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Nonholonomic Mechanical Systems with Symmetry

Bloch, Anthony M. and Krishnaprasad, P. S. and Marsden, Jerrold E. and Murray, Richard M. (1996) Nonholonomic Mechanical Systems with Symmetry. Archive for Rational Mechanics and Analysis, 136 (1). pp. 21-99. ISSN 0003-9527.

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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 ldquobody reference framerdquo 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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Murray, Richard M.0000-0002-5785-7481
Additional Information:© Springer-Verlag 1996, Accepted: 24 July 1995. We thank JOEL BURDICK, SCOTT GRAY, PHILIP HOLMES, WANG-SANG KOON, HARVEY LAM, NAOMI LEONARD, ANDREW LEWIS, OLIVER O’REILLY, JIM OSTROWSKI, JUERGENSCHEURLE, andDIMITRY ZENKOV for very helpful comments concerning this paper. In particular, the snakeboard example studied by OSTROWSKI, LEWIS, and BURDICK was a crucial ingredient in the theoretical development of the present paper. We also thank NEIL GETZ for his comments on the paper and on the controlled bicycle, and HANS DUISTERMAAT and JOST HERMANS for their very helpful remarks, both scientific and historical. BLOCH’s researchwas partially supported by the National Science Foundation PYI grant DMS-91-57556, AFORSR grant F49620-1-0037, and MSRI; KRISHNAPRASAD’s research was partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation’s Engineering Research Centers Program NSD CDS 8803012;MARSDEN’s research was partially supported by the NSF,DOE, a Fairchild Fellowship at the California Institute of Technology, and the Fields Institute for Research in the Mathematical Sciences; and MURRAY’s research was partially supported by NSF, NASA, and the Powell Foundation. During the time of this research BLOCH was a visiting Professor at the Department of Mathematics, University of Michigan, Ann Arbor.
Funding AgencyGrant Number
Air Force Office of Scientific ResearchF49620-1-0037
Mathematical Sciences Research Institute (MSRI)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR). University Research Initiative ProgramAFOSR-87-0073
Air Force Office of Scientific Research (AFOSR) University Research Initiative ProgramAFOSR-90-0105
NSF Engineering Research Center ProgramNSD CDS 8803012
Department of EnergyUNSPECIFIED
Fairchild Fellowship at the California Institute of TechnologyUNSPECIFIED
Powell FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20100713-091226984
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19017
Deposited By: Ruth Sustaita
Deposited On:13 Jul 2010 20:40
Last Modified:18 Mar 2015 23:21

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