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

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.