The N-5 Scaling Law: Topological Dimensionality Reduction in the Optimal Design of Fully-actuated Multirotors

TL;DR

This paper uncovers the topological structure of optimal rotor configurations in fully-actuated multirotors, revealing a phase transition and a predictable N-5 scaling law that enables continuous reconfiguration while maintaining optimal control.

Contribution

It introduces a topological analysis of the design landscape, discovering a phase transition and the N-5 Scaling Law that characterize optimal configurations for various multirotor geometries.

Findings

Optimal configurations form discrete points or continuous curves depending on chassis regularity.

The N-5 Scaling Law predicts the number of topological branches for N <= 10.

Reconfiguration along these branches preserves optimal isotropic control authority.

Abstract

The geometric design of fully-actuated and omnidirectional N-rotor aerial vehicles is conventionally formulated as a parametric optimization problem, seeking a single optimal set of N orientations within a fixed architectural family. This work departs from that paradigm to investigate the intrinsic topological structure of the optimization landscape itself. We formulate the design problem on the product manifold of Projective Lines \RP^2^N, fixing the rotor positions to the vertices of polyhedral chassis while varying their lines of action. By minimizing a coordinate-invariant Log-Volume isotropy metric, we reveal that the topology of the global optima is governed strictly by the symmetry of the chassis. For generic (irregular) vertex arrangements, the solutions appear as a discrete set of isolated points. However, as the chassis geometry approaches regularity, the solution space undergoes a critical phase transition, collapsing onto an N-dimensional Torus of the lines tangent at the vertexes to the circumscribing sphere of the chassis, and subsequently reducing to continuous 1-dimensional curves driven by Affine Phase Locking. We synthesize these observations into the N-5 Scaling Law: an empirical relationship holding for all examined regular planar polygons and Platonic solids (N <= 10), where the space of optimal configurations consists of K=N-5 disconnected 1D topological branches. We demonstrate that these locking patterns correspond to a sequence of admissible Star Polygons {N/q}, allowing for the exact prediction of optimal phases for arbitrary N. Crucially, this topology reveals a design redundancy that enables optimality-preserving morphing: the vehicle can continuously reconfigure along these branches while preserving optimal isotropic control authority.