-1
[36] Vera, J.A.: Dynamics of a triaxial gyrostat at a Lagrangian equilibrium of a binary asteroid. Astrophys. Space Sci. 323, 375–382 (2009) · Zbl 1178.85006 · doi:10.1007/s10509-009-0085-8
[37] Vera, J.A.: Eulerian equilibria of a triaxial gyrostat in the three-body problem: rotational Poisson dynamics in Eulerian equilibria with oblateness. Acta Astronaut. 65, 755–765 (2009) · Zbl 1200.70007 · doi:10.1016/j.actaastro.2009.03.014
[38] Vera, J.A.: On the dynamics of a gyrostat on Lagrangian equilibria in the three body problem. Multibody Syst. Dyn. 23(3), 263–291 (2010) · Zbl 1376.70027 · doi:10.1007/s11044-009-9183-1
[39] Vera, J.A., Vigueras, A.: Reduction, relative equilibria and stability for a gyrostat in the n-body problem. Monogr. Semin. Mat. García Galdeano 31, 257–271 (2004)
[40] Vera, J.A., Vigueras, A.: Hamiltonian dynamics of a gyrostat in the n-body problem: relative equilibria. Celest. Mech. Dyn. Astron. 94, 289–315 (2006) · Zbl 1175.70015 · doi:10.1007/s10569-005-5910-y
[41] Volterra, V.: Sur la theorie des variations des latitudes. Acta Math. 22, 201–358 (1899) · JFM 29.0650.01 · doi:10.1007/BF02417877
[42] Wang, L.-S., Cheng, S.-F.: Dynamics of two spring-connected masses in orbit. Celest. Mech. Dyn. Astron. 63, 289–312 (1996) · Zbl 0887.70017 · doi:10.1007/BF00692292
[43] Wang, L.-S., Chern, S.-J., Shin, C.-W.: On the dynamics of a tethered satellite system. Arch. Ration. Mech. Anal. 127, 297–318 (1994) · Zbl 0817.70017 · doi:10.1007/BF00375018
[44] Wang, L.-S., Krishnaprasad, P.S.: Relative equilibria for two rigid bodies connected by a ball-in-socket joint. In: Proceeding of the 28th IEEE Conference on Decision and Control, Tampa, Florida, December 1989, pp. 692–697 (1989)
[45] Wang, L.-S., Krishnaprasad, P.S.: Gyroscopic control and stabilization. J. Nonlinear Sci. 2, 367–415 (1992) · Zbl 0800.93563 · doi:10.1007/BF01209527
[46] Wang, L.-S., Krishnaprasad, P.S., Maddocks, J.H.: Hamiltonian dynamics of a rigid body in a central gravitational field. Celest. Mech. Dyn. Astron. 50, 349–386 (1991) · Zbl 0737.70003 · doi:10.1007/BF02426678
[47] Wang, L.-S., Lian, K.-Y., Chen, P.-T.: Steady motions of gyrostat satellites and their stability. IEEE Trans. Autom. Control 40(10), 1732–1743 (1995) · Zbl 0841.93057 · doi:10.1109/9.467678
[48] Wang, L.-S., Maddocks, J.H., Krishnaprasad, P.S.: Steady rigid-body motions in a central gravitational field. J. Astronaut. Sci. 40, 449–478 (1992)
[49] Wang, Y., Xu, S.: Gravitational orbit-rotation coupling of a rigid satellite around a spheroid planet. J. Aerosp. Eng. (2012, accepted). doi: 10.1061/(ASCE)AS.1943-5525.0000222
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.