Shepherding control and herdability in complex multiagent systems

TL;DR

This work uncovers the existence of a critical threshold of the density of the targets, below which the number of herders needed for success significantly increases, and derives and analyses a PDE describing the herders dynamics in a simplified one-dimensional setting.

Abstract

We study the shepherding control problem where a group of "herders" need to orchestrate their collective behaviour in order to steer the dynamics of a group of "target" agents towards a desired goal. We relax the strong assumptions of targets showing cohesive collective behavior in the absence of the herders, and herders owning global sensing capabilities. We find scaling laws linking the number of targets and minimum herders needed, and we unveil the existence of a critical threshold of the density of the targets, below which the number of herders needed for success significantly increases. We explain the existence of such a threshold in terms of the percolation of a suitably defined herdability graph and support our numerical evidence by deriving and analysing a PDE describing the herders dynamics in a simplified one-dimensional setting. Extensive numerical experiments validate our methodology.

Figures (5)

• Figure 1: Representative snapshots of the system configuration (with herders represented by blue diamonds and targets by magenta circles) at (a) the initial time $t=0$ with the agents uniformly distributed in $\Omega_{0}$ (yellow shaded disk), (b) at an intermediate time during shepherding control when herders surround all targets and (c) when the task is successfully achieved with all the targets in $\Omega_{G}$ (dark blue circle). (d) Schematic of the herders' and targets' sensing (magenta shaded disks) and repulsion (blue shaded disk) regions of radius $\xi$ and $\lambda$ respectively. The solid black arrows represent the direction of motion of the herders when moving in the absence of nearby targets ($\textbf{H}_1$), selecting the target to chase $\textbf{T}^*$ with the largest distance from to goal ($\textbf{H}_2$), or when the herder pushes a selected target towards the goal region ($\textbf{H}_3$).
• Figure 2: Values of the fraction $\chi$ of successfully herd targets obtained for different values of $M$ and $N$ when $R=50$. Results are averaged over $50$ simulations; the increments of $N$ and $\sqrt{M}$ have values $\Delta N=1$, $\Delta \sqrt{M}=1$. The level curve for $\chi^*=0.99$ is depicted by the white curve. The left panel shows the case of infinite sensing ($\xi=\infty$) where $N^*\propto\sqrt{M}$ while the right panel the case of limited sensing ($\xi<\infty$) where we recover $N^*\propto\sqrt{M}$ only above a critical threshold $M>M^{\text{low}}$ (white vertical line).
• Figure 3: Two representative configurations of targets and the structure of the corresponding herdability graph $\mathcal{G}$ (whose edges are depicted as solid black lines) (a) below and (b) above the critical percolation threshold $\widehat{M^{\text{low}}}$. Green arrows show possible paths the herder could potentially navigate to reach the furthermost targets, denoted as $\textbf{T}^{\star}$, showing that when the graph is too sparse [panel (a)] more distant targets can be lost.
• Figure 4: Scaling of the critical threshold $M^{\text{low}}$ as a function of $\xi/R$. The numerically observed values of $M^{\text{low}}$ (scatter dots), evaluated by direct inspection, are compared with the theoretical estimate $\widehat{M^{\text{low}}}$ (dashed line) for different values of $\xi$ and $R$ (see Tab... in the SM for the values of $\xi$ and $R$ selected). Error bars represent the maximum precision of the computation given the stepsize $\Delta \sqrt{M}=1$ used in the simulations. Results for $\chi^*\in\{0.90,\,0.95\}$ are reported in section I of the Supplementary Material confirming the observed scaling. For the same $\xi/R$ value, scatter points were shifted on the $x$-axis to increase visibility.
• Figure 5: Stationary distributions of the targets, $\rho^T$, over a one-dimensional domain together with the corresponding potential $V$ (red line) computed by \ref{['eqn:PDE']} showing global stability when no regions exist where $\rho^T|_{x\in\Delta}=0$ with $|\Delta| > \xi$ (top panel) and local stability otherwise (bottom panel). See supplementary video 3 in the SM for a simulation of \ref{['eqn:PDE_dyn']} in the above scenarios.