  In this work, we design distributed control laws for spatial
self-organization of multi-agent swarms in 1D and 2D spatial domains. The
objective is to achieve a desired density distribution over a simply-connected
spatial domain. Since individual agents in a swarm are not themselves of
interest and we are concerned only with the macroscopic objective, we view the
network of agents in the swarm as a discrete approximation of a continuous
medium and design control laws to shape the density distribution of the
continuous medium. The key feature of this work is that the agents in the swarm
do not have access to position information. Each individual agent is capable of
measuring the current local density of agents and can communicate with its
spatial neighbors. The network of agents implement a Laplacian-based
distributed algorithm, which we call pseudo-localization, to localize
themselves in a new coordinate frame, and a distributed control law to converge
to the desired spatial density distribution. We start by studying
self-organization in one-dimension, which is then followed by the
two-dimensional case.
