Each robot partitions the arena with the other robots by nearest-distance: every point belongs to whichever robot is closest to it, forming that robot's Voronoi cell. In coverage-control mode, each robot only needs to know its own cell — the standard distributed law drives every robot toward the weighted centroid of its own cell, no robot ever 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; flip to "Hotspots" and the swarm visibly pulls extra robots toward the bright zones, exactly the property a naive scheme lacks.
- Voronoi coverage — this Lloyd's-algorithm-style law provably drives H downhill every step and settles into a centroidal Voronoi tessellation: a near-optimal spatial division of labor with zero global coordination.
- Naive spread — the baseline: robots only repel from nearby neighbors with no notion of φ(q) at all, so they end up roughly evenly spaced regardless of where coverage actually matters — watch H stay high and hotspots go under-served.
- Convergence gain — how aggressively robots chase their cell centroid; too high and they overshoot and oscillate, too low and convergence crawls.
Real-world relevance: this is the same locational-cost gradient law (Cortés, Martínez, Bullo & Karatas — coverage control for mobile sensing networks) behind search-and-rescue drone spread, environmental-monitoring buoy fleets and warehouse-floor sensor placement.