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Geometric discretization of nonholonomic systems with symmetries

Kobilarov, Marin and Marsden, Jerrold E. and Sukhatme, Gaurav S. (2010) Geometric discretization of nonholonomic systems with symmetries. Discrete and Continuous Dynamical Systems Series S, 3 (1). pp. 61-84. ISSN 1937-1632 http://resolver.caltech.edu/CaltechAUTHORS:20100607-151910834

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Abstract

The paper develops discretization schemes for mechanical systems for integration and optimization purposes through a discrete geometric approach. We focus on systems with symmetries, controllable shape (internal variables), and nonholonomic constraints. Motivated by the abundance of important models from science and engineering with such properties, we propose numerical methods specifically designed to account for their special geometric structure. At the core of the formulation lies a discrete variational principle that respects the structure of the state space and provides a framework for constructing accurate and numerically stable integrators. The dynamics of the systems we study is derived by vertical and horizontal splitting of the variational principle with respect to a nonholonomic connection that encodes the kinematic constraints and symmetries. We formulate a discrete analog of this principle by evaluating the Lagrangian and the connection at selected points along a discretized trajectory and derive discrete momentum equation and discrete reduced Euler-Lagrange equations resulting from the splitting of this principle. A family of nonholonomic integrators that are general, yet simple and easy to implement, are then obtained and applied to two examples-the steered robotic car and the snakeboard. Their numerical advantages are confirmed through comparisons with standard methods.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.3934/dcdss.2010.3.61 DOIUNSPECIFIED
http://aimsciences.org/journals/displayArticles.jsp?paperID=4804PublisherUNSPECIFIED
Additional Information:© AIMS 2009. Received: September 2008. Revised: May 2009. Published: December 2009. JEM and MK were partially supported by AFOSR contract FA9550-08-1-0173. The authors would like to thank M. Desbrun, J. C. Marrero, D. Mart´ın De Diego, and S. Ferraro for their valuable input and advise.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR) FA9550-08-1-0173
Subject Keywords:Nonholonomic systems, discrete mechanics, variational integrators, symmetries, reduction
Classification Code:2000 Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35
Record Number:CaltechAUTHORS:20100607-151910834
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100607-151910834
Official Citation:Geometric discretization of nonholonomic systems with symmetries. Marin Kobilarov, Jerrold E. Marsden and Gaurav S. Sukhatme Pgs. 61-84
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18592
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:23 Jun 2010 16:08
Last Modified:26 Dec 2012 12:07

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