Path planning for permutation-invariant multi-robot formations

In this paper we demonstrate path planning for our formation space that represents permutation-invariant multi-robot formations. Earlier methods generally pre-assign roles for each individual robot, rely on local planning and behaviors to build emergent behaviors, or give robots implicit constraints to meet. Our method first directly plans the formation as a set, and only afterwards determines which robot takes which role. To build our representation of this formation space, we make use of a property of complex polynomials: they are unchanged by permutations of their roots. Thus we build a characteristic polynomial whose roots are the robot locations, and use its coefficients as a representation of the formation. Mappings between work spaces and formation spaces amount to building and solving polynomials. In this paper, we construct an efficient obstacle collision detector, and use it in a local planner. From this we construct a basic roadmap planner. We thus demonstrate that our polynomial-based representation can be used for effective permutation-invariant formation planning.