The Formation Metric

A swarm of agents — satellites, drones, robots — has a configuration: the set of their positions. Two configurations are the same formation if one can be obtained from the other by rotation, translation, or relabeling of agents. The formation space is the quotient: all configurations modulo ambient symmetries and agent permutations.

The paper constructs a metric on this space and proves it has the right mathematical properties for monitoring swarm reconfiguration.

The formation matching metric optimizes a worst-case assignment error over all ambient symmetries and all relabelings. It’s a structured relaxation of Gromov-Hausdorff distance — the standard metric on spaces of metric spaces — specialized to the swarm setting where the ambient space (Euclidean, spherical, toroidal) provides additional structure that Gromov-Hausdorff ignores.

The key property is persistence stability. Composing the formation metric with Vietoris-Rips persistence produces topological signatures — barcodes — that change continuously as the swarm reconfigures. Small changes in formation produce small changes in the signature. This means the signature can monitor reconfiguration in real time without false alarms from noise.

The metric geometry of the formation space is rich. Under compactness and completeness assumptions, the space is compact, complete, and geodesic. It has stratified singularities along collision strata (where agents coincide) and symmetry strata (where the formation has non-trivial stabilizer). These singularities correspond to physically meaningful configurations — collisions and symmetric formations — and the metric respects their geometric structure.

A conditional inverse theorem in a phase-circle model proves the signature is locally bi-Lipschitz to the formation metric. The topological invariant doesn’t just detect changes — it measures them, up to explicit constants.