Stable emergent formations for a swarm of autonomous car-like vehicles

Abstract

A biological swarm is an ideal multi-agent system that collectively self-organizes into bounded, if not stable, formations. A mathematical model, developed appropriately from some principle of swarming, should enable one, therefore, to study formation strategies for multiple autonomous robots. In this article, based on the hypothesis that swarming is an interplay between long-range attraction and short-range repulsion between the individuals in the swarm, a planar individual-based or Lagrangian swarm model is constructed using the Direct Method of Lyapunov. While attraction ensures the swarm is cohesive, meaning that the individuals in the swarm remain close to each other at all times, repulsion ensures that the swarm is well-spaced, meaning that no two individuals in the swarm occupy the same space at the same time. Via a novel Lyapunov-like function with attractive and repulsive components, the article establishes the global existence, uniqueness, and boundedness of solutions about the centroid. This paves the way to prove that the swarm model, governed by a system of first-order ordinary differential equations (ODEs), is cohesive and well-spaced. The article goes on to show that the artificial swarm can collectively self-organize into two stable formations: (i) a constant arrangement about the centroid when the system has equilibrium points, and (ii) a highly parallel formation when the system does not have equilibrium points. Computer simulations not only illustrate these but also reveal other emergent patterns such as swirling structures and random-like walks. As an application, we tailor the model accordingly and propose new autonomous steering laws giving rise to pattern-forming for multiple nonholonomic car-like vehicles.