Optimal control of stochastic coverage strategies for robotic swarms

This paper addresses a trajectory planning and task allocation problem for a swarm of resource-constrained robots that cannot localize or communicate with each other and that exhibit stochasticity in their motion and task-switching policies. We model the population dynamics of the robotic swarm as a set of advection-diffusion-reaction partial differential equations (PDEs), a linear parabolic PDE model that is bilinear in the robots´ velocity and task-switching rates. These parameters constitute a set of time-dependent control variables that can be optimized and broadcast to the robots prior to their deployment. The planning and allocation problem can then be formulated as a PDE-constrained optimization problem, which we solve using techniques from optimal control. Simulations of a commercial pollination scenario validate the ability of our control approach to drive a robotic swarm to achieve predefined spatial distributions of activity over a closed domain, which may contain obstacles.