Archive for Rational Mechanics and Analysis

, Volume 136, Issue 1, pp 21–99 | Cite as

Nonholonomic mechanical systems with symmetry

  • Anthony M. Bloch
  • P. S. Krishnaprasad
  • Jerrold E. Marsden
  • Richard M. Murray
Article

Abstract

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control-theoretical applications. The basic methodology is that of geometric mechanics applied to the Lagrange-d'Alembert formulation, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration-space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a “body reference frame” relates part of the momentum equation to the components of the Euler-Poincaré equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory.

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References

  1. Abraham, R. &J. E. Marsden [1978]Foundations of Mechanics. Second Edition, Addison-Wesley.Google Scholar
  2. Abraham, R., J. E. Marsden &T. S. Ratiu [1988]Manifolds, Tensor Analysis, and Applications. Second Edition, Springer-Verlag.Google Scholar
  3. Sánchez de Alvarez, G. [1989] Controllability of Poisson control systems with symmetry.Contemp. Math. Amer. Math. Soc. 97, 399–412.Google Scholar
  4. Arnold, V. [1988]Dynamical Systems III. Springer-Verlag.Google Scholar
  5. Arnold, V. I. [1989]Mathematical Methods of Classical Mechanics. Second Edition, Springer-Verlag.Google Scholar
  6. Bates, L. &J. Sniatycki [1993] Nonholonomic reduction.Reports on Math. Phys. 32, 99–115.Google Scholar
  7. Bloch, A. M. &P. E. Crouch [1992] On the dynamics and control of nonholonomic systems on Riemannian Manifolds.Proceedings of NOLCOS '92, Bordeaux, 368–372.Google Scholar
  8. Bloch, A. M. &P. E. Crouch [1994] Nonholonomic control systems on Riemannian manifolds.SIAM J. Control Optim. 33, no. 1, 126–148Google Scholar
  9. Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden &T. S. Ratiu [1994] Dissipation Induced Instabilities.Ann. Inst. H. Poincaré, Analyse Nonlinéaire.11, 37–90.Google Scholar
  10. Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden &T. S. Ratiu [1996] The Euler-Poincaré equations and double bracket dissipation,Comm. Math. Phys. 175, 1–42.Google Scholar
  11. Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden &G. Sánchez de Alvarez [1992] Stabilization of rigid body dynamics by internal and external torques.Automatica.28, 745–756.Google Scholar
  12. Bloch, A. M., J.E. Marsden, &G. Sánchez de Alvarez [1996] Feedback stabilization of relative equilibria for mechanical systems with symmetry.Preprint, California Institute of Technology.Google Scholar
  13. Bloch, A. M., M. Reyhanoglu &H. McClamroch [1992] Control and stabilization of nonholonomic systems.IEEE Trans. Automat. Control. 37, 1746–1757.Google Scholar
  14. Bondi, H. [1986] The rigid-body dynamics of unidirectional spin.Proc. Roy. Soc. Lond. 405, 265–274.Google Scholar
  15. Brockett, R. W. &L. Dai [1992] Nonholonomic kinematics and the role of elliptic functions in constructive controllability, inNonholonomic Motion Planning,Z. Li &J. F. Canny, Kluwer, 1–22, 1993.Google Scholar
  16. Bryant, R. &P. Griffiths [1983] Reduction for constrained variational problems and ∝ κ2/2ds.Amer. J. Math. 108, 525–570.Google Scholar
  17. Burdick, J., B. Goodwine &J. Ostrowski [1994] The rattleback revisited. Preprint, California Institute of Technology.Google Scholar
  18. Cardin, F. &M. Favretti [1996] On Chetaev and vakonomic dynamics of nonholonomic mechanical systems.J. Geom. and Phys. 18, 295–325.Google Scholar
  19. Cartan, E. [1928] Sur la représentation géométrique des systèmes matèriels non holonomes.Atti. Cong. Int. Matem. 4, 253–261.Google Scholar
  20. Chaplygin, S. A. [1897a] On the motion of a heavy body of revolution on a horizontal plane (in Russian).Physics Section of the Imperial Society of Friends of Physics, Anthropology and Ethnographics, Moscow9, 10–16. (Reproduced inChaplygin [1954, 413–425].)Google Scholar
  21. Chaplygin, S. A. [1897b] On some feasible generalization of the theorem of area, with an application to the problem of rolling spheres (in Russian).Mat. Sbornik 20, 1–32. (Reproduced inChaplygin [1954, 434–454].)Google Scholar
  22. Chaplygin, S. A. [1903] On a rolling sphere on a horizontal plane (in Russian).Mat. Sbornik 24, 139–168. (Reproduced inChaplygin [1949, 72–99] andChaplygin [1954, 455–471].)Google Scholar
  23. Chaplygin, S. A. [1911] On the theory of the motion of nonholonomic systems. Theorem on the reducing factor (in Russian).Mat. Sbornik 28, 303–314. (Reproduced inChaplygin [1949, 28–38] andChaplygin [1954, 426–433].)Google Scholar
  24. Chaplygin, S. A. [1949]Analysis of the Dynamics of Nonholonomic Systems (in Russian). Classical Natural Sciences, Moscow.Google Scholar
  25. Chaplygin, S. A. [1954]Selected Works on Mechanics and Mathematics (in Russian). State Publ. House, Technical-Theoretical Literature, Moscow.Google Scholar
  26. Crabtree, H. [1909]Spinning Tops and Gyroscopic Motion. Chelsea.Google Scholar
  27. Cushman, R., J. Hermans, &D. Kemppainen [1995] The rolling disc. InNonlinear Dynamical Systems and Chaos (Groningen, 1995), Progr. Nonlinear Differential Equations Appl.,19, Birkhäuser, Basel, 21–60.Google Scholar
  28. Cushman, R., D. Kemppainen, J. Śniatycki &L. Bates [1995] Geometry of nonholonomic constraints.Rep. Math. Phys. 36, 275–286.Google Scholar
  29. Enos, M. J. (ed.) [1993]Dynamics and Control of Mechanical Systems, Fields Inst. Commun., Amer. Math. Soc.1.Google Scholar
  30. Getz, N. H. [1993] Control of nonholonomic systems with dynamically decoupled actuators.Proc. 32nd IEEE Control & Decision Conf., San Antonio, December 1993.Google Scholar
  31. Getz, N. H. [1994] Control of balance for a nonlinear nonholonomic non-minimum phase model of a bicycle.Proc. Amer. Control Conf., Baltimore, June, 1994.Google Scholar
  32. Getz, N. H. &J. E. Marsden [1994 Symmetry and dynamics of the rolling disk. Preprint, 630, Center for Pure and Applied Mathematics, Univ. California, Berkeley.Google Scholar
  33. Getz, N. H. &J. E. Marsden [1995] Control for an autonomous bicycle.IEEE Intern. Conf. on Robotics and Automation, Nagoya, Japan, May, 1995Google Scholar
  34. Hermans, J. [1995] A symmetric sphere rolling on a surface,Nonlinearity 8, 1–23.Google Scholar
  35. Hermans, J. [1995]Rolling Rigid Bodies, with and without Symmetries, Ph.D. Thesis, University of Utrecht.Google Scholar
  36. Hamel, G. [1904] Die Lagrange-Eulerschen Gleichungen der Mechanik.Z. f. Math. u. Phys. 50, 1–57.Google Scholar
  37. Jalnapurkar, S. [1995] The role of forces in nonholonomic systems, Preprint, Univ. California, Berkeley.Google Scholar
  38. Jurdjevic, V. [1993] The geometry of the plate-ball problem.Arch. Rational Mech. Anal. 124, 305–328.Google Scholar
  39. Karapetyan, A. V. [1994] On the specific character of the application of Routh's theory to systems with differential constraints.J. Appl. Math. Mech. 58, 387–392. (See alsoJ. Appl. Math. Mech. 51 (1987), 431–436.)Google Scholar
  40. Karapetyan, A. V. &V. V. Rumyantsev [1990] Stability of conservative and dissipative systems, inApplied Mechanics: Soviet Reviews 1, G.K. Mikhailov and V.Z. Parton (eds.), Hemisphere NY.Google Scholar
  41. Kelly, S. D. &R. M. Murray [1995] Geometric phases and robotic locomotion.J. Robotic Systems 12, no. 6, 417–431Google Scholar
  42. Kobayashi, S. &K. Nomizu [1963]Foundations of Differential Geometry. WileyGoogle Scholar
  43. Koiller, J. [1992] Reduction of some classical nonholonomic systems with symmetry.Arch. Rational Mech. Anal. 118, 113–148.Google Scholar
  44. Koon, W-S. &J.E. Marsden [1996a] Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction.SIAM J. Control and Optim. (to appear).Google Scholar
  45. Koon, W-S. &J.E. Marsden [1996b] The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems.Preprint, California Institute of Technology.Google Scholar
  46. Kozlov, V.V. &N.N. Kolesnikov [1978] On theorems of dynamics.Prikl. Mat. Mekh. 42, 28–33.Google Scholar
  47. Krishnaprasad, P.S. [1989] Eulerian many-body problems.Contemp. Math. Amer. Math. Soc. 97, 187–208.Google Scholar
  48. Krishnaprasad, P. S. [1990] Geometric phases and optimal reconfiguration for multibody systems.Proc. Amer. Control Conf., 2440–2444.Google Scholar
  49. Krishnaprasad, P. S., W. Dayawansa &R. Yang [1992] The geometry of nonholonomic constraints. Preprint, University of Maryland.Google Scholar
  50. Lam, S. H. [1994] Lagrangian dynamics and its control formulation. Preprint, MAE 1993, Mechanical Engineering, Princeton University.Google Scholar
  51. Lewis, A. &R. M. Murray [1994] Variational principles in constrained systems: theory and experiments,Intern. J. Nonlinear Mech. 30, 793–815.Google Scholar
  52. Lewis, A., J. P. Ostrowski, R. M. Murray &J. Burdick [1994] Nonholonomic mechanics and locomotion: the snakeboard example.IEEE Intern. Conf. on Robotics and Automation.Google Scholar
  53. Marle, C.-M. [1995] Reduction of constrained mechanical systems and stability of relative equilibria.Comm. Math. Phys. 174, 295–318.Google Scholar
  54. Marsden, J. E., P. S. Krishnaprasad &J. C. Simo (eds.) [1989]Dynamics and Control of Multibody Systems. Contemp. Math., Amer. Math. Soc.97.Google Scholar
  55. Marsden, J. E. [1992]Lectures on Mechanics. Cambridge University Press.Google Scholar
  56. Marsden, J. E., R. Montgomery &T. S. Ratiu [1990]Reduction, Symmetry, and Phases in Mechanics. Mem. Amer. Math. Soc.436.Google Scholar
  57. Marsden, J. E., G. W. Patrick &W. F. Shadwick, (eds.) [1996]Integration Algorithms and Classical Mechanics. Fields Inst. Commun.,10, Am. Math. Soc.Google Scholar
  58. Marsden, J. E. &T. S. Ratiu [1994]An Introduction to Mechanics and Symmetry. Springer-Verlag.Google Scholar
  59. Marsden, J. E. &T. S. Ratiu [1986] Reduction of Poisson Manifolds.Lett. Math. Phys. 11, 161–170.Google Scholar
  60. Marsden, J. E. &J. Scheurle [1993a] Lagrangian reduction and the double spherical pendulum.Z. Agnew. Math. Phys. 44, 17–43.Google Scholar
  61. Marsden, J. E. &J. Scheurle [1993b] The reduced Euler-Lagrange equations,Fields Inst. Commun., Amer. Math. Soc.1, 139–164.Google Scholar
  62. Murray, R. M., Z. Li &S. S. Sastry [1994]A Mathematical Introduction to Robotic Manipulation. CRC Press.Google Scholar
  63. Murray, R. M. &S. S. Sastry [1993] Nonholonomic motion planning: steering using sinusoids.IEEE Trans. Automat. Control 38, 700–716.Google Scholar
  64. Neimark, Ju. I. &N. A. Fufaev [1966] On stability of stationary motions of holonomic and nonholonomic systems.J. Appl. Math. (Prikl. Math. Mekh.) 30, 293–300.Google Scholar
  65. Neimark, Ju. I. &N. A. Fufaev [1972]Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs, Amer. Math. Soc.,33.Google Scholar
  66. O'Reilly, O. M. [1996] The dynamics of rolling disks and sliding disks.Nonlinear Dynamics,10, 287–305.Google Scholar
  67. Ostrowski, J. [1995]Geometric Perspectives on the Mechanics and Control of Undulatory Locomotion. Ph.D. Thesis, California Institute of Technology.Google Scholar
  68. Ostrowski, J., J. W. Burdick, A. D. Lewis &R. M. Murray [1995] The mechanics of undulatory locomotion: The mixed kinematic and dynamic case.IEEE Intern. Conf. on Robotics and Automation 1945–1951.Google Scholar
  69. Poincaré, H. [1901 Sur une forme nouvelle des equations de la mecanique.C. R. Acad. Sci. 132, 369–371.Google Scholar
  70. Rosenberg, R. M. [1977]Analytical Dynamics of Discrete Systems. Plenum Press, NY.Google Scholar
  71. Routh, E. J. [1860]Treatise on the Dynamics of a System of Rigid Bodies. MacMillan, London.Google Scholar
  72. San Martin, L. &P. E. Crouch [1984] Controllability on principal fibre bundles with compact structure group.Systems Control Lett. 8, 35–40.Google Scholar
  73. Sarlett, W., F. Cantrijn andDJ J. Suanders [1995] A geometrical framework for the study of non-holonomic Lagrangian systems.J. Phys. A: Math. Gen. 28, 3253–3268.Google Scholar
  74. Simo, J. C., D. Lewis &J. E. Marsden [1991] Stability of relative equilibria I: The reduced energy momentum method.Arch. Rational Mech. Anal. 115, 15–59.Google Scholar
  75. Sumbatov, A. S. [1992] Developments of some of Lagrange's ideas in the works of Russian and Soviet mechanicians.La mécanique analytique de Lagrange et son héritage, Atti della Accademia delle Scienze di Torino, Suppl.2, 126, 169–200.Google Scholar
  76. Tsikiris, D. P. [1995]Motion control and planning for nonholonomic kinematic chains. Ph.D. Thesis, Systems Research Institute, University of Maryland.Google Scholar
  77. van der Schaft, A. J. &P. E. Crouch [1987] Hamiltonian and self-adjoint control systems.Systems Control Lett. 8, 289–295.Google Scholar
  78. van der Schaft, A. J. &B. M. Maschke [1994] On the Hamiltonian formulation of nonholonomic mechanical systems.Rep. Math. Phys. 34, 225–233.Google Scholar
  79. Vershik, A. M. &L. D. Faddeev [1981] Lagrangian mechanics in invariant form.Sel. Math. Sov. 1, 339–350.Google Scholar
  80. Vershik, A. M. &V. Ya. Gershkovich [1994] Nonholonomic dynamical systems, geometry of distributions and variational problems.Dynamical Systems VII,V. Arnold &S. P. Novikov, eds., 1–81. Springer-Verlag.Google Scholar
  81. Vierkandt, A. [1892] Über gleitende und rollende Bewegung.Monats. der Math. u. Phys. 3, 31–54.Google Scholar
  82. Walker, G. T. [1896] On a dynamical top.Quart. J. Pure Appl. Math. 28, 175–184.Google Scholar
  83. Wang, L. S. &P. S. Krishnaprasad [1992] Gyroscopic control and stabilization.J. Nonlin. Sci. 2, 367–415.Google Scholar
  84. Weber, R. W. [1986] Hamiltonian systems with constraints and their meaning in mechanics.Arch. Rational Mech. Anal. 91, 309–335.Google Scholar
  85. Whittaker, E. T. [1937]A Treatise on the Analytical Dynamics of Particles and Rigid Bodies Fourth Edition, Cambridge University Press.Google Scholar
  86. Yang, R. [1992]Nonholonomic Geometry, Mechanics and Control. Ph.D. Thesis, Systems Research Institute, Univ. of Maryland.Google Scholar
  87. Yang, R., P. S. Krishnaprasad &W. Dayawansa [1993] Chaplygin dynamics and Lagrangian reduction.Proc. 2nd Intern. Cong. on Nonlinear Mechanics,W-Z. Chien, Z. H. Guo &Y. Z. Guo, eds., Peking University Press, 745–749.Google Scholar
  88. Zenkov, D. V. [1995] The Geometry of the Routh Problem,J. Nonlin. Sci. 5, 503–519.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Anthony M. Bloch
    • 1
  • P. S. Krishnaprasad
    • 2
  • Jerrold E. Marsden
    • 3
  • Richard M. Murray
    • 3
  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor
  2. 2.Institute for Systems ResearchUniversity of MarylandCollege Park
  3. 3.Control and Dynamical SystemsCalifornia Institute of TechnologyPasadena

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