Dr. Antonio Franchi, a researcher at the Italian Institute of Technology (IIT), has published a study that explores the optimal design of fully-actuated multirotor aerial vehicles, such as drones. The research, titled “The N-5 Scaling Law: Topological Dimensionality Reduction in the Optimal Design of Fully-actuated Multirotors,” was published in the journal IEEE Transactions on Robotics.
In this study, Franchi departs from conventional design methods that focus on finding a single optimal set of rotor orientations within a fixed architectural family. Instead, he investigates the intrinsic topological structure of the optimization landscape itself. By doing so, he reveals that the topology of the global optima is governed strictly by the symmetry of the chassis.
Franchi formulates the design problem on the product manifold of Projective Lines, 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, he discovers that 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.
This transition collapses onto an N-dimensional Torus of the lines tangent at the vertexes to the circumscribing sphere of the chassis, and subsequently reduces to continuous 1-dimensional curves driven by Affine Phase Locking. Franchi synthesizes 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. The research demonstrates 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. For the energy industry, particularly in the realm of drone technology and aerial vehicles, this research offers practical applications. Understanding the optimal design and configuration of multirotor aerial vehicles can lead to more efficient and effective energy use, improved performance, and enhanced versatility in various applications, such as renewable energy inspections, environmental monitoring, and disaster response. By leveraging the N-5 Scaling Law, designers and engineers can create drones that are not only optimally configured but also capable of morphing and adapting to different operational requirements without compromising performance. This article is based on research available at arXiv.