Abstract
We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borisov, A. V., Bolsinov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132 [Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318].
Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Regular & Chaotic Dynamics, 2005 (Russian).
Borisov, A. V. and Mamaev, I. S., Rolling of a Non-Homogeneous Ball over a Sphere without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.
Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Selected Problems on Nonholonomic Mechanics, Epreprint, 2005 (Russian) http://ics.org.ru/doc?book=30&dir=r.
Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148 [Russian original: Matem. Sb., 1903, vol. 24, no. 1, pp. 139–168; reprinted in: Collected Works: Vol. 1, Moscow-Leningrad: GITTL, 1948, pp. 76–101].
Chung, W., Nonholonomic Manipulators, Springer Tracts in Advanced Robotics, vol. 13, Berlin-New York: Springer, 2004.
Duistermaat, J. J., Chaplygin’s Sphere, arXiv:math/0409019v1
Eisenhart, L.P., Separable Systems of Stäckel, Ann. of Math. (2), 1934, vol. 35, no. 2, pp. 284–305.
Fedorov, Yu.N., The Motion of a Rigid Body in a Spherical Support, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1988, no. 5, pp. 91–93 (Russian).
Fedorov, Yu.N., Two Integrable Nonholonomic Systems in Classical Dynamics, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1989, no. 4, pp. 38–41 (Russian).
Jovanović, B., LR and L + R Systems, J. Phys. A, 2009, vol. 42, 225202, 18 pp.
Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.
Koiller, J. and Ehlers, K., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127–152.
Kumagai, M. and Ochiai, T., Development of a Robot Balancing on a Ball, Proc. Internat. Conf. on Control, Automation and Systems (Seoul, Oct. 14–17, 2008), pp. 433–438.
Lauwers, T. B., Kantor, G. A. and Hollis, R. L., A Dynamically Stable Single-Wheeled Mobile Robot with Inverse Mouse-Ball Drive, Proc. IEEE Intern. Conf. on Robotics and Automation (Orlando, FL, May 15–19, 2006), pp. 2884–2889.
Markeev, A.P., Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Izv. Akad. Nauk SSSR. Mekhanika tverdogo tela, 1986, vol. 21, no. 1, pp. 64–65 [English transl.: Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 149–151].
Nagarajan, U., Kantor, G., and Hollis, R. L., Trajectory Planning and Control of an Underactuated Dynamically Stable Single Spherical Wheeled Mobile Robot, Proc. IEEE Intern. Conf. on Robotics and Automation (Kobe, Japan, May 12–17, 2009), pp. 3743–3748.
Shen, J., Schneider, D.A., and Bloch, A.M., Controllability and Motion Planning of a Multibody Chaplygin’s Sphere and Chaplygin’s Top, Internat. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905–945.
Veselov, A.P. and Veselova, L. E., Integrable Nonholonomic Systems on Lie Groups, Mat. Zametki, 1988, vol. 44, no. 5, pp. 604–619 [Math. Notes, 1988, vol. 44, nos. 5–6, pp. 810–819].
Veselova, L.E., New Cases of the Integrability of the Equations of Motion of a Rigid Body in the Presence of a Nonholonomic Constraint, in Geometry, Differential Equations and Mechanics, Moscow: Mosk. Gos. Univ., 1986, pp. 64–68 (Russian).
Wilson, J. L., Mazzoleni, A.P., DeJarnette, F.R., Antol, J., Hajos, G.A., and Strickland, C. V., Design, Analysis, and Testing of Mars Tumbleweed Rover Concepts, J. Spacecraft Rockets, 2008, vol. 45, no. 2, pp. 370–382.
Xu, Ya. and Ou, Yo., Control of Single Wheel Robots, Springer Tracts in Advanced Robotics, vol. 20, Berlin-New York: Springer, 2005.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Borisov, A.V., Kilin, A.A. & Mamaev, I.S. Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support. Regul. Chaot. Dyn. 17, 170–190 (2012). https://doi.org/10.1134/S1560354712020062
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354712020062