  This paper presents a method to control the probability density distribution
of a swarm of vehicles via velocity fields. The proposed approach synthesizes
smooth velocity fields, which specify a desired velocity as a function of time
and position in a decentralized manner i.e., each agent calculates the desired
velocity locally by utilizing the number of agents within a prescribed
communication distance. Swarm converges to the desired/commanded density
distribution by following the velocity field. The local information consists,
only, of agents' positions and it is utilized to estimate the density around
each agent. Local density estimation is performed by using kernel density
estimation. Then the local density estimates and the desired density are
utilized to generate the velocity field, which propagates the swarm probability
density distribution via the well-known heat equation. The key advantage of
using smooth velocity fields to control swarm density with respect to our
earlier Markov chain based probabilistic control methods is that the agents
move more smoothly in an organized manner and their relative velocities go to
zero as they get closer to each other, i.e., they facilitate conflict/collision
avoidance. The desired density distribution, which is commanded to the local
controllers, can be computed by using a Markov chain that propagates the
desired density distribution within prescribed requirements. Along with
convergence and stability analysis, the effectiveness of the approach is
illustrated via numerical examples.
