A model of collective behavior based purely on vision

Abstract

Classical models of collective behavior often take a "bird's-eye perspective," assuming that individuals have access to social information that is not directly available (e.g., the behavior of individuals outside of their field of view). Despite the explanatory success of those models, it is now thought that a better understanding needs to incorporate the perception of the individual, i.e., how internal and external information are acquired and processed. In particular, vision has appeared to be a central feature to gather external information and influence the collective organization of the group. Here, we show that a vision-based model of collective behavior is sufficient to generate organized collective behavior in the absence of spatial representation and collision. Our work suggests a different approach for the development of purely vision-based autonomous swarm robotic systems and formulates a mathematical framework for exploration of perception-based interactions and how they differ from physical ones.

Fig. 1. Geometry of the system.

(A) A set of disks with diameter BL (body length) is considered. Each disk is propelled in the direction ψ_i with a velocity v_i(t). (B) A co-moving referential can be defined following the movement of the disk j. This referential is centered on the position of the disk, (x_j,y_j), and oriented so that the vertical axis is aligned with the direction ψ_j. The position of other objects can be recovered through their left-right (d_LR) and front-back (d_FB) distances relative to the disk j. ϕ represents the swiping angle. (C) Representation of the visible field of the pink disk through ray casting. The position of the eye is considered to be at the center of the disk with a fully circular point of view, i.e., no blind angles. (D) The projection of the visual field in two dimensions (2D) is given by a 1D function. On top, objects can be represented by their colors. However, on the bottom part, a binary visual field is given. It is not possible to distinguish individuals.

Fig. 2. Effects of the terms of Eq. 3 on a focal observer according to the relative position of another disk.

(A and B) The white disk is looking straight at the blue disk with an eye positioned in the center. When the object is far (A), the subtended angle of the object on the projection of the visual field, V (ϕ), is smaller than when the object is close (B). When integrating with a cosine function, the subtended angle of the object (orange) results in a larger integration for a closer object, while the edges (purple) sum larger elements of the cosine when the object is far. (C) For different relative positions between both disks (Fig. 1B), the subtended angle of the object produces a short-range interaction, while the edges create a long-range interaction. The difference of those two terms can create a short-range repulsion (with the subtended angle of the object)/long-range attraction (with the edges of the object).

Fig. 3. Different collective behaviors observed in the model for N = 50 individuals.

Unless stated otherwise, ${\mathrm{\alpha }}_1^{-1}={\mathrm{\beta }}_1^{-1}=12.5\text{BL}$. (A) Polarized on a line perpendicular to the movement (α₀ = 0.2 and β₀ = 0.01; movie S1). (B) Polarized in a circular shape (α₀ = 0.5 and β₀ = 0.1; movie S2). (C) Rotating. No preferred direction is chosen here, so individuals are turning in both directions at the same time (α₀ = 0.1 and β₀ = 0.02; movie S3). (D) Swarm behavior where individuals are moving freely in the swarm (α₀ = 0.5 and β₀ = 1; movie S4). (E) Crystal-like configuration (α₀ = 0.1 and β₀ = 10; movie S5). (F) Tube-like configuration (${\mathrm{\alpha }}_1^{-1}={\mathrm{\beta }}_1^{-1}=5\text{BL}$, α₀ = 0.5, and β₀ = 1; movie S6).

Fig. 4. A qualitative phase diagram describing the modes of collective movement for the lowest-order vision-based model in the (α0,β0) parameter plane.

In the zone with dashed lines, collisions are always observed. Outside of this zone, collisions can be avoided.

Fig. 5. Results of the simulation.

(Top to bottom) The average closest neighbor distance, the polarization of the swarm, and the minimal distance observed in the simulations (42) as function of α₀ and β₀. For different numbers of individuals (from left to right), N = 2, 10, 20, 50, and 100, and for two different values of the equilibrium distance, α₁ = β₁ = 25BL (top row) and α₁ = β₁ = 5BL (bottom row) (BL corresponds here to the diameter of the disk). For the minimal distance, dashed lines represent distances that are less than one BL, so the objects are colliding. Together with Fig. 4, those results can be used as a map to navigate the collection of video simulations of the model (42). The phase diagram gives a global overview, while this figure provides a more detailed, quantitative view on the system behavior. The letters a, b, c, d, e, and f indicate the parameters of the corresponding panels in Fig. 3.

Fig. 6. Collective movements in 3D.

(A) A set of spheres with diameter BL is considered. Each sphere is propelled with a velocity v_i(t) = vψ_i(t)eψ(t) + v_zi(t)_ez. (B) A set of colored spheres is randomly distributed in space. The point of view of the red sphere is considered here with an idealized omnidirectional field pointing inside the image. (C) The projection of the visual field in 3D is given by a 2D surface. On top, objects can be represented by their colors. However, on the bottom part, a binary visual field is given. It is not possible to distinguish individuals. Because of the spherical nature of the projection, objects seem more deformed when they are further away from the horizon (θ = 0). (D) Top to bottom: The average closest neighbor distance and the polarization of the swarm and the minimal distance observed in the simulations (42) as function of λ₁. For different numbers of individuals, from left to right, N = 2, 10, 20, 50, and 100. α₁ = β₁ = 0.1, α₀ = 5.0, β₀ = 2.0, and λ₀ = 10.0.