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Mathematical Problems in Engineering
Volume 2014, Article ID 674910, 11 pages
http://dx.doi.org/10.1155/2014/674910
Research Article

Formulation and Simulations of the Conserving Algorithm for Feedback Stabilization on Rigid Body Rotations

1Department of Occupational Safety and Health, Chang Jung Christian University, Tainan City 71101, Taiwan
2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455-0170, USA

Received 19 February 2014; Accepted 29 March 2014; Published 27 April 2014

Academic Editor: Her-Terng Yau

Copyright © 2014 Yong-Ren Pu and Thomas A. Posbergh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. D. Holm, Geometric Mechanics—Part II: Rotating, Translating and Rolling, Imperial College Press, London, UK, 2008. View at MathSciNet
  2. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 1999. View at Publisher · View at Google Scholar
  3. D. Kim, J. D. Turner, and H. Leeghim, “Reorientation of asymmetric rigid body using two controls,” Mathematical Problems in Engineering, Article ID 708935, 8 pages, 2013. View at Google Scholar · View at MathSciNet
  4. J. M. Selig, Geometric Fundamentals of Robotics, Monographs in Computer Science, Springer, New York, NY, USA, 2nd edition, 2005. View at MathSciNet
  5. R. A. Labudde and D. Greenspan, “Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. II. Motion of a system of particles,” Numerische Mathematik, vol. 26, no. 1, pp. 1–16, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. A. Austin, P. S. Krishnaprasad, and L. S. Wang, “Almost poisson integration of rigid body systems,” Journal of Computational Physics, vol. 107, no. 1, pp. 105–117, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P. J. Channell, “Symplectic integration algorithms,” Los Alamos National Laboratory Internal Report AT-6:ATN-83-9, 1983. View at Google Scholar
  8. C. Linton, W. Holderbaum, and J. Biggs, “Rigid body trajectories in different 6D spaces,” ISRN Mathematical Physics, vol. 2012, Article ID 467520, 21 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A. M. Bloch and J. E. Marsden, “Stabilization of rigid body dynamics by the energy-Casimir method,” Systems & Control Letters, vol. 14, no. 4, pp. 341–346, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and G. S. de Alvarez, “Stabilization of rigid body dynamics by internal and external torques,” Automatica, vol. 28, no. 4, pp. 745–756, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Zhao and T. A. Posbergh, “Feedback stabilization of uniform rigid body rotation,” Systems & Control Letters, vol. 22, no. 1, pp. 39–45, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. C. Simo and K. K. Wong, “Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum,” International Journal for Numerical Methods in Engineering, vol. 31, no. 1, pp. 19–52, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. Lewis and J. C. Simo, “Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups,” Journal of Nonlinear Science, vol. 4, no. 3, pp. 253–299, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. C. Simo, N. Tarnow, and K. K. Wong, “Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics,” Computer Methods in Applied Mechanics and Engineering, vol. 100, no. 1, pp. 63–116, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. Abraham and J. E. Marsden, Foundations of Mechanics, AMS Chelsea, Rhode Island, RI, USA, 2nd edition, 2008.
  16. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, NY, USA, 1997. View at MathSciNet
  17. G. Zhong and J. E. Marsden, “Lie-poisson Hamilton-Jacobi theory and lie-poisson integrators,” Physics Letters A, vol. 133, no. 3, pp. 134–139, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  18. M. A. Austin, P. S. Krishnaprasad, and L. S. Wang, “Almost poisson integration of rigid body systems,” Journal of Computational Physics, vol. 107, no. 1, pp. 105–117, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. Zhao and T. A. Posbergh, “Robust stabilization of a uniformly rotating rigid body,” in Proceedings of the American Control Conference, pp. 2408–2412, June 1993. View at Scopus
  20. T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, Mineola, NY, USA, 2000. View at MathSciNet
  21. J. C. Simo and L. Vu-Quoc, “On the dynamics in space of rods undergoing large motions—a geometrically exact approach,” Computer Methods in Applied Mechanics and Engineering, vol. 66, no. 2, pp. 125–161, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet