Nonconservativity and noncommutativity in locomotion

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.