Nonlinear Generalizations of Diffusion-Based Coverage by Robotic Swarms

In this paper, we generalize our diffusion-based approach [8] to achieve coverage of a bounded domain by a robotic swarm according to a target probability density that is a function of a locally measurable scalar field. We generalize this approach in two different ways. First, we show that our method can be extended in a natural way to scenarios where the robots' state space is a compact Riemannian manifold, which is the case if the robots are confined to a surface or if their configuration space is non-Euclidean due to dynamical constraints such as those present in most mechanical systems. Then, we establish the stability properties of a weighted variation of the porous media equation, a nonlinear partial differential equation (PDE). Coverage strategies based on these nonlinear PDEs have the advantage that the robots stop moving once the equilibrium probability density is reached, in contrast to our original approach [8]. We establish long-time stability properties of the target probability densities using semigroup theoretic arguments. We validate our theoretical results through stochastic simulations of a linear diffusion-based coverage strategy on a 2-dimensional sphere and numerical solutions of the weighted porous media equation on the 2-dimensional torus.