Vision-based collective motion: A locust-inspired reductionist model

Naturally occurring collective motion is a fascinating phenomenon in which swarming individuals aggregate and coordinate their motion. Many theoretical models of swarming assume idealized, perfect perceptual capabilities, and ignore the underlying perception processes, particularly for agents relying on visual perception. Specifically, biological vision in many swarming animals, such as locusts, utilizes monocular non-stereoscopic vision, which prevents perfect acquisition of distances and velocities. Moreover, swarming peers can visually occlude each other, further introducing estimation errors. In this study, we explore necessary conditions for the emergence of ordered collective motion under restricted conditions, using non-stereoscopic, monocular vision. We present a model of vision-based collective motion for locust-like agents: elongated shape, omni-directional visual sensor parallel to the horizontal plane, and lacking stereoscopic depth perception. The model addresses (i) the non-stereoscopic estimation of distance and velocity, (ii) the presence of occlusions in the visual field. We consider and compare three strategies that an agent may use to interpret partially-occluded visual information at the cost of the computational complexity required for the visual perception processes. Computer-simulated experiments conducted in various geometrical environments (toroidal, corridor, and ring-shaped arenas) demonstrate that the models can result in an ordered or near-ordered state. At the same time, they differ in the rate at which order is achieved. Moreover, the results are sensitive to the elongation of the agents. Experiments in geometrically constrained environments reveal differences between the models and elucidate possible tradeoffs in using them to control swarming agents. These suggest avenues for further study in biology and robotics.


Measurement of Distance to Neighbors in Monocular, Non-stereoscopic Vision
In Fig A, the computation of distance from the subtended angle is shown for idealized circular-shaped agents.It can be seen that for circles, there exists a one-to-one relationship, given by r = d 2 sin 0.5α between the subtended angle and the distance.There is no ambiguity; hence the distance calculation is exact in the circular case.We know, from Eq. ( 4), that by differentiating Eq. ( 4) with respect to time t, we see (Eq. ( 6)) that where α denotes the time derivative of the subtended angle.
Expressing d = 2r j tan ( αj 2 ) from Eq. ( 4) and substituting d into Eq.( 6) results in the following derivation of the radial velocity v j,r , which is what appears in Eq. ( 7).
We experimentally set the value of N in different arenas, to values that proved informative in the sense that they highlighted and clarified the differences between different strategies or other parameters.Below, we highlight the procedure and results used to set the values of N used in the torus arena.For other arenas, a similar experimental analysis was carried out.
We begin with the values that have already been set for the rest of the parameters: R = 3[BL], η = 0.01, length-to-width ratio set at 3. We then vary N .Figs Ca-Cb, show the results of 50 independent trials.For small N sizes, convergence to an ordered state is slow (at best) and probably nonexistent.In the context of range-limited vision (R = 3[BL]), the sparse density undoubtedly inhibits overall convergence to ordered flocking.Increasing N leads to a higher long-term order parameter (Fig Cb ), though clearly the rate of convergence differs (Fig Ca).Based on these results, we typically use population sizes N = 60, 120, 180 in the experiments in the torus arena.
Continuing from the aforementioned analysis, the choice of N = 100 for our principal experiments is further justified by the observed trends in Figs S3(a) and S3(b).These figures underscore a saturation effect in the order parameter beyond N = 100, where increasing the population size yields negligible improvements in collective behavior fidelity.This plateau suggests that N = 100 is a representative value for dense swarm simulations, providing a realistic portrayal of swarming dynamics without incurring disproportionate computational costs.Therefore, N = 100 was selected as it aligns with our dual objectives of accurately capturing the emergent properties of swarming behavior and maintaining computational feasibility within our simulation framework.

Influence of the Length-to-Width Ratio
We examine the influence of elongation on the emergence of order in flocking.In nature, nymph lengths vary as they grow to adulthood.A question also arises (when discussing the visual perception of the locust) regarding the inclusion of out-stretching legs in the perceived image.Based on our own measurements of the locust in our laboratory, we have chosen a body length-to-width ratio of 3 (i.e., length is three times the body width) as a baseline, which was used in the experiments reported above.
In addition, we sought to examine whether these settings influence the results.In particular, different models exist for vision-based flocking, for the case where agents are circular [51,56,58].If the models introduced in this study are insensitive to the length-to-width ratio, then perhaps these other models could be just as useful in informing our understanding of how vision-based flocking may work in locusts (or other species that are elongated).
We, therefore, experimented with other ratios: a ratio of 1:1 (agents are perfect square), a 3:1 ratio, and a ratio of 6:1.Figs D and E report on the results from these experiments, in all environments (with the other parameters set as before N = 100, R = 3, η = 0.01, etc.).In all, we tested the principal model, as well as all three occlusion-handling strategies.As before, we conducted 50 independent trials in each setting and presented the means and standard errors.The figures show that the evolution of order, over time, is greatly influenced by the body length-to-width ratio.This is generally true for all the models; thus, we conclude that the body length ratio is an important factor in the convergence rate.Qualitatively, we note that higher ratios (elongated morphology) improve the rate of order increase in both the torus and corridor arenas.We believe these less-constrained arenas are closer to natural environments than the small ring-shaped arena (where greater elongation reduces--possibly eliminates-the rate of order increases over time.

Figure A :Figure B :
Figure A: Exact distance r computation from subtended angle α for circular morphology.Assumption of a circular (or spherical in 3D) shape of the agents, enables precise computation of distance r to a neighbor while employing the non-stereoscopic visual parameter of subtended angle α.The small circle on the lower left depicts the focal agent's visual sensor and the large circle on the right depicts a circular neighboring agent.Green lines represent the extreme rays toward the neighbor, as seen by the focal agent.The angle between the radius and the tangent extreme ray is always 90°by geometrical definition.Therefore, as shown r = d 2 sin 0.5α where d is the circle's diameter and α the subtended angle measured.

Figure
Figure C: (a) Time-dependent and (b) long-term sensitivity analysis for visual range N , in the torus arena.Means and standard errors are shown for 50 trials.
Fig D shows the order parameter evolving over time in various settings.The subfigures are arranged by columns (different ratios) and rows (different arenas).The left column of the figures (Figs (a), (d), (g)) shows the results for ratio 1:1, the middle column (Figs (b), (e), (h)) show the results for the baseline ratio 3:1 used in the main group of experiments as reported above, and the rightmost column of figures (Figs (c), (f), (i)) show the results for ratio 6:1.The top row shows the results from the toroidal arena, the middle row shows results for the narrow corridor arena, and the bottom row shows the results for the narrow ring arena.
Fig E displays the long-term order parameter (ϕ at t = 3000) across different arenas and length-to-width ratios, offering a snapshot of Fig D at t = 3000.The torus arena shows more consistent model performance across various ratios, unlike other arenas.Notably, the COMPLID model generally excels in different settings, though this requires further study.As with Fig D, these findings underscore the importance of considering the length-to-width ratio in model evaluations for specific species.