Hearing the shape of an arena with spectral swarm robotics

TL;DR

Spectral swarm robotics is introduced where robots diffuse information to their neighbors to emulate the Laplacian operator - enabling them to "hear" the spectrum of their arena.

Abstract

Swarm robotics promises adaptability to unknown situations and robustness against failures. However, it still struggles with global tasks that require understanding the broader context in which the robots operate, such as identifying the shape of the arena in which the robots are embedded. Biological swarms, such as shoals of fish, flocks of birds, and colonies of insects, routinely solve global geometrical problems through the diffusion of local cues. This paradigm can be explicitly described by mathematical models that could be directly computed and exploited by a robotic swarm. Diffusion over a domain is mathematically encapsulated by the Laplacian, a linear operator that measures the local curvature of a function. Crucially the geometry of a domain can generally be reconstructed from the eigenspectrum of its Laplacian. Here we introduce spectral swarm robotics where robots diffuse information to their neighbors to emulate the Laplacian operator - enabling them to "hear" the spectrum of their arena. We reveal a universal scaling that links the optimal number of robots (a global parameter) with their optimal radius of interaction (a local parameter). We validate experimentally spectral swarm robotics under challenging conditions with the one-shot classification of arena shapes using a sparse swarm of Kilobots. Spectral methods can assist with challenging tasks where robots need to build an emergent consensus on their environment, such as adaptation to unknown terrains, division of labor, or quorum sensing. Spectral methods may extend beyond robotics to analyze and coordinate swarms of agents of various natures, such as traffic or crowds, and to better understand the long-range dynamics of natural systems emerging from short-range interactions.

Figures (18)

• Figure 1: Framework of spectral swarm robotics.a: Description of our contribution: a swarm of robots computes the spectral fingerprint of its arena in a distributed way, so that each individual robot has its own fingerprint estimate; the robots reach a consensus on this value across the entire swarm; finally, this value is used to classify the shape in which the swarm is located. Our approach is validated using swarms of Kilobots rubenstein2012kilobotrubenstein2014programmable. b: Comparison of results from the continuous and graph Laplace operators over 7 geometric shapes. Color codes correspond to the local components of the second eigenfunction (in the continuous case) and eigenvector (in the graph case). The graph cases have 250 nodes randomly distributed in the shapes. $\lambda_2$ is averaged over 64 runs with different seeds and shown with its standard deviation. To easily compare the continuous and graph cases, values of the second eigenvalue $\lambda_2$ are normalized to have a value of 1 for disk arenas. Images are re-scaled for better visualization, computations are performed with arenas of the same surface. c: Workflow of the spectral swarm algorithm (see Supplementary Sec. 3 for full details) for decentralized shape classification. Kilobots will diffuse their internal state, resulting in a partition. Then the convergence rate of the diffusion process is used to estimate $\lambda_2$. The robots reach a consensus on this value across the entire swarm and show a different LED color depending on its value, resulting in a classification decision.
• Figure 2: Representative time-lapses of the behavior of the proposed algorithm applied to 7 arenas in simulations.Left: diffusion stage, resulting in a partitioning of the shapes (color code: sign of internal state $s_i^n$). The second eigenvalue $\lambda_2$ of the Laplacian matrix of the communication graph between robots corresponds to the convergence rate of diffusion: the information diffusion converges at different rates depending on the topology of the arena. Right: in turn, robots use the observed convergence rate to compute a local estimate of this value iteratively refined at each iteration of the algorithm. After 30 iterations, the estimations have converged and formed a consensus. Based on the value of $\lambda_2$, the robots display a color code corresponding to the detected arena shape. Images are re-scaled for visualization; during simulations, arenas surface are normalized to be $500000\texttt{mm}^2$, with $\tau = 1/15$ seconds corresponding to the amount of time between two steps of diffusion. We use the same parameters as regime r3 of Fig. \ref{['fig:phase_diagrams']}, with $N = 300$ robots and $\sigma = 85 \texttt{mm}$. The percentages of correct $\lambda_2$ are computed over 64 runs.
• Figure 3: Classifying regimes of the system computed in simulations over 64 runs.a: Influence of the number of robots $N$ and of the field of perception $\sigma$ on the accuracy. We filter the accuracy score to only show results where the simulated diffusion sessions converge: empty bins correspond to results with more than $50\%$ of cases with algorithmic instabilities from over-connected graphs and numerical errors (i.e., without exponential decay of $s_i^{n}$). We identify five representative regimes: r1 to r5. Examples of regime r3 behavior are shown in Fig. \ref{['fig:timelapses']}. The best-performing results are found on the hyperbola $N = 17 S / (\pi \sigma^2)$. Top-right corner: examples of robot distribution in the annulus arena; colored disks around each robot represent their field of perception $\sigma$. b: Evolution of estimated $\lambda_2$ values over 30 iterations of the algorithm, for each arena. Line colors correspond to the color code in Fig. \ref{['fig:timelapses']}. Only r1, r3, r5 are shown (r4 is similar to r3, and r2 is an over-connected case). The order of $\lambda_2$ values for each arena is generally conserved across regimes, except in r5, where the order is inverted compared to r1, r3, r4. c: Confusion matrices and accuracy scores for regimes r1, r3, r5, showing the performance of the system to accurately classify the shapes. Each row represents the instances of an actual class (i.e., the shapes the robots are situated in) while each column represents the instances of a predicted class (i.e., the shapes the robots report using LED color). The diagonal elements of the matrix (from top-left to bottom-right) represent the number of points for which the predicted class is equal to the true class, i.e., the correct predictions; perfect accuracy corresponds to the identity matrix. Off-diagonal elements in the matrix describe the misclassifications, i.e., the instances where the model predicted a class that differs from the true class.
• Figure 4: Comparison between theoretical, simulated, and experimental results on a two classes classification setting with two shapes (disk and annulus) using one iteration of the algorithm. a,b: Examples of arena partition in both the continuous and discrete (graph) cases. Colored zones and nodes correspond to the sign of the local component of the Fiedler vector $\textbf{v}_2$ of the Laplacian (red: negative, blue: positive). In both cases, global information (i.e., access to the full Laplacian matrix) is used to compute $\textbf{v}_2$. c: Evolution of the internal state $| s_i^n |$ of each robot $i$ during a diffusion session (i.e., using only local information) in simulations (64 runs). The slope of each respective curve is used to estimate the local value of $\lambda_2$ on each robot $i$. d: Computation of $\lambda_2$ using global information directly from the entire graph in b. e: Distribution of $\lambda_2$ values estimated on each robot (violin plots) in simulations, compared to the value computed using global information from the continuous Laplacian (red lines). Values are normalized so that the mean of disk values is $1.0$. f: Time-lapses of the algorithm, in simulations and in experiments, with 25 robots. Experimental photos are blurred to ease the visualization of LED colors. Left: diffusion, resulting in partitioning (the color of each robot corresponds to a local component of the Fiedler vector $\textbf{v}_2$). Right: individual $\lambda_2$ estimation on each robot from the convergence rate of diffusion, and consensus (collective averaging) over the entire swarm. Robots show a LED color-code according to the value of $\lambda_2$: cyan when a disk is detected, violet for an annulus. Images are re-scaled for better visualization; both simulations and experiments have arenas surface approximately equal to $70000\texttt{mm}^2$. g: Confusion matrices and accuracy scores of simulations (over 64 runs per arena) and experiments (over 15 runs per arena).
• Figure 5: Time-lapses of the experiments, with 25 robots, respectively for the disk (a) and annulus (b) arenas. Experiments use the same parameters as simulations and experiments of Fig. \ref{['fig:simuVsExpes']}. Only the results of 16 runs are shown (over 30 considered runs). Time-lapses of all runs are presented in Supplementary movie 1. Experimental photos are blurred to ease the visualization of LED colors (except left column). Images are re-scaled for better visualization; experiments have arenas surface approximately equal to $70000\texttt{mm}^2$. Left: Initial state of the robots before the start of experiments. Middle: diffusion, resulting in partitioning (the color of each robot corresponds to a local component of the Fiedler vector $\textbf{v}_2$). In the studied configuration, robots typically converge in the domain $n\in [60, 110]$ for the disk arenas and in the domain $n\in [180, 210]$ for the annulus arenas. In rare cases, the robots may still change state after convergence, due to computation errors, or message transmission errors -- in such case, the partitioning the robots converged into may be broken (e.g., last experiment on the annulus arena). The spectral swarm robotics algorithm will detect and ignore divergent cases after convergence (cf details in Supplementary Sec. 3). Right: individual $\lambda_2$ estimation on each robot from the convergence rate of diffusion, and consensus (collective averaging) over the entire swarm. Robots show a LED color-code according to the value of $\lambda_2$: cyan when a disk is detected, violet for an annulus. The robots detected the correct shape on 22 runs over 30 runs (15 runs on each arena), which translates into an accuracy score of $22/30 = 73\%$.