The N-5 Scaling Law: Topological Dimensionality Reduction in the Optimal Design of Fully-actuated Multirotors
arXiv:2512.23619v3 Announce Type: replace 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 sym
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arXiv:2512.23619v3 Announce Type: replace
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 Originally published at arxiv.org.Source
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Source: https://arxiv.org/abs/2512.23619