Recent years have seen an increased interest in using mean-field density based modelling and control strategy for deploying robotic swarms. In this paper, we study how to dynamically deploy the robots subject to their physical constraints to efficiently measure and reconstruct certain unknown spatial field (e.g. the air pollution index over a city). Specifically, the evolution of 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 variance function that represents the confidence of the prediction. We formulate a PDE constrained optimization problem based on this variance function to dynamically generate a reference density signal which guides the robots to uncertain areas to collect new data, and design mean-field feedback-based control laws such that the robots density converges to this reference signal. We also show that the proposed feedback law is robust to density estimation errors in the sense of input-to-state stability. Simulations are included to verify the effectiveness of the algorithms.
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