The Navigation Transformation: Point worlds, time abstractions and towards tuning-free navigation

This paper proposes a novel solution to the navi gation 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 Trans formation". The properties of this transformation are studied in this paper as well as its capability to provide a solution to the path and motion planning problems. A construction for such a transformation is proposed. A feedback controller based on the Navigation Transformation is designed that guarantees almost global exponential stability. It is shown that by utilizing the Navigation Transformation, the time for a motion task can be abstracted by means of an appropriately designed time abstracting controller. In addition to the theoretical guarantees, a set of simulation case studies on a star shaped world are provided to demonstrate the effectiveness of the methodology.