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

Abstract

We investigate the topological structure of the optimal actuation landscape for fully-actuated N-rotor aerial vehicles. By formulating the design problem on the 2N-dimensional product manifold of projective lines (RP^2)^N and minimizing a rotation-invariant Log-Volume isotropy metric, we map how optimal rotor orientations evolve across diverse polyhedral chassis. The results establish that global optimality is strictly bounded by geometric symmetry. While irregular chassis yield discrete, isolated optimal configurations, regular geometries induce a structural phase transition: the optimal space initially collapses onto an N-dimensional tangent torus, then systematically reduces to continuous configurations governed by affine phase coordination. These collapses define the "N-5 Scaling Law." For N <=7, the optimal landscape fundamentally forms exactly K= N-5 disconnected 1D closed loops. For N >=8, these 1D trajectories expand into core backbones embedded within multi-dimensional flat optimal hypersurfaces. Furthermore, for regular planar geometries, we theoretically unify these trajectories by demonstrating a strict geometric isomorphism to star polygons {N/q}(2 <q<N-2), yielding exact analytical predictions of the N-5-branched optimal manifolds for arbitrary N. By shifting the classical framework from single and static optimal points to continuous optimal manifolds, this work establishes the mathematical foundations for optimality-preserving morphing. The identified topological branches provide continuous null spaces, introducing to the field the concept of design redundancy, and allowing symmetric multi-rotor systems to dynamically reconfigure their thrust vectors without degrading their perfectly isotropic control authority.

Accepted for publication, final version before production