Velocity Field Generation for Density Control of Swarms using Heat Equation and Smoothing Kernels

TL;DR

This paper introduces a decentralized method for controlling swarm density using smooth velocity fields derived from the heat equation, enabling organized, collision-free movement based on local density estimates.

Contribution

The paper presents a novel decentralized approach that synthesizes smooth velocity fields for swarm density control using heat equation and kernel density estimation, improving movement smoothness and collision avoidance.

Findings

Swarm converges to desired density distribution effectively.

Agents move smoothly with velocities decreasing as they approach each other.

The method ensures conflict avoidance and organized movement.

Abstract

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.