## A Blob Method for Advection-Diusion-Reaction Systems with Application to Robotic Swarms

- Author(s): Zhang, Fangbo
- Advisor(s): Bertozzi, Andrea L
- et al.

## Abstract

In this dissertation, a blob method of measuring spatial coverage by a swarm of agents is presented, and two models are introduced: a macroscopic model, consisting of a system of Advection-Diffusion-Reaction equations that govern the spatial distribution of the swarm at the population level; and a microscopic model, represented by a stochastic differential equation, describing the individual behavior of each agent in the swarm. Depending on different tasks, multiple control frameworks are proposed to drive the swarm to a target distribution, and are proved to be both valid in theory and robust in real world case.

First, we briefly review the history of Advection-Diffusion-Reaction (ADR) system in the literature. The ADR system models the agents' motion as drifted brownian motion, and the agents' reaction as probabilistic transition between different states. The key idea of the ADR model lies in the fact that the green's function of Advection-Diffusion Equation is the conditional probability density function of the corresponding drifted brownian motion. The proposed point mass approach verifies that the error between the macroscopic and the microscopic model converges with respect to the inverse of the square root of the swarm size.

Next, we explores a stochastic approach for controlling swarms of independent robots toward a target distribution in a bounded domain. The robots are resource-constraint: they lack both

communication and localization capabilities, and can only gather local information by measuring a scalar field (e.g. concentration of a chemical) from the environment. A simple control law is introduced to govern the diffusion of each robot, so that the distribution of the swarm converges to a pre-defined target distribution over time. The key point behind the control framework is that the solution of heat equation converges to the first-order eigenvector of the Laplacian operator. We further confirm the robustness of the control law in real world case by conducting simulations in computer and test bed experiments in multiple cases of target distributions, where the swarm achieves the theorized convergence to the target distribution despite deviations from assumptions underpinning the theory.

Finally, we presents a novel procedure for computing parameters of a robotic swarm that guarantee coverage performance by the swarm within a specified error from a target spatial distribution. The main contribution is the analysis of the dependence of this error on two key parameters: the number of robots in the swarm and the robot sensing radius, in which we view each robot as a blob instead of a point mass. We model the population dynamics of the swarm as an advection-diffusion-reaction partial differential equation (PDE) with time-dependent advection and reaction terms. We derive rigorous bounds on the discrepancies between the target distribution and the coverage achieved by individual-based and PDE models of the swarm. We use these bounds to select the swarm size that will achieve coverage performance within a given error and the corresponding robot sensing radius that will minimize this error. We also apply the optimal control approach to compute the robots' velocity field and task-switching rates. We validate our procedure through simulations of a scenario in which a robotic swarm must achieve a specified density of pollination activity over a crop field.