Local Motion Planning for Nonholonomic Control Systems Evolving on Principal Bundles

1 Motivation Nonholonomic mechanical systems naturally occur whenthere are rolling constraints [4] or Lagrangian symmetries leading to momentum constraints [1]. Examples includekinematic wheeled vehicles, free floating satellites with appendages, and simplified models of biomimetic loco-motion. This work considers the local motion planning problem for a specific class of nonholonomic systems--those whose configuration space is the total space of a principal fibre bundle and whose equations of motion area connection on that bundle. The state variables of these systems naturally split into two classes. One class is theset of base or shape variables that describe the internal configuration of the system. The other variables take val-ues in a Lie group G, and are termed group or fibre vari-ables. They typically describe the position of the system via the displacement of a reference frame in the mov-ing system with respect to a fixed frame. Motion in the position variables can often be realized though periodicmotion of the shape variables. The governing equations of such nonholonomic controlsystems locally take the form.g =-gAi(x)ui.xi = ui (1) where x 2 M the shape manifold, g 2 G a Lie groupwith Lie algebra g, and Ai: T M! g is termed the lo-cal form of the connection. We assume that we have complete control over M. We seek to control the groupvariables through actuation of the shape variables. In the case of the kinematic car, the shape space consists of thewheel rolling and turning angles, while the car's position in SE(2) defines the group. The connection describes theno slip constraint between the wheels and the ground. For a more complete review of these ideas see [4].The outline and contributions of this paper are as follows. First, we construct an expansion for the system'sgroup displacement that arises from small periodic motions in the base space. This expansion is a generaliza-tion of the work of Leonard and Krishnaprasad [6], who developed an analogous formula for case when the loca