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.
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References
Abraham, R. & J. E. Marsden,Foundations of Mechanics, 2nd ed., Reading: Benjamin/Cummings, 1978.
Alvarez, G. S.,Geometric Methods of Classical Mechanics Applied to Control Theory, University of California, Berkeley, Ph.D. Dissertation, 1986.
Antman, S. S.,Nonlinear Problems of Elasticity, in preparation, 1991.
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.
Arnold, V. I.,Mathematical Methods of Classical Mechanics, New York: Springer-Verlag, 1978.
Arnold, V. I., ed.,Dynamical Systems III, Encyclopedia of Mathematical Sciences,3, Berlin: Springer-Verlag, 1988.
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.
Bloch, A. M. & J. E. Marsden, “Controlling Homoclinic Orbits,”Theoretical and Computational Fluid Mechanics (1989), pp. 179–190.
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.
Brockett, R. W., “Feedback Invariants for Nonlinear Systems,”6th IFAC Congress, Helsinki (1978), pp. 1115–1120.
Chetayev, N. G.,The Stability of Motion, Pergamon Press, 1961.
Crouch, P. E. & Van Der Schaft,Variational and Hamiltonian Control Systems, Springer-Verlag, 1987.
Godbillon, C.,Géometrie différentielle et mécanique analytique, Hermann, Paris, 1969.
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.
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.
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.
Guillemin, V. & S. Sternberg,Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.
Han, J. H.,Symmetries in Nonlinear Control Systems and Their Applications, Ph.D. Dissertation, Electrical Engineering Department, University of Maryland, College Park, 1986.
Hermann, R.,Differential Geometry and the Calculus of Variation, 2nd ed., Math. Sci. Press, 1977.
Holm, D., J. E. Marsden, T. Ratiu, & A. Weinstein, “Nonlinear Stability of Fluid and Plasma Equilibria,”Physics Report,123(1985), pp. 1–116.
Isidori, A.,Nonlinear Control Systems, 2nd ed., Springer-Verlag, 1989.
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.
Krishnaprasad, P. S., “Lie-Poisson Structures, Dual-spin Spacecraft and Asymptotic Stability,”Nonlinear Analysis: Theory, Methods, and Applications,7(1984), pp. 1011–1035.
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.
Krishnaprasad, P. S. & C. A. Berenstein, “On the Equilibria of Rigid Spacecraft with Rotors,”Systems & Control Letters,4(1984), pp. 157–163.
Lanczos, C.,The Variational Principles of Mechanics, 4th ed., University of Toronto Press, 1970.
Lewis, D., “Lagrangian Block Diagonalization,”J. Dynamics and Differential Equations (1991), (to appear).
Libermann, P. & C-M. Marie,Symplectic Geometry and Analytical Mechanics, Dordrecht: D. Reidel Publ., 1987.
Maddocks, J. H., “Stability and Folds,”Archive for Rational Mechanics and Analysis,99(4) (1987), pp. 301–328.
Marmo, G., E. J. Saletan, & A. Simoni, “Reduction of Symplectic Manifolds Through Constants of the Motion,”Nuovo Cimento,50B(1) (1979).
Marsden, J. E. & T. Ratiu, “Reduction of Poisson Manifolds,”Letters in Math. Phys.,11 (1986), pp. 161–169.
Marsden, J. E. & A. Weinstein, “Reduction of Symplectic Manifolds with Symmetry,”Reports in Math. Phys.,5(1974), pp. 121–130.
Nijmeijer, H. & Van Der Schaft,Nonlinear Dynamical Control Systems, Springer-Verlag, 1990.
Nomizu, K.,Lie Groups and Differential Geometry, The Mathematical Society of Japan, 1956.
Palais, R. S., “The Principle of Symmetric Criticality,”Comm. in Math. Physics,69(1) (1979), pp. 19–30.
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.
Routh, E.,The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, 4th ed., Dover Publications, Inc., 1884.
Siam,Report on the Panel on Future Directions in Control Theory: a Mathematical Perspective, Philadelphia, 1988.
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).
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.
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).
Smale, S., “Topology and Mechanics I, II,”Inventiones Mathematicae,10,11 (1970), pp. 305–331, 45–64.
Souriau, J. M.,Structure des systémes dynamique, Dunod, Paris, 1970.
Sternberg, S.,Lectures on Celestial Mechanics: Iand II, Addison-Wesley, 1969.
Vershik, A. M. & L. D. Faddeev, “Lagrangian Mechanics in Invariant Form,”Sel. Math. Sov.,1, 4 (1981), pp. 339–350.
Wang, L.-S.,Geometry, Dynamics and Control of Coupled Systems, Ph.D. Dissertation, Electrical Engineering Department, University of Maryland, College Park, August, 1990.
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.
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.
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.
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.
Whittaker, E. T.,A Treatise on the Analytical Dynamics of Panicles and Rigid Bodies, 4th ed., Cambridge University Press, Cambridge, 1959.
Willmore, T. J., “The Definition of Lie Derivative,”Proc. Edin. Math. Soc.,12(2) (1960), pp. 27–29.
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.
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).
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Communicated by Jerrold Marsden
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Wang, L.S., Krishnaprasad, P.S. Gyroscopic control and stabilization. J Nonlinear Sci 2, 367–415 (1992). https://doi.org/10.1007/BF01209527
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DOI: https://doi.org/10.1007/BF01209527