Eur. Phys. J. Special Topics 224, 3141-3174 (2015)
https://doi.org/10.1140/epjst/e2015-50085-y

Regular Article

Nonconservativity and noncommutativity in locomotion

Geometric mechanics in minimum-perturbation coordinates

¹ School of Mechanical, Industrial, and Manufacturing Engineering, Oregon State University, USA
² Robotics Institute, Carnegie Mellon University, USA

^a e-mail: Ross.Hatton@oregonstate.edu

Received: 14 April 2015
Revised: 2 November 2015
Published online: 15 December 2015

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 in'uence the e↑ectiveness of these approaches. This property appears at ↓rst to run counter to the principle that di↑erential 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 con↓guration space. Nonconservative constraint variations can be integrated geometrically via Stokes' theorem, but noncommutative e↑ects 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.