Distributed path integral feedback control based on Kalman smoothing for unicycle formations

We consider the problem of a team of non-holonomic agents that must independently compute a discounted, infinite-horizon optimal feedback control from a Hamilton-Jacobi-Bellman equation in order to drive the team into a distance-based formation, without explicit communication. In this work, the uncertainty in a neighboring agent's control input is modeled by Brownian motion, which allows the solution to the Hamilton-Jacobi-Bellman equation to be written as a path integral over the agents' future trajectories. We show how this representation allows the formation control problem to be transformed into independent Kalman smoothing problems, avoiding the computational issues that are typically associated with computing an optimal feedback control for high-dimensional systems. A numerical example in which five agents form a regular pentagon is provided.