  In this study, we present and analyze a framework for geometric and
topological estimation for mapping of unknown environments. We consider agents
mimicking motion behaviors of cyborg insects, known as biobots, and exploit
coordinate-free local interactions among them to infer geometric and
topological information about the environment, under minimal sensing and
localization constraints. Local interactions are used to create a graphical
representation referred to as the encounter graph. A metric is estimated over
the encounter graph of the agents in order to construct a geometric point cloud
using manifold learning techniques. Topological data analysis (TDA), in
particular persistent homology, is used in order to extract topological
features of the space and a classification method is proposed to infer robust
features of interest (e.g. existence of obstacles). We examine the asymptotic
behavior of the proposed metric in terms of the convergence to the geodesic
distances in the underlying manifold of the domain, and provide stability
analysis results for the topological persistence. The proposed framework and
its convergences and stability analysis are demonstrated through numerical
simulations and experiments.
