Local Motion Planning for Nonholonomic Control Systems Evolving on Principal Bundles

Nonholonomic mechanical systems naturally occur when there are rolling constraints [4] or Lagrangian symmetries leading to momentum constraints [1]. Examples include kinematic wheeled vehicles, free floating satellites with appendages, and simplified models of biomimetic locomotion. 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 are a connection on that bundle. The state variables of these systems naturally split into two classes. One class is the set of base or shape variables that describe the internal configuration of the system. The other variables take values in a Lie group G, and are termed group or fibre variables. They typically describe the position of the system via the displacement of a reference frame in the moving system with respect to a fixed frame. Motion in the position variables can often be realized though periodic motion of the shape variables. The governing equations of such nonholonomic control systems locally take the form