Dynamics of multibody systems in planar motion in a central gravitational field

Dynamical Systems

Volumn 19, Issue 4, 2004, Pages 303-343

  Sanyal, Amit K ^a   Bloch, Anthony ^a   Mcclamroch, N Harris ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; EQUATIONS OF MOTION; GRAVITATIONAL EFFECTS; HAMILTONIANS; INTEGRATION; PERTURBATION TECHNIQUES; SYSTEM STABILITY;

DYNAMIC COUPLING; MULTIBODY SYSTEMS; PLANAR MOTION; ROUTH REDUCTION;

MOTION CONTROL;

EID: 11144297251     PISSN: 14689367     EISSN: None     Source Type: Journal    
DOI: 10.1080/14689360412331309160     Document Type: Article

References (28)

1. Arnold, V. I. (Ed.), Kozlov, V. V., and Neishtadt, A. I., 1988, Dynamical Systems III, Vol. 3 of Series in Encyclopaedia of Mathematical Sciences (Berlin: Springer-Verlag).
2. Birkhoff, G. D., 1927, Dynamical Systems, 2nd edn, AMS Colloq. Publications, No. 9 (Providence, RI: American Mathematical Society).
3. Bloch, A. M., 2003, Basic concepts in Geometric Mechanics. Nonholonomic Mechanics and Control, Vol. 24 of Series in Interdisciplinary Applied Mathematics (New York: Springer-Verlag), pp. 119-174.
4. Fassò, F., Guzzo, M., and Benettin, G., 1998, Nekhoroshev- stability of elliptic equilibria of Hamiltonian systems. Communications in Mathematical Physics, 197(2): 347-360.
5. Giorgilli, A., 1988, Rigorous results on the power expansions for the integrals of a Hamiltonian system near an equilibrium point. Annales Inst. Henri Poincaré, 48(4): 423-439.
6. Giorgilli, A., Delshams, A., Fontich, E., Galgani, L., and Simó, C., 1989, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem. Journal of Differential Equations, 77: 167-198.
7. Hairer, E., Lubich, C., and Wanner, G., 2002, Geometric Numerical Integration, Vol. 31 of Series in Computational Mathematics (Berlin: Springer-Verlag).
8. Ito, H., 1989, Convergence of Birkhoff normal forms for integrable systems. Commentarii Mathematics Helvetica, 64: 412-461.
9. Ito, H., 1992, Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Mathematische Annalen, 292(3): 411-444.
10. Jalnapurkar, S. M., Leok, M., Marsden, J. E., and West, M., 2003, Discrete Routh reduction. Preprint.
11. Jalnapurkar, S. M. and Marsden, J. E., 2000, Stabilization of relative equilibria. IEEE Transactions on Automatic Control, 45(8): 1483-1491.
12. Jorba, À. and Masdemont, J., 1999, Dynamics in the centre manifold of the collinear points of the Restricted Three Body problem. Physica D, 132: 189-213.
13. Kane, C., Marsden, J. E., and Ortiz, M., 1999, Symplectic-energy-momentum preserving variational integrators. Journal of Mathematical Physics, 40(7): 3353-3371.
14. Marsden, J. E. and Ratiu, T. S., 1999, Introduction to Mechanics and Symmetry, 2nd edn (New York: Springer-Verlag).
15. Marsden, J. E., Ratiu, T. S., and Scheurle, J., 2000, Reduction theory and the Lagrange-Routh equations. Journal of Mathematical Physics, 41(6): 3379-3429.
16. Marsden, J. E. and West, M., 2001, Discrete mechanics and variational integrators. Acta Numerica, 10: 357-514.
17. Oh, Y.-G., Sreenath, N., Krishnaprasad, P. S., and Marsden, J. E., 1988, The dynamics of coupled planar rigid bodies. Part I: reduction, equilibria, and stability. Dynamics and Stability of Systems, 3(1-2): 25-49.
18. Oh, Y.-G., Sreenath, N., Krishnaprasad, P. S., and Marsden, J. E., 1989, The dynamics of coupled planar rigid bodies. Part II: bifurcations, periodic solutions, and chaos. Journal of Dynamics and Differential Equations, 1(3): 269-298.
19. Patrick, G. W., 1995, Two axially symmetric coupled rigid bodies: relative equilibria, stability, bifurcations, and a momentum preserving symplectic integrator. PhD thesis, Department of Mathematics, University of California, Berkeley.
20. Sakai, T., 1996, Riemannian Geometry, Vol. 149 of Translations of Mathematical Monographs (Providence: American Mathematical Society).
21. Sanyal, A. K., 2004, Dynamics and control of multibody systems in central gravity. PhD thesis, Department of Aerospace Engineering, University of Michigan, Ann Arbor.
22. Sanyał, A. K., Shen, J., and McClamroch, N. H., 2003, Dynamics and control of an elastic dumb-bell spacecraft in a central gravitational field. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, pp. 2798-2803.
23. Sanyal, A. K., Shen, J., and McClamroch, N. H., 2004a, Control of a dumb-bell spacecraft using attitude and shape control inputs only. Proceedings of the 2004 American Control Conference, Boston, MA, pp. 1014-1018.
24. Sanyal, A. K., Shen, J., and McClamroch, N. H., 2004b, Variational integrators for mechanical systems with cyclic generalized coordinates. Computer Methods in Applied Mechanics and Engineering (submitted for publication).
25. Sanyal, A. K., Shen, J., McClamroch, N. H., and Bloch, A. M., 2004c, Stability and stabilization of relative equilibria of a dumb-bell body in central gravity. AIAA Journal of Guidance, Control and Dynamics (submitted for publication).
26. Simo, J. C., Tarrow, N., and Wong, K. K., 1992, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100: 63-116.
27. Verhulst, F., 1996, Symmetry and integrability in Hamiltonian normal forms. Symmetry and Perturbation Theory 1996, proceedings of the conference held at I.S.I., Villa Gualino Torino, Italy, edited by D. Bambusi and G. Gaeta (Rome: Consiglio Nazionale delle Ricerche (CNR)), Quadermo GNFM-CNR n. 54, pp. 245-284. Also available at http://web.tiscali.it/SPT2001/SPT96.html.
28. Zung, N. T., 2003, Convergence versus integrability in Birkhoff normal form, http://arxiv.org/pdf/math.DS/0104279.