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The European Physical Journal Special Topics

, Volume 224, Issue 17–18, pp 3141–3174 | Cite as

Nonconservativity and noncommutativity in locomotion

Geometric mechanics in minimum-perturbation coordinates
  • R.L. HattonEmail author
  • H. Choset
Regular Article Physics of Locomotion
Part of the following topical collections:
  1. Dynamics of Animal Systems

Abstract

Geometric mechanics techniques based on Lie brackets provide high-level characterizations of the motion capabilities of locomoting systems. In particular, they relate the net displacement they experience over cyclic gaits to area integrals of their constraints; plotting these constraints thus provides a visual “landscape” that intuitively captures all available solutions of the system’s dynamic equations. Recently, we have found that choices of system coordinates heavily influence the effectiveness of these approaches. This property appears at first to run counter to the principle that differential geometric structures should be coordinate-invariant. In this paper, we provide a tutorial overview of the Lie bracket techniques, then examine how the coordinate-independent nonholonomy of these systems has a coordinate-dependent separation into nonconservative and noncommutative components that respectively capture how the system constraints vary over the shape and position components of the configuration space. Nonconservative constraint variations can be integrated geometrically via Stokes’ theorem, but noncommutative effects can only be approximated by similar means; therefore choices of coordinates in which the nonholonomy is primarily nonconservative improve the accuracy of the geometric techniques.

Keywords

European Physical Journal Special Topic Position Space Body Frame Shape Space Exterior Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© EDP Sciences and Springer 2015

Authors and Affiliations

  1. 1.School of Mechanical, Industrial, and Manufacturing Engineering, Oregon State UniversityCorvallisUSA
  2. 2.Robotics Institute, Carnegie Mellon UniversityPittsburghUSA

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