Hearing the shape of an arena with spectral swarm robotics

Leo Cazenille; Nicolas Lobato-Dauzier; Alessia Loi; Loona Macabre; Mika Ito; Olivier Marchal; Soo-Hyeon Kim; Nathanael Aubert-Kato

Figure 1: Framework of spectral swarm robotics. a: 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. b: Main mathematical tricks allowing each robot to extract

\lambda_{2}

from diffusion of information. c: Exponential decays associated to first five eigenvalues. After a vanishing time, only

e^{-\lambda_{2}\tilde{t}}

prevails. Schematics show how a random initial condition, consisting of a superposition of multiple modes converges towards a partition along the 2nd eigenvector.

Figure 2: Workflow of the spectral swarm algorithm for decentralized shape classification. Random walks and handshake messages allow for robots to change and identify their neighbors. Pre-diffusion ensures the applicability of the ”zero-mean trick”: from a random initial condition Kilobots will diffuse their internal state, resulting in an homogeneous state, the mean. This value will be subtracted to the initial condition of the diffusion step. The Diffusion step ensures the applicability of the ”vanishing trick”: as depicted in Fig2c, robots converge to a partition along the 2nd eigenvector. The ”log trick” allows then to estimate

\lambda_{2}

directly from the convergence rate of the diffusion process . 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 3: 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. 4 , with

N=300

robots and

\sigma=85\texttt{mm}

. The percentages of correct

\lambda_{2}

are computed over 64 runs.

Figure 4: 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 overconnected 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. 3 . The best-performing results are found on the hyperbola

N=17S/(\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. 3 . Only r1, r3, r5 are shown (r4 is similar to r3, and r2 is an overconnected 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 5: Comparison between theoretical, simulated, and experimental results on a two-class classification setting with two shapes (disk and annulus) using one iteration of the algorithm. (A) and (B) Examples of arena partition in 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 (access to the full Laplacian matrix) is used to compute

\textbf{v}_{2}

. (C) and (D) Evolution of the internal state

|s_{i}^{n}|

of each robot

i

(Kilobots in C, Pogobots in D) during a diffusion session in simulations, using only local information. Thin curves show individual robot trajectories; the bold black curve shows their mean in that run. The slope of each individual curve is used to estimate a local value of

\lambda_{2}

. (E) and (F) Distributions of estimated

\lambda_{2}

values for disk (E) and annulus (F) arenas in Kilobot simulations, Pogobot simulations, and Pogobot experiments. Symbols represent run- or experiment-level mean estimates. Horizontal central marks indicate the mean, and vertical bars span the minimum and maximum. Each condition comprised

N=64

simulation runs or

n=15

experimental runs per arena, with 25 robots per run; each point is the mean estimate across the 25 robots in one run. Values were normalized such that the mean of the corresponding disk condition is 1.0; red horizontal lines indicate the normalized continuous-Laplacian reference values. Disk and annulus distributions were compared separately for Kilobot simulations, Pogobot simulations, and Pogobot experiments using Mann–Whitney–Wilcoxon tests with Bonferroni correction for multiple comparisons (adjusted

P<10^{-4}

for each comparison).

Figure 6: Comparison between simulated and experimental results on a two classes classification setting with two shapes (disk and annulus) using one iteration of the algorithm. a & c: Time-lapses of the algorithm, in simulations and in experiments, with 25 robots (Kilobots in a , Pogobots in c ). The times exclude the prediffusion stage for simplicity, so actual durations are nearly double. 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}

for Kilobots, and

150000\texttt{mm}^{2}

for Pogobots (both setups have roughly the same density). b & d: Confusion matrices and accuracy scores of simulations (Kilobots in b , Pogobots in d ) over 64 runs per arena, and experiments (over 15 runs per arena).

Figure 7: Time-lapses of the experiments with Pogobots , with 25 Pogobot robots, respectively for the disk ( a ) and annulus ( b ) arenas. Experiments use the same parameters as simulations and experiments of Fig. 6 . 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

150000\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[10,20]

for the disk arenas and in the domain

n\in[40,80]

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. 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. The robots detected the correct shape on 30 runs over 30 runs (15 runs on each arena), which translates into an accuracy score of

30/30=100\%

.

Variable name	Unit	Description
$N$	-	Number of agents.
$H$	-	Mean number of neighbors of agents in a given configuration.
$\sigma$	$[m]$	Radius of the field of perception of the agents.
$S$	$[m^{2}]$	Arena surface.
$\mathcal{G}$	-	Graph of communication flows between agents
A	-	Adjacency matrix of $\mathcal{G}$ . $A_{i,j}=1$ if the agents $i$ and $j$ are neighbor, and 0 otherwise (by convention a robot is not its own neighbor).

Table S1: Nomenclature of all considered variables

Figure S1: General overview of the experimental setups used for the robotics experiments. Top-left : general view of the experimental setup, which is closed using light-occluding black curtains. The robot area is 152 cm per 72 cm, within which a smaller area (88 cm per 72 cm) is physically delimited. The green floor (D) placed on the robot area is a cutting map on which stands a single Kilobot robot. Two Pixel K80 professional LED photography light projectors (A) are placed on either side of the setup, pointing upwards with an angle of

45\deg

, and complemented with two rails of LED (White 5700K) placed on the ceiling, for a global illumination of 506-584 lux (B). A Logitech C920s HD Pro Webcam (C) and a Pixelink PL-D734MU camera (not used here) are placed in the center of the ceiling (115 cm above the robot area) and point downwards. The ceiling is covered in white matte paper Top-right : a Kilobot robot (diameter

=33

mm). The Kilobot features two vibrating motors (A), one 3.3V rechargeable battery, three metallic rods (C) and an infrared LED transmitter and photodiode receiver (D) placed underneath the robot and pointing downwards. Middle-left : the annulus (left) and disk (right) configurations, using

25

Kilobots. The disk configuration has a radius of 150mm and a surface of

\approx 70685.8

mm 2 . The annulus configuration has an external radius of

200mm

, internal radius of

133mm

and a surface of

\approx 70092.1\texttt{mm}^{2}

. Middle-right : same as above, acquired from the overhead webcam. A matte white floor is used, as well as a black disk shape for the annulus configuration used to physically forbid communication between robots from the opposite region of the annulus. Bottom-left : a Pogobot robot (diameter

=53

mm). The Pogobot features modular motors and 4 high-throughput directional IR emitters and receivers placed around the robot body. Bottom-right : the annulus configuration with 25 Pogobot robots. Both the annulus and disk arenas have a normalized surface (accessible to robots, within walls) of

\approx 1.5e5

mm 2 .

Property	Kilobot	Pogobot
Diameter	33 mm	53 mm
Mass	$\sim$ 16 g	$\sim$ 43–56 g
Locomotion	Vibration-based (slip–stick)	Modular: wheels or vibration-based
Max linear speed	$\sim$ 1 cm/s	up to 25 cm/s (wheels); $\sim$ 7 cm/s (TPU legs)
Max angular speed	$\sim$ 0.78 rad/s	up to 15 rad/s (nylon vibration mode)
Communication type	Infrared (1 emitter + 1 receiver, downward reflection)	4 directional IR emitter/receiver pairs

Table S2: Comparison between Kilobot and Pogobot platforms [ 51 , 52 ] .

	LED color	Description
Dispersion	Red	Tumble behavior
Dispersion	Green	Run behavior
Diffusion	White	Pre-diffusion stage
	Blue	$\texttt{sign(}s_{i}^{t}\texttt{)}>0$
	Red	$\texttt{sign(}s_{i}^{t}\texttt{)}<0$
	Grey	$\texttt{sign(}s_{i}^{t}\texttt{)}=0$

Table S3: Table of LED colors displayed by the robots depending on their current behavior.

Handshake	Variable name	Type	Size (bits)	Description
	kilo_uid	uint16	16	Kilobot unique identifier
	known_uid0	uint16	16	Unique identifier of already known neighbor
	known_uid1	uint16	16	Unique identifier of already known neighbor
	known_uid2	uint16	16	Unique identifier of already known neighbor
	-	-	8	Empty
	type	uint8	8	Type of message ( handshake , diffusion, coll. averaging, etc)

Table S4: Description of the data packets transmitted from robot to robot during the handshake , diffusion and collective averaging stages of the spectral swarm robotics algorithm. The maximal size of kilobots messages payload ( i.e., data, without type and CRC) is 72 bits (9 bytes).

Case name	Robot type	Simulations or experiments	Dispersion alg.	Surface of arenas $S$	Arenas	Nr. of runs	Nr. of agents $N$	Lenght scale $l$	Nr. of iterations $I$
sim-large-7	Kilobots	Simulations	run-and-tumble	$500000mm^{2}$	Disk, Square, Arrow, Star, Triangle, Stop, Annulus	64	$50-550$	$100.0-30.15$	1-30
sim-large-2	Kilobots	Simulations	run-and-tumble	$500000mm^{2}$	Disk, Annulus	64	$50-550$	$100.0-30.15$	1-30
sim-small-2-dispersion	Kilobots	Simulations	run-and-tumble	$70000mm^{2}$	Disk, Annulus	64	$10-80$	$83.67-29.58$	1-10
sim-small-2-random	Kilobots	Simulations	None - immobile robots initialized at random positions	$70000mm^{2}$	Disk, Annulus	64	$10-80$	$83.67-29.58$	1-10
sim-small-2-uniform	Kilobots	Simulations	None - immobile robots initialized uniformly in the arenas	$70000mm^{2}$	Disk, Annulus	64	$10-80$	$83.67-29.58$	1-10
sim-2	Kilobots	Simulation	None - immobile robots initialized uniformly in the arena	$70000mm^{2}$	Disk, Annulus	64	25	$52.91$	1

Table S5: List of all considered cases in simulation and experiments.

	Parameter Name	sim-large-7	sim-large-2	sim-small-2-dispersion	sim-small-2-random	sim-small-2-uniform	sim-2	expe-2	pogo-sim-2	pogo-expe-2
Base	Robot type	Kilobots							Pogobots
	Nr. of runs	$64$						15	64	15
	Arena Surface $S$	$500000mm^{2}$		$70000mm^{2}$					$150000mm^{2}$
	Agent diameter	$33$ mm							$53$ mm
	Field of Perception $\sigma$ (center to center)	$30$ mm- $320$ mm					$\approx 85$ mm		$\approx 135$ mm
	Nr. of agents $N$	50-550		10-80			25

Table S6: List of parameter values used with all cases. One kilotick (kt) corresponds to

1/31\approx 0.03

seconds.

Figure S2: Evolution of the accuracy with respect to the field of perception of each agent and the number of agents, for the case sim-large-7 . Here, we consider 7 arenas with a surface

S=500000mm^{2}

. All results are computed over 64 simulation runs. We adjust the accuracy score to only show results where the simulated diffusion sessions in the swarm did not diverge: empty bins correspond to results where there are more than

50\%

of runs exhibiting divergences. Top: results obtained after only 1 iteration. Bottom: results obtained after 30 iterations.

Figure S3: Top: Mean number of neighbors of agents ( i.e., mean degree of the communication graphs), and Bottom : mean number of connected graph components in the sim-large-7 case, after 30 iterations.

Figure S4: Distribution of degrees of the regimes r1, r3, r5 , as defined in Fig. 5 of the Main Text, and for the sim-large-7 case, after 30 iterations. Results are obtained over 32 runs, with the variability in values represented by the error bars in each histogram bin. In each plot, an orange dotted line corresponds to the distribution mean. Values of mean and skewness are provided for each plot. Regimes r1 and r3 both show different values mean and skewness for each arena. The skewness of the distribution is affected by the prevalence of borders in the arenas: agents close to a wall will have less neighbors than agents far for them. In the case of the disk, the disparity in degree is particularly high, with agents close to walls having far less neighbors than agents in the center. Conversely, the annulus has far less disparity in degree, as most agents will be close either to the outer wall or the inner wall. As such, there are no real center. The other arenas degree distributions fall between the two extremes cases of the disk and annulus.

Figure S5: Density of agents in the arenas after the initial seeding, for cases with arena surface

S=70000\texttt{mm}^{2}

and for two different configurations (either 25 or 45 agents). Results were obtained over 64 runs.

Figure S6: Examples of position of the agents after the initial seeding, for cases with arena surface

S=70000\texttt{mm}^{2}

and for two different configurations (either 25 or 45 agents).

Figure S7: Statistical analysis of the different types of agent dispersion in arenas of surface

S=70000\texttt{mm}^{2}

. We consider the sim-small-2-dispersion , sim-small-2-random , and sim-small-2-uniform cases – named in the figure “dispersion”, “random” and “uniform” respectively. Two configurations are considered, using 25 and 45 agents respectively. Statistical significance is computed using the two-sided Mann-Whitney-Wilcoxon test with Bonferroni correction. Significant pairs are designated as follows: ns:

0.05<p\leq 1.00

, *:

0.01<p\leq 0.05

, **:

0.001<p\leq 0.01

, ***:

0.0001<p\leq 0.001

, ****:

p\leq 0.0001

.

Figure S8: Difference in accuracy scores between cases sim-small-2-dispersion , sim-small-2-random and sim-small-2-uniform , depending on the number of agents and their field of perception. All results are obtained in simulations over 10 iterations and over 64 runs. Each bin of the 2D histograms corresponds to the difference between the accuracy score of sim-small-2-dispersion and that of either sim-small-2-random (top) or sim-small-2-uniform (bottom). We adjust the accuracy score to only show results where the simulated diffusion sessions in the swarm do not diverge: empty bins correspond to results where there are more than

50\%

of the runs exhibiting divergences.

Figure S9: Evolution of the accuracy depending on the number of agents and their field of perception, for the case in simulation sim-small-2-dispersion . In this case, we consider 2 arenas (disk vs annulus) of surface

S=70000mm^{2}

and the robots use the run-and-tumble dispersion algorithm to move throughout the arenas between each iteration. All results are computed over 64 runs of simulation. We adjust the accuracy score to only show results where the simulated diffusion sessions in the swarm do not diverge: empty bins correspond to results where there are more than

50\%

of runs exhibiting divergences. Top: results with only 1 iteration. Middle: results obtained after 10 iterations. Bottom: Ranking of the order of values of the Final

\lambda_{2}

centroids for the disk arena.

Figure S10: Evolution of the accuracy depending on the number of agents and their field of perception, for the case in simulation sim-small-2-random . In this case, we consider 2 arenas (disk vs annulus) of surface

S=70000mm^{2}

and the robots always stay immobile during the simulation. The initial position of the robots inside the arenas are random (2D uniform distribution). All results are computed over 64 runs of simulation. We adjust the accuracy score to only show results where the simulated diffusion sessions in the swarm do not diverge: empty bins correspond to results where there are more than

50\%

of runs exhibiting divergences. Top: results after 1 iteration. Middle: results after 10 iterations. Bottom: Ranking of the values of the Final

\lambda_{2}

centroids for the disk arena.

Figure S11: Evolution of the accuracy with respect to the number of agents and their field of perception, for the case in simulation sim-small-2-uniform . In this case, we consider 2 arenas (disk vs annulus) of surface

S=70000mm^{2}

and the robots always stay immobile during the simulation. Robots are distributed uniformly in the arenas, so that they are equidistant to their neighbors (obtained through Voronoi tessellation). All results are computed over 64 runs of simulation. We adjust the accuracy score to only show results where the simulated diffusion sessions in the swarm do not diverge: empty bins correspond to results where there are more than

50\%

of runs exhibiting divergences. Top: results after one iteration. Middle: results after 10 iterations. Bottom: Ranking of the values of the Final

\lambda_{2}

centroids for the disk arena.

Figure S12: Impact of communication noise on accuracy , in simulations of one iteration of the SSR algorithm (32 runs), with 25 simulated Pogobot robots, and the same algorithm parameters as results in main text Figs. 6, 7 and 8. We vary two parameters: 1) the probability of a focal robot to correctly receive neighbors’ messages; 2) the amount of time the neighbors messages are retained in the focal robot (after that, they are discarded).

Figure S13: Examples of experimental state at the end of an iteration , with 25 Pogobot robots that light LED colors that correspond to their detected arena geometry: purple for the annulus, yellow for the triangle, cyan for the disk. The photos are blurred to ease visualization of robot LED colors.

Figure S14: Comparison between theoretical, simulated, and experimental results on a three-class classification setting with three shapes: disk, annulus and triangle. a: Distribution of

\lambda_{2}

values estimated on each robot after one iteration of the algorithm in simulations (pogo-sim-2 case), compared to the value computed using global information from the continuous Laplacian (red lines) and Pogobot experiments (pogo-expe-2 case). Values are normalized so that the mean of disk values is

1.0

. Comparisons used Mann–Whitney–Wilcoxon tests with Bonferroni correction; adjusted

P

values and comparison groups are provided in data file S1 accompanying the published article. b: Distribution of

\lambda_{2}

values estimated on each robot after three iterations of the algorithm in simulations, compared to the value computed using global information from the continuous Laplacian and Pogobot experiments. Values are normalized so that the mean of disk values is

1.0

. Comparisons used Mann–Whitney–Wilcoxon tests with Bonferroni correction; adjusted

P

values and comparison groups are provided in data file S1 accompanying the published article. c: Confusion matrix of simulations over 64 runs per arena, after one iteration of the algorithm. Accuracy:

87.9\%

. d: Confusion matrix of experiments over 15 runs per arena, after one iteration of the algorithm. Accuracy:

75.4\%

. e: Confusion matrix of simulations over 64 runs per arena, after three iterations of the algorithm. Accuracy:

97\%

. f: Confusion matrix of experiments over 15 runs per arena, after three iterations of the algorithm. Accuracy:

93\%

.

Figure S15: Confusion matrices of the baselines , in 64-run simulations of 25 Pogobots robots with parameters from the pogo-sim-2 case, across three arenas: disk, annulus and triangle and two initial formations: random and Lloyd-relaxed (uniform). Rows correspond to the two initial formations. Columns correspond to the four classification methods: SSR, average degree, degree statistics and convergence time.

Baseline name	Random formation accuracy	Lloyd-relaxed formation accuracy
SSR (base algorithm)	$\mathbf{93.9\%}$	$\mathbf{96.7\%}$
average degree	$80.1\%$	$83.2\%$
degree statistics	$80.6\%$	$84.5\%$
convergence time	$63.7\%$	$46.5\%$

Table S7: Table of accuracy scores of baselines compared to the SSR algorithm (in bold), across 64-run simulations using the parameters of the pogo-sim-2 case, after 3 iterations.

Figure S16: Time-lapses of the experiments with Kilobots , with 25 Kilobot robots, respectively for the disk ( a ) and annulus ( b ) arenas. Experiments use the same parameters as simulations and experiments of main text Figs. 6, 7 and 8. 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\%

.

Exp. number	Associated video file	Disk classification	Annulus classification	Additional notes
0	2022-07-21_14-16-04.mp4	Disk	Annulus	Disk LED color is orange instead of cyan (early version of the color set), code, parameters, centroids are the same as other experiments.
1	2022-07-21_15-47-12.mp4	Disk	Annulus	-
2	2022-07-22_10-18-21.mp4	Disk	Annulus	-
3	2022-07-22_11-50-10.mp4	Annulus	Annulus	-
4	2022-07-22_13-33-07.mp4	Disk	Disk	-
5	2022-07-22_15-25-54.mp4	Disk	Annulus	-

Table S8: List of Kilobots experimental runs and names of the associated video files. Each experiment involved two arenas: Disk (color code: cyan) and Annulus (color code: violet). Correctly detected arenas are in bold. Note that experiment 0 used the color code “orange” rather than cyan, as we initially intended to use the former to indicate a disk arena. We changed the color code to cyan to ease visualization. However, the results of this experiment remain correct, as they used the same code, parameters and centroids as the other 14 experiments. Raw video files can be found at https://doi.org/10.6084/m9.figshare.31069147

Hearing the shape of an arena with spectral swarm robotics

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