Towards simplicial coverage repair for mobile robot teams

In this note, we present initial results towards developing a distributed algorithm for repairing topological holes in the sensor cover of a mobile robot team. Central to our approach is the melding of recent advances in the application of computational homology (a sub-discipline of algebraic topology) to static sensor networks with relative metric information (i.e. relative pose). More precisely, we consider a greedy, hybrid (discrete-continuous) algorithm whereby a desired Cěch complex, the simplicial complex that captures the underlying topology of the sensing cover, is iteratively generated using local rules (between multi-hop neighbors) and agents are driven towards achieving this topology via a gradient-ascent simplicial control law. Convergence of the proposed algorithm is established as a function of the convergence of the underlying simplicial control law, and the relationship of the latter to the spectrum of the combinatorial Laplacian is considered. Simulation results for teams operating in ℝ² are presented.