In this paper, a new online robot motion planner is developed for systematically exploring unknown environ¬ments by intelligent mobile robots in real-time applications. The algorithm takes advantage of sensory data to find an obstacle-free start-to-goal path. It does so by online calculation of the Generalized Voronoi Graph (GVG) of the free space, and utilizing a combination of depth-first an...

We introduce the geodesic complexity of a metric space, inspired by topological space. Both them are numerical invariants, but, while TC only depends on homotopy type, GC is an invariant under isometries. show that in many cases they coincide but we also develop tools to distinguish two range examples. To this end, study what denote total cut locus, which does not appear have been explicitly co...