Towards locally computable polynomial navigation functions for convex obstacle workspaces

In this paper we present a polynomial Navigation Function (NF) for a sphere world that can be constructed almost locally, with partial knowledge of the environment. The presented navigation function is C² and as a result the computational complexity is very low, while the construction uses local knowledge and information. Moreover, an almost locally computable diffeomorphism between convex obstacles and spheres is presented, allowing the NF scheme to be used in a workspace populated by convex obstacles. Our approach is not strictly local in the ε sense, i.e. the field around a point is not influenced only by an ε region around the point, but rather it is local in the sense that the NF around each obstacle is influenced only by the obstacle and the adjacent obstacles. In particular, we require, in the vicinity of an obstacle, the distance between the obstacle and the adjacent obstacles. Simulations are presented to verify this approach.