Emergent Underactuation in Fully Actuated Multirotors by Minimizing a Homogeneous Allocation Cost

2026-10-03

AI Summary

This study addresses the fundamental question of why underactuated characteristics emerge in the optimal control allocation of fully actuated multirotors when tracking position trajectories alone. To investigate this, we propose a fiber-level infimum approach that establishes a mapping between homogeneous costs and the wrench space. By integrating positive homogeneous norms, pseudoinverse allocation, and landscape analysis along zero-torque directions, this work quantifies the additional cost incurred by non-optimal attitudes and delineates the minimum-cost equivalence classes. Our results demonstrate that fully actuated platforms spontaneously exhibit underactuation-like behaviors during nominal motion, revealing that their principal advantages lie primarily in disturbance rejection and transient wrench generation. These findings provide a novel paradigm for understanding the optimal behavior of fully actuated aerial systems.

Abstract

Full actuation is commonly valued for independent position and attitude tracking. This work establishes a complementary principle: when only a position trajectory is prescribed, minimizing a positively homogeneous allocation cost selects underactuated-like behavior on a fully actuated multirotor. This cost class includes norms and power laws, including standard minimum-norm pseudoinverse allocation. We use fiberwise infimization to transport the actuator-output cost to wrench space, yielding a homogeneous infimal allocation cost. Restriction to unit zero-moment forces defines a directional landscape whose minimizing directions are independent of force magnitude. We show that aligning any fixed minimizing direction with the required world-frame force minimizes instantaneous and integrated zero-moment allocation costs. The resulting body force remains in a fixed one-dimensional subspace without transverse components. We derive an exact optimality-gap decomposition quantifying the additional cost of every other attitude trajectory and characterizing the minimum-cost class. When a disturbance or prescribed-motion change modifies the required-force direction, a fully actuated platform can immediately generate the modified force using transverse authority and reorient toward the new minimum-cost family. A structurally underactuated platform must reorient before generating that force. Thus, homogeneous allocation-cost minimization explains why optimal nominal motion can be underactuated-like, while full actuation remains valuable for disturbance rejection and transient force generation.

Problem

• fully actuated multirotor
• underactuation
• allocation cost minimization
• position trajectory tracking
• disturbance rejection

Innovation

• Fully actuated multirotors
• Homogeneous allocation cost
• Fiberwise infimization
• Optimality-gap decomposition
• Emergent underactuation