SIGEST

Abstract

In recent years, almost everyone has heard about autonomous vehicles. The well-known DARPA grand challenges of the last two years concerned autonomous vehicles navigating challenging courses and terrain. The intent of developing this technology is that one day vehicles will be able to operate autonomously in a variety of dangerous situations, from military scenarios to disaster relief. How will these vehicles communicate and deploy to cover a space evenly, in an environment where they may only have local knowledge of other vehicles? This is an example of the problem that motivates this issue's SIGEST paper. The paper “Nonsmooth Coordination and Geometric Optimization via Distributed Dynamical Systems,” by Jorge Cortés and Francesco Bullo, first appeared in the SIAM Journal on Control and Optimization, 44 (2005), pp. 1543–1574. In this seminal work, the authors design distributed (i.e., reliant on local knowledge) coordination algorithms for dynamic networks, and provide formal verification of the correctness of these algorithms. One of the important and challenging aspects of the problem they study is that they assume that the communication topology of the network may change as the system evolves, as opposed to remaining fixed. The paper considers two different scenarios for “even spacing” in the dynamic network, which are easy to describe informally if we continue to consider the example of vehicles in a convex polygonal region. The first is to cause each point in the region to be as close as possible to the nearest vehicle (akin in a static sense to positioning post offices in a town to minimize the longest distance of any home to the nearest post office); the second is to require each vehicle to be as distant as possible from all other vehicles, and the boundary of the region. In the case of just one vehicle, the first scenario corresponds to finding the center of the smallest circle entirely containing the polygon, whereas the second corresponds to finding the center of the largest circle fully contained within the polygon. It turns out that these disk-covering and sphere-packing concepts are related to the solutions in general. One of the fascinating and beautiful aspects of this paper is that it closely combines three rather distinct areas of applied mathematics: geometry and in particular Voronoi diagrams; dynamical systems where relationships can evolve and change; and optimization, specifically nonsmooth optimization, meaning that the objective functions are not continuously differentiable. Indeed, the authors show that dynamical system strategies based upon nonsmooth gradients have geometric interpretations related to Voronoi diagrams. The authors have modified an already very nicely written original paper to make it even more accessible to a general SIAM audience. We invite readers to enjoy this very important and illuminating view into the world of dynamical systems and optimization. Who knows, one day your car may incorporate technology stemming from this research!