Topological Landmark-based Navigation and Mapping

Algebraic-topological criteria for domain coverage, hole-detection, andmapping are given in the context of dynamic multi-agent systems navigating with respect to beacons or landmark visibility. The key constructs involve dual pairs of nerves of the relevant visibility and observation covers, based on a sensing relation, either binary or via distance/signal-strength. This, in turn, is used to construct simplicial approximations to the regions explored as a topological map of the domain. We prove equivalence of various simplicial approximations (based on landmarks or observation points), show how to incorporate distance-correlated data with persistent homology, and demonstrate these methods via simulations.