This is a Research Experiences for Undergraduates (REU) project organized by the Mathematics Department at the University of California, Los Angeles (UCLA)
In this report, we analyze a decentralized stochastic control law used to transport a robotic swarm toward a desired distribution. We first review relevant background information, including the error metric used to evaluate the swarm’s performance. We then develop a definition of “steady state” for this stochastic system using an exponential decay model. Next, we compute the optimal configuration of robots with respect to the desired distribution to develop a reference point for our error metric. We further study the error metric, proving that for randomly chosen robot configurations drawn from the desired distribution, its probability density function converges to a normal distribution. We show that for robots distributed according to the desired distribution, the error metric approaches zero as the number of robots goes to infinity and their size goes to zero. Furthermore, we show that if the desired distribution is uniform, the control law preserves and tends towards uniformity. By using a simple “bounceback” boundary behavior, convergence toward the desired distribution is achieved, aside from slight warping in corner regions of the domain. Finally, we explore the use of a deterministic control law, and its accuracy in approximating a macroscopic diffusion model is demonstrated using a one-dimensional simulation.
B. Anderson, E. Loeser, M. Gee, F. Ren, S. Biswas, O. Turanova, M. Haberland, and A. L. Bertozzi.