PDE-Based Optimization for Stochastic Mapping and Coverage Strategies using Robotic Ensembles

This paper presents a novel partial differential equation (PDE)-based\nframework for controlling an ensemble of robots, which have limited sensing and\nactuation capabilities and exhibit stochastic behaviors, to perform mapping and\ncoverage tasks. We model the ensemble population dynamics as an\nadvection-diffusion-reaction PDE model and formulate the mapping and coverage\ntasks as identification and control problems for this model. In the mapping\ntask, robots are deployed over a closed domain to gather data, which is\nunlocalized and independent of robot identities, for reconstructing the unknown\nspatial distribution of a region of interest. We frame this task as a convex\noptimization problem whose solution represents the region as a\nspatially-dependent coefficient in the PDE model. We then consider a coverage\nproblem in which the robots must perform a desired activity at a programmable\nprobability rate to achieve a target spatial distribution of activity over the\nreconstructed region of interest. We formulate this task as an optimal control\nproblem in which the PDE model is expressed as a bilinear control system, with\nthe robots' coverage activity rate and velocity field defined as the control\ninputs. We validate our approach with simulations of a combined mapping and\ncoverage scenario in two environments with three target coverage distributions.\n