Regular and Chaotic Dynamics

, Volume 20, Issue 3, pp 205–224 | Cite as

The dynamics of systems with servoconstraints. I

Article

Abstract

The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.

Keywords

servoconstraints symmetries Lie groups left-invariant constraints systems with quadratic right-hand sides 

MSC2010 numbers

34D20 70F25 70Q05 

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References

  1. 1.
    Béghin, M. H., Étude théorique des compas gyrostatiques Anschütz et Sperry, Paris: Impr. nationale, 1921; see also: Annales hydrographiques, 1921.Google Scholar
  2. 2.
    Appel, P., Traité de Mécanique rationnelle: Vol. 2. Dynamique des systèmes. Mécanique analytique, 6th ed., Paris: Gauthier-Villars, 1953.Google Scholar
  3. 3.
    Kozlov, V.V., Principles of Dynamics and Servoconstraints, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1989, no. 5, pp. 59–66 (Russian).Google Scholar
  4. 4.
    Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Borisov, A.V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics–Uspekhi, 2003, vol. 46, no. 4, pp. 393–403; Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 407–418.MathSciNetGoogle Scholar
  6. 6.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465–483.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Borisov, A.V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Borisov, A.V., Kazakov, A.O., and Sataev, I.R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718–733.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277–328.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Dynamics and Control of an Omniwheel Vehicle, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 153–172.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Poincaré, H., Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci., 1901, vol. 132, pp. 369–371.MATHGoogle Scholar
  13. 13.
    Kozlov, V.V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003.Google Scholar
  14. 14.
    Kozlov, V.V. and Furta, S.D., Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations, Springer Monogr. Math., Heidelberg: Springer, 2013.Google Scholar
  15. 15.
    Kozlov, V.V., Exchange of Stabilities in the Euler-Poincaré-Suslov Systems under the Change of the Direction of Motion, Nonlinear Dynamics & Mobile Robotics, 2014, vol. 2, no. 2, pp. 199–211.Google Scholar
  16. 16.
    Kozlov, V.V., On the Integration Theory of Equations of Nonholonomic Mechanics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 161–176.MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Hamiltonicity and Integrability of the Suslov Problem, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 104–116.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Andronov, A.A., Leontovich, E.A., Gordon, I. I., and Maier, A.G., Theory of Bifurcations of Dynamic Systems on a Plane, New York: Wiley, 1973.Google Scholar
  19. 19.
    Kozlov, V.V., Invariant Measures of the Euler–Poincaré Equations on Lie Algebras, Funct. Anal. Appl., 1988, vol. 22, no. 1, pp. 58–59; see also: Funktsional. Anal. i Prilozhen., 1988, vol. 22, no. 1, pp. 69–70.MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 1972.Google Scholar
  21. 21.
    Borisov, A.V. and Mamaev, I. S., The Dynamics of a Chaplygin Sleigh, J. Appl. Math. Mech., 2009, vol. 73, no. 2, pp. 156–161; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 2, pp. 219–225.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955.MATHGoogle Scholar
  23. 23.
    Kirgetov, V. I., The Motion of Controlled Mechanical Systems with Prescribed Constraints (Servoconstraints), J. Appl. Math. Mech., 1967, vol. 31, no. 3, pp. 465–477; see also: Prikl. Mat. Mekh., 1967, vol. 31, no. 3, pp. 433–446.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Golubev, Yu. F., Mechanical Systems with Servoconstraints, J. Appl. Math. Mech., 2001, vol. 65, no. 2, pp. 205–217; see also: Prikl. Mat. Mekh., 2001, vol. 65, no. 2, pp. 211–224.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rumiantsev, V.V., On the Motion of Controllable Mechanical Systems, J. Appl. Math. Mech., 1976, vol. 40, no. 5, pp. 719–729; see also: Prikl. Mat. Mekh., 1976, vol. 40, no. 5, pp. 771–781.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Grdina, Ya. I., Notes on the Dynamics of Living Organisms, Ekaterinoslav: Izd. Ekaterinosl. Gorn. In-ta, 1916 (Russian).Google Scholar
  27. 27.
    Blajer, W., Seifried, R., and Kołodziejczyk, K., Servo-Constraint Realization for Underactuated Mechanical Systems, Arch. Appl. Mech., (2015); open access http://link.springer.com/article/10.1007%2Fs00419-014-0959-2 (DOI 10.1007/s 00419-014-0959-2).Google Scholar
  28. 28.
    Utkin, V. I., Sliding Modes and Their Application in Variable Structure Systems, Moscow: Mir, 1978.MATHGoogle Scholar
  29. 29.
    Arnol’d, V. I., Kozlov, V.V., and Neïshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.Google Scholar
  30. 30.
    Kharlamova-Zabelina, E. I., Rapid Rotation of a Rigid Body around a Fixed Point in the Presence of a Non-Holonomic Constraint, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1957, no. 6, pp. 25–34 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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