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Recent years have seen an increased interest in using mean-field densitybased modelling and control strategy for deploying robotic swarms. In thispaper, we study how to dynamically deploy the robots subject to their physicalconstraints to efficiently measure and reconstruct certain unknown spatialfield (e.g. the air pollution index over a city). Specifically, the evolutionof the robots' density is modelled by mean-field partial differential equations(PDEs) which are uniquely determined by the robots' individual dynamics.Bayesian regression models are used to obtain predictions and return a variancefunction that represents the confidence of the prediction. We formulate a PDEconstrained optimization problem based on this variance function to dynamicallygenerate a reference density signal which guides the robots to uncertain areasto collect new data, and design mean-field feedback-based control laws suchthat the robots' density converges to this reference signal. We also show thatthe proposed feedback law is robust to density estimation errors in the senseof input-to-state stability. Simulations are included to verify theeffectiveness of the algorithms.

Field Estimation using Robotic Swarms through Bayesian Regression and Mean-Field Feedback