G-Snakes: Nonholonomic Kinematic Chains on Lie Groups

Abstract

We consider kinematic chains evolving on a finite-dimensional Lie group G under nonholonomic constraints, where snake-like global motion is induced by shape variations of the system. In particular, we consider the case when the evolution of the system is restricted to a subspace h of the corresponding Lie algebra g, where h is not a subalgebra of g and it can generate the whole algebra under Lie bracketing. Such systems are referred to as G- snakes. Away from certain singular configurations of the system, the constraints specify a (partial) connection on a principal fiber bundle, which in turn gives rise to a geometric phase under periodic shape variations. This geometric structure can be exploited in order to solve the nonholonomic motion planning problem for such systems.G-snakes generalize the concept of nonholonomic Variable Geometry Truss assemblies, which are kinematic chains evolving on the Special Euclidean group SE (2) under nonholonomic constraints imposed by idler wheels. We examine in detail the cases of 3-dimensional groups with real non-abelian Lie algebras such as the Heisenberg group H(3), the Special Orthogonal group SO (3) and the Special Linear group SL(2).