Generalized multicircumcenter trajectories for optimal design under near-independence

This work deals with trajectory optimization for a network of robotic sensors sampling a spatio-temporal random field. We examine the problem of minimizing over the space of network trajectories the maximum predictive variance of the estimator. This is a high-dimensional, multi-modal, nonsmooth optimization problem, known to be NP-hard even for static fields and discrete design spaces. Under an asymptotic regime of near-independence between distinct sample locations, we show that the solutions to a novel generalized disk-covering problem are solutions to the optimal sampling problem. This result transforms the search for the optimal trajectories into a geometric optimization problem. Constrained versions of the latter are also of interest as they can accommodate trajectories that satisfy a maximum velocity restriction on the robots. We characterize the solution for the unconstrained and constrained versions of the problem as generalized multicircumcenter trajectories, and provide distributed algorithms to find them.