The Navigation Transformation

This work introduces a novel approach to the solution of the navigation problem by mapping an obstacle-cluttered environment to a trivial domain called the point world, where the navigation task is reduced to connecting the images of the initial and destination configurations by a straight line. Due to this effect, the underlying transformation is termed the “navigation transformation.” The properties of the navigation transformation are studied in this work as well as its capability to provide-through the proposed feedback controller designs-solutions to the motion- and path-planning problems. Notably, the proposed approach enables the construction of temporal stabilization controllers as detailed herein, which provide a time abstraction to the navigation problem. The proposed solutions are correct by construction and, given a diffeomorphism from the workspace to a sphere world, tuning free. A candidate construction for the navigation transformation on sphere worlds is proposed. The provided theoretical results are backed by analytical proofs. The efficiency, robustness, and applicability of the proposed solutions are supported by a series of experimental case studies.