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

TL;DR

This work reframes the design of fully actuated MRAVs as a global topological problem, modeling rotor lines of action on the product manifold $(\mathbb{R}P^2)^N$ and optimizing a smooth Log-Volume objective to maximize wrench isotropy. Through a three-phase methodology—massive multi-start optimization, abductive topological extraction, and theoretical unification—the authors reveal a topological phase transition: irregular chassis yield discrete optima, while regular chassis exhibit a collapse onto an $N$-dimensional Tangent Torus, further organizing into exactly $K=N-5$ 1D branches governed by affine phase locking. They establish a Star Polygon Isomorphism that analytically predicts optimal phase patterns for arbitrary $N$, tying the solution structure to harmonic geometric resonances. The resulting N-5 law and tangent-torus framework imply design redundancy and morphing capabilities that preserve optimal isotropy, offering a principled pathway for rapid, topology-aware synthesis of robust, reconfigurable MRAVs. The work also provides an open-source software suite to reproduce the results and explore higher-order regular geometries, with future directions including rigorous group-theoretic proofs and robustness analyses under symmetry imperfectioins.

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.

Figures (21)

• Figure 1: Visual dictionary of representative chassis. The designs range from planar regular polygons (Top Row) to complex 3D polyhedra (Bottom Rows). Each sub-figure displays the chassis geometry (grey faces and black edges) and the points of application of the propeller forces (orange vertices). The labels indicate the unique Identifier (ID) used throughout the paper and the number of rotors ($N$). See Table \ref{['tab:chassis_zoo']} for full specifications.
• Figure 2: Geometric construction of the $\mathbb{R}P^2$ Disc Model.(Left) A 3D view of the upper hemisphere of $S^2$ and the equatorial disc. A rotor's line of action (dashed line) passes through the origin, intersects the hemisphere at a red dot, and projects orthographically to a blue square on the disc. (Right) The resulting 2D disc representation. Topological Loops: We also visualize a full rotation of a line about a fixed axis (effectively a plane of lines passing through the origin). On the hemisphere, this traces an orange great semi-circle connecting two antipodal black dots. When projected onto the disc, this forms a purple curve. Geometrically, this curve is a semi-ellipse (degenerating to a straight diameter or a circular arc depending on the axis tilt). Because the antipodal boundary points are topologically identified, this purple curve connects back to itself, forming a continuous closed loop in $\mathbb{R}P^2$.
• Figure 3: The disc dashboard view of $\mathcal{M}=(\mathbb{R}P^2)^N$: a graphical model of the manifold of all force directions for two representative chassis. Configuration spaces for a regular hexagon (Top-Left) and an octahedron (Top-Right), where $N=6$. Each chassis vertex is equipped with a local copy of the $\mathbb{R}P^2$ disc model defined in Fig. \ref{['fig:disc_concept']}, representing the manifold of possible force directions at that specific rotor location. Generic Configuration: The red markers (1--6) represent the projections of a generic set of rotor directions (analogous to the blue square in Fig. \ref{['fig:disc_concept']}). The bottom sub-figures provide a dashboard view of these six discs. The Tangent Torus: To visualize the subspace of tangential forces lines represented by ${^\mathcal{C}\mathbb{T}}^N_\perp$, we illustrate the intersection of the local upper hemispheres with planes tangent to the sphere centered at the chassis origin and passing through the vertices of the chassis. These intersections project as semi-ellipses (orange curves) on the discs, geometrically identical to the purple loops described in Fig. \ref{['fig:disc_concept']}. A configuration on $\mathcal{M}$, e.g., a solution of \ref{['eq:optimization']}, lies on the Tangent Torus if all the $N$ markers in each disc lie on their respective orange curves.
• Figure 4: Stochastic initialization of the search space for an $N=10$ chassis. Each disc of the dashboard represents the projection on the local orientation manifold $\mathbb{R}P^2_i$ of the $i$-th rotor. The scatter plots depict the $2$-dimensional projections on the discs of $10^3$ random starting $2N$-dimensional configurations sampled uniformly across the product manifold $\mathcal{M} = (\mathbb{R}P^2)^N$. The uniform coverage confirms a state of maximal configuration entropy, ensuring an unbiased initialization of the global optimization landscape.
• Figure 5: Sensitivity Analysis of Isotropy ($\kappa$) and Strength ($\sigma_{min}$) with respect to the characteristic length ratio $L_c / R$. Results are shown for two representative chassis: a regular polygon ($N=12$, top) and a dodecahedron ($N=20$, bottom). Left: The Condition Number is minimized exactly at the unit ratio ($L_c = R$), indicating maximal balance between force and moment subspaces. Right: The Minimum Singular Value is maximized and stable for $L_c \le R$, but degrades for $L_c > R$ as the system becomes moment-limited. The vertical dashed line indicates the nominal design choice $L_c=R$.

Theorems & Definitions (4)

• Definition 1: The Chassis
• Definition 2: The Tangent Torus
• Definition 3: Affine Phase Locking
• Conjecture 1: Resolution Limit