Planar Herding of Multiple Evaders with a Single Herder

TL;DR

This paper introduces a herding strategy using a single pursuer with two trajectory classes to control multiple evaders around a target, ensuring they are confined within small limit cycles.

Contribution

It proposes novel feedback control laws for herding evaders using spiral and circular pursuer trajectories with stability analysis.

Findings

All evaders can be herded into small limit cycles.

The pursuer's trajectory converges to a circular orbit.

Explicit formulas for limit cycle radii are provided.

Abstract

A planar herding problem is considered, where a superior pursuer herds a flock of non-cooperative, inferior evaders around a predefined target point. An inverse square law of repulsion is assumed between the pursuer and each evader. Two classes of pursuer trajectories are proposed: (i) a constant angular-velocity spiral, and (ii) a constant angular-velocity circle, both centered around the target point. For the spiraling pursuer, the radial velocity is dynamically adjusted based on a feedback law that depends on the instantaneous position of the evader, which is located at the farthest distance from the target at the start of the game. It is shown that, under suitable choices of the model parameters, all the evaders are herded into an arbitrarily small limit cycle around the target point. Meanwhile, the pursuer also converges onto a circular trajectory around the target. The conditions for the stability of these limit cycles are derived. For the circling pursuer, similar guarantees are provided along with explicit formulas for the radii of the limit cycles.