  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.
