Target-density formation in swarms with stochastic sensing and dynamics

TL;DR

This work develops a stochastic, physics-inspired model for autonomous swarms to form a prescribed target density using local, uncertain sensing and movement on a network of patches. Through mean-field, small-fluctuation, and finite-$N$ analyses, it derives the density dynamics, mode-wise relaxation, and a closed-form target-error measure, revealing how convergence speed and accuracy depend on measurement noise, agent count, and interaction rules, including collisions. It identifies a finite-$N$ crossover and a regime where strong repulsion can prevent exact target formation, highlighting the trade-offs between sensing precision, swarm size, and target spatial complexity. The framework offers principled guidance for designing robust, minimally controlled swarms and suggests directions for experimental validation with robotic and biological systems.

Abstract

An important goal for swarming research is to create methods for predicting, controlling and designing swarms, which produce collective dynamics that solve a problem through emergent and stable pattern formation, without the need for constant intervention, and with a minimal number of parameters and controls. One such problem involves a swarm collectively producing a desired (target) density through local sensing, motion, and interactions in a domain. Here, we take a statistical physics perspective and develop and analyze a model wherein agents move in a stochastic walk over a networked domain, so as to reduce the error between the swarm density and the target, based on local, random, and uncertain measurements of the current density by the swarming agents. Using a combination of mean-field, small-fluctuation, and finite-number analysis, we are able to quantify how close and how fast a swarm comes to producing a target as a function of sensing uncertainty, stochastic collision rates, numbers of agents, and spatial variation of the target.

Figures (4)

• Figure 1: Swarm evolving toward a target density through local sensing and dynamics. (a) The target known to the agents. (b) An example patch and its four neighbors. A randomly selected agent shown in red, makes two uncertain density measurements-- one at its local patch $i$, and another at a randomly selected neighboring patch $j$. (c) The probability density (Pr) for how far-off the densities of the patches are from their target, $\hat{z}_{ij}=\overline{y}_{i}\!-\!\hat{y}_{i}\!-\!\overline{y}_{j}\!+\!\hat{y}_{j}$, as estimated by the red agent. In the example, the probability density is Gaussian with mean $\overline{y}_{i}\!-\!y_{i}\!-\!\overline{y}_{j}\!+\!y_{j}$ and variance $2\sigma^{2}$. If the difference between the neighboring patch's estimated density and the target is greater than the difference for its current patch, the agent moves to the neighboring patch. (d) swarm density mean-squared error (divided by the target variance) versus time for: $N=$$10^4$ (magenta), $10^5$ (red), and $10^6$ (blue). The solution of Eq.(\ref{['eq:MFpatternFormation']}) is plotted with a black line. Pr($\hat{z}_{ij}$) is the same as (c). Other parameters are: $M\!=\!60^{2}$, $\sigma M\!=\!0.5$, and $\beta\!=\!1$.
• Figure 2: Dynamics of density formation. (a) Three target densities, each plotted with a different color. All panels follow the same color convention. (b) Stochastic simulations with $N\!=\!10^{4}$. Lines correspond to solutions of Eqs.(\ref{['eq:MFpatternFormation']}). The mean-sqarred error is plotted versus time. (c) Projection of the dynamics onto the Fiedler mode. Solutions of Eqs.(\ref{['eq:MFpatternFormation']}) for each target compared to Eqs.(\ref{['eq:LinearSolution1']}-\ref{['eq:LinearSolution3']}) for $c_{M-1}(t)$. Other parameters are: $M\!=\!100$, $\sigma M\!=\!0.1$, and $\beta\!=\!1$.
• Figure 3: Finite-$N$ steady-state error. (a) $\text{MSE}$ versus $\sigma$ for: $N\!=\!10^4,$ (diamonds), $N\!=\!10^5,$ (squares), $N\!=\!10^6,$ (x's), and mean-field (circles). Equation (\ref{['eq:SFsteady']}) is plotted with a black line, while Eq.(\ref{['eq:FiniteNSimple']}) predictions are plotted with dashed lines. The target density is shown in Fig.\ref{['fig1']} (a). (b) $\text{MSE}$ versus $N$ for two disconnected target densities with $\sigma\!=\!0$: blue circles ($S\!=\!9$) and red squares ($S\!=\!24$). Equation (\ref{['eq:FiniteN']}) predictions are plotted with solid and dashed lines, respectively. $\beta\!=\!0$ for both panels.
• Figure 4: Dependence of the error on repulsion and spatial variation of the target. (a) Two example targets with small (top, $p\!=\!0.1$) and large (bottom, $p\!=\!1.0$) spatial variation. (b) $\text{MSE}$ versus $\sigma$ from Eq.(\ref{['eq:MFpatternFormation']}) for two values of repulsion: $\beta=0.2$ (red) and $\beta=1.0$ (blue). Plot markers correspond to $p=0.1$ (circles), $p=0.4$ (squares), $p=0.7$ (x's), and $p=1.0$ (triangles).