Skip to main content
SpringerLink
Log in

Gyroscopic control and stabilization

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Summary

In this paper, we consider the geometry of gyroscopic systems with symmetry, starting from an intrinsic Lagrangian viewpoint. We note that natural mechanical systems with exogenous forces can be transformed into gyroscopic systems, when the forces are determined by a suitable class of feedback laws. To assess the stability of relative equilibria in the resultant feedback systems, we extend the energy-momentum block-diagonalization theorem of Simo, Lewis, Posbergh, and Marsden to gyroscopic systems with symmetry. We illustrate the main ideas by a key example of two coupled rigid bodies with internal rotors. The energy-momentum method yields computationally tractable stability criteria in this and other examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R. & J. E. Marsden,Foundations of Mechanics, 2nd ed., Reading: Benjamin/Cummings, 1978.

    Google Scholar 

  2. Alvarez, G. S.,Geometric Methods of Classical Mechanics Applied to Control Theory, University of California, Berkeley, Ph.D. Dissertation, 1986.

    Google Scholar 

  3. Antman, S. S.,Nonlinear Problems of Elasticity, in preparation, 1991.

  4. Arnold, V. I., “Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications á l'hydrodynamique des fluides parfaits,”Annales de l'Institut Fourier,16(1) (1966), pp. 319–361.

    Google Scholar 

  5. Arnold, V. I.,Mathematical Methods of Classical Mechanics, New York: Springer-Verlag, 1978.

    Google Scholar 

  6. Arnold, V. I., ed.,Dynamical Systems III, Encyclopedia of Mathematical Sciences,3, Berlin: Springer-Verlag, 1988.

    Google Scholar 

  7. Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden, & G. Sanchez de Alvarez, “Stabilization of Rigid Body Dynamics by Internal and External Torques,” 1990, preprint.

  8. Bloch, A. M. & J. E. Marsden, “Controlling Homoclinic Orbits,”Theoretical and Computational Fluid Mechanics (1989), pp. 179–190.

  9. Bloch, A. M. & J. E. Marsden, “Stabilization of Rigid Body Dynamics by the Energy-Casimir Method,”Systems and Control Letters,14, (1990), pp. 341–346.

    Google Scholar 

  10. Brockett, R. W., “Feedback Invariants for Nonlinear Systems,”6th IFAC Congress, Helsinki (1978), pp. 1115–1120.

  11. Chetayev, N. G.,The Stability of Motion, Pergamon Press, 1961.

  12. Crouch, P. E. & Van Der Schaft,Variational and Hamiltonian Control Systems, Springer-Verlag, 1987.

  13. Godbillon, C.,Géometrie différentielle et mécanique analytique, Hermann, Paris, 1969.

    Google Scholar 

  14. Gotay, M. J. & J. M. Nester, “Presymplectic Lagrangian Systems I: the constraint algorithm and the equivalence theorem,”Ann. Inst. Henri Poincaré-Section A,30(2) (1979), pp. 129–142.

    Google Scholar 

  15. Gotay, M. J. & J. M. Nester, “Presymplectic Lagrangian Systems II: the second-order equation problem,”Ann. Inst. Henri Poincaré-Section A,32(1) (1980), pp. 1–13.

    Google Scholar 

  16. Grossman, R., P. S. Krishnaprasad, & J. E. Marsden, “The Dynamics of Two Coupled Three Dimensional Rigid Bodies,” inDynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, F. Salam & M. Levi, eds., SIAM, Philadelphia, 1988, pp. 373–378.

    Google Scholar 

  17. Guillemin, V. & S. Sternberg,Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.

    Google Scholar 

  18. Han, J. H.,Symmetries in Nonlinear Control Systems and Their Applications, Ph.D. Dissertation, Electrical Engineering Department, University of Maryland, College Park, 1986.

    Google Scholar 

  19. Hermann, R.,Differential Geometry and the Calculus of Variation, 2nd ed., Math. Sci. Press, 1977.

  20. Holm, D., J. E. Marsden, T. Ratiu, & A. Weinstein, “Nonlinear Stability of Fluid and Plasma Equilibria,”Physics Report,123(1985), pp. 1–116.

    Google Scholar 

  21. Isidori, A.,Nonlinear Control Systems, 2nd ed., Springer-Verlag, 1989.

  22. Jakubczyk, B., “Hamiltonian Realizations of Nonlinear Systems,” inTheory and Applications of Nonlinear Control Systems, C. I. Byrnes & A. Lindquist, eds., North-Holland, 1986, pp. 261–271.

  23. Krishnaprasad, P. S., “Lie-Poisson Structures, Dual-spin Spacecraft and Asymptotic Stability,”Nonlinear Analysis: Theory, Methods, and Applications,7(1984), pp. 1011–1035.

    Google Scholar 

  24. Krishnaprasad, P. S., “Eulerian Many-Body Problems,” Systems Research Center, University of Maryland, College Park, SRC TR-89-15, 1989, also inContemp. Math., Vol. 97, pp. 187–208, AMS.

    Google Scholar 

  25. Krishnaprasad, P. S. & C. A. Berenstein, “On the Equilibria of Rigid Spacecraft with Rotors,”Systems & Control Letters,4(1984), pp. 157–163.

    Google Scholar 

  26. Lanczos, C.,The Variational Principles of Mechanics, 4th ed., University of Toronto Press, 1970.

  27. Lewis, D., “Lagrangian Block Diagonalization,”J. Dynamics and Differential Equations (1991), (to appear).

  28. Libermann, P. & C-M. Marie,Symplectic Geometry and Analytical Mechanics, Dordrecht: D. Reidel Publ., 1987.

    Google Scholar 

  29. Maddocks, J. H., “Stability and Folds,”Archive for Rational Mechanics and Analysis,99(4) (1987), pp. 301–328.

    Google Scholar 

  30. Marmo, G., E. J. Saletan, & A. Simoni, “Reduction of Symplectic Manifolds Through Constants of the Motion,”Nuovo Cimento,50B(1) (1979).

  31. Marsden, J. E. & T. Ratiu, “Reduction of Poisson Manifolds,”Letters in Math. Phys.,11 (1986), pp. 161–169.

    Google Scholar 

  32. Marsden, J. E. & A. Weinstein, “Reduction of Symplectic Manifolds with Symmetry,”Reports in Math. Phys.,5(1974), pp. 121–130.

    Google Scholar 

  33. Nijmeijer, H. & Van Der Schaft,Nonlinear Dynamical Control Systems, Springer-Verlag, 1990.

  34. Nomizu, K.,Lie Groups and Differential Geometry, The Mathematical Society of Japan, 1956.

  35. Palais, R. S., “The Principle of Symmetric Criticality,”Comm. in Math. Physics,69(1) (1979), pp. 19–30.

    Google Scholar 

  36. Posbergh, T, J. C. Simo & J. E. Marsden, “Stability Analysis of a Rigid Body with Attached Geometrically Nonlinear Appendage by the Energy-Momentum Method,” inDynamics and Control of Multibody Systems, J. E. Marsden, P. S. Krishnaprasad, & J. C. Simo, eds., AMS, 1988, pp. 371–397,Contemporary Mathematics, Vol. 97.

  37. Routh, E.,The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, 4th ed., Dover Publications, Inc., 1884.

  38. Siam,Report on the Panel on Future Directions in Control Theory: a Mathematical Perspective, Philadelphia, 1988.

  39. Simo, J. C., D. Lewis, & J. E. Marsden, “Stability of Relative Equilibria, Part I: The Reduced Energy-Momentum Method,” Division of Applied Mechanics, Stanford University, SUDAM Report No. 89-3, 1989, (to appear inArchive for Rational Mechanics and Analysis, 1990).

  40. Simo, J. C., T. Posbergh & J. E. Marsden, “Nonlinear Stability of Geometrically Exact Rods by the Energy-Momentum Method,” Stanford University, Division of Applied Mechanics, preprint, 1989.

  41. Simo, J. C., T. Posbergh & J. E. Marsden, “Stability of Relative Equilibria, Part II: Application to Nonlinear Elasticity,” Stanford University, Division of Applied Mechanics, preprint, 1989, (to appear inArchive for Rational Mechanics and Analysis, 1990).

  42. Smale, S., “Topology and Mechanics I, II,”Inventiones Mathematicae,10,11 (1970), pp. 305–331, 45–64.

    Google Scholar 

  43. Souriau, J. M.,Structure des systémes dynamique, Dunod, Paris, 1970.

    Google Scholar 

  44. Sternberg, S.,Lectures on Celestial Mechanics: Iand II, Addison-Wesley, 1969.

  45. Vershik, A. M. & L. D. Faddeev, “Lagrangian Mechanics in Invariant Form,”Sel. Math. Sov.,1, 4 (1981), pp. 339–350.

    Google Scholar 

  46. Wang, L.-S.,Geometry, Dynamics and Control of Coupled Systems, Ph.D. Dissertation, Electrical Engineering Department, University of Maryland, College Park, August, 1990.

    Google Scholar 

  47. Wang, L.-S. & P. S. Krishnaprasad, “Relative Equilibria of Two Rigid Bodies connected by a Ball-in-Socket Joint,”Proc. of the 1989 IEEE Conference on Decision and Control, Tampa, FL (Dec. 1989), pp. 692–697.

  48. Wang, L.-S. & P. S. Krishnaprasad, “A Multibody Analog of the Dual-Spin Problems,”Proc. of the 29th IEEE Conference on Decision and Control, Honolulu, Hawaii (Dec. 1990), pp. 1294–1299.

  49. Wang, L.-S., P. S. Krishnaprasad & J. H. Maddocks, “Hamiltonian Dynamics of a Rigid Body in a Central Gravitational Field,”Celestial Mechanics and Dynamical Astronomy,50(4) (1991), pp. 349–386.

    Google Scholar 

  50. Weinstein, A., “Stability of Poisson-Hamilton Equilibria,” inFluids and Plasmas: Geometry and Dynamics, J. E. Marsden, ed., 1984, in seriesContemporary Mathematics,28, pp. 3–13, AMS, Providence.

    Google Scholar 

  51. Whittaker, E. T.,A Treatise on the Analytical Dynamics of Panicles and Rigid Bodies, 4th ed., Cambridge University Press, Cambridge, 1959.

    Google Scholar 

  52. Willmore, T. J., “The Definition of Lie Derivative,”Proc. Edin. Math. Soc.,12(2) (1960), pp. 27–29.

    Google Scholar 

  53. Wong, S. K., “Field and Particle Equations for the Classical Yang-Mills Field and Particles with Isotopic Spin,”Nuovo Cimento,65A (1970), pp. 689–693.

    Google Scholar 

  54. Yang, R. & P. S. Krishnaprasad, “On the Dynamics of Four-Bar Linkages, Part II: Bifurcations of Relative Equilibria,”Proc. of the 1990 IEEE Conference on Decision and Control, Honolulu, Hawaii (Dec. 1990).

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan, R.O.C.

    L. -S. Wang

  2. Electrical Engineering Department & Systems Research Center, University of Maryland, 20742, College Park, MD, USA

    P. S. Krishnaprasad

Authors
  1. L. -S. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. S. Krishnaprasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Jerrold Marsden

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, L.S., Krishnaprasad, P.S. Gyroscopic control and stabilization. J Nonlinear Sci 2, 367–415 (1992). https://doi.org/10.1007/BF01209527

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01209527

Key words

Search

Navigation