A top-down view of the same distributed law as the 3D model: every robot only needs to know its own Voronoi cell — the region of the arena closer to it than to any other robot — and steers toward that cell's weighted centroid. No robot needs the full swarm's state or a central planner.
cell(i) = { q : ‖q−pᵢ‖ ≤ ‖q−pⱼ‖ ∀j }
centroid(i) = ∫∫_cell(i) q·φ(q) dq / ∫∫_cell(i) φ(q) dq
ṗᵢ = k · ( centroid(i) − pᵢ )
H = Σᵢ ∫∫_cell(i) ‖q−pᵢ‖² φ(q) dq (lower = better covered)
- φ(q) — the importance/density field being covered; switch to "Hotspots" and the swarm visibly pulls extra robots toward the bright zones.
- Voronoi coverage — this Lloyd's-algorithm-style law drives H downhill every step, settling into a centroidal Voronoi tessellation.
- Naive spread — robots only repel from nearby neighbors with no notion of φ(q), so they space out evenly regardless of where coverage matters — H stays high and hotspots go under-served.
- Convergence gain — how aggressively robots chase their cell centroid; too high and they oscillate, too low and convergence crawls.