Alignment Distance in Path Control

In this paper we discuss alignment distance for measuring path deviation between curves. We compare properties of the alignment distance to both p-norms and the Haus-dorff distance to argue its superiority for use in path following problems. While problems of finding an optimal parameterization of a fixed input curve to a tracking system are not new, typical formulations focus on parameterizations that minimize transversal time while respecting certain system constraints; on-line governors can then be employed that choose a path velocity in real time, trading off computational complexity and time-optimality. By explicitly characterizing the error measure implicit in path control problems, we revisit the off-line, open loop parameterization problem to explore the inherit tradeoffs between command shape, command parameterization, and system dynamics. The utility of the alignment distance as a tool for elucidating these fundamental tradeoffs is demonstrated on a simple PD-controlled mass system.