A Hopf-Lax Type Formula for Multi-Agent Path Planning with Pattern Coordination

We present an algorithm for a multi-agent path planning problem with pattern coordination based on dynamic programming and a Hamilton-Jacobi-Bellman equation. This falls broadly into the class of partial differential equation (PDE) based optimal path planning methods, which give a black-box-free alternative to machine learning hierarchies. Due to the high-dimensional state space of multi-agent planning problems, grid-based methods for PDE which suffer from the curse of dimensionality are infeasible, so we instead develop grid-free numerical methods based on variational Hopf-Lax type representations of solutions to Hamilton-Jacobi Equations. Our formulation is amenable to nonlinear dynamics and heterogeneous agents. We apply our method to synthetic examples wherein agents navigate around obstacles while attempting to maintain a prespecified formation, though with small changes it is likely applicable to much larger classes of problems.