Emergent Underactuation in Fully Actuated Multirotors by Minimizing a Homogeneous Allocation Cost
Authors: Antonio Franchi, Barbara Bazzana
Organizations: Robotics and Mechatronics, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, Enschede, The Netherlands. · Department of Computer, Control and Management Engineering, Sapienza University of Rome, Rome, Italy.
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.
Figures & tables
Fig. 1 : Dependency map of the main results. Each box states the control-theoretic content in accessible terms and identifies the formal result that establishes it. Solid arrows denote formal dependence; dashed arrows denote architectural interpretation or numerical evidence.
Fig. 2 : Synopsis of the actuation and allocation construction. The physical task map factors through the actuator and allocation maps; fiberwise infimization defines the infimal allocation cost and minimum-output map; and the zero-moment embedding followed by restriction to unit force directions defines the directional infimal allocation cost and its minimizing directions.
Fig. 3 : From physical actuation to fiberwise optimization. The propeller-state-to-wrench map factors through the actuator and allocation maps. Minimizing the actuator-output cost over each allocation fiber defines the infimal allocation cost and the minimum-output map.
Fig. 4 : Restriction of the infimal allocation cost to zero-moment wrenches and unit force directions. The resulting directional infimal allocation cost determines the minimum directional cost and the set of minimizing force directions.
c:X→R≥0
Fig. 5 : Complete geometric construction. Fiberwise infimization transports the actuator-output cost from the actuator-output space to the wrench space. Restriction to zero-moment wrenches and unit force directions yields the directional infimal allocation cost. Positive homogeneity makes every spherical section of the zero-moment force cone a radially scaled copy of the unit section, and optimal actuator outputs are recovered by lifting the minimizing wrenches through the minimum-output map.
Fig. 6 : Planar comparison of the three actuation realizations. The dotted vertical lines delimit the external-force interval. The generated-force cost reflects the closed-loop force actually applied, whereas the cumulative cost evaluates the common force required by the prescribed motion and therefore isolates the allocation effect of attitude selection.
Allocation schemes that greedily maximize a readiness metric over the actuator fiber bundle of an overactuated multirotor produce commands that jump between disconnected optimal strata, demanding actuator rates no motor can deliver; effort-minimizing schemes are continuous but cannot guarantee that wrench-rate authority stays above any certified level. We reconcile the two by treating authority as a forward-invariant quantity: a control barrier function on the log-determinant of the drag-aware actuator-authority co-metric, enforced at torque level by a quadratic program in the allocation null space. A single design inequality renders the certified set compact and strictly interior to the actuator box, with the readiness cost of any rotor deactivation given in closed form as ln(n/(n−m)) for symmetric designs. Tracking is sacrificed only through an explicit alignment ratio, with wrench error bounded by O(ρ−1/2) and a robust variant handles motor-parameter uncertainty with a closed-form floor shift independent of the airframe matrix. On a hexarotor and a fully-actuated octorotor the closed-form gap matches simulation to machine precision; in the authority-scarce regime greedy maximization violates the certified floor and commits wrench errors up to eighty times larger than the proposed filter, which holds invariance of the certified set at negligible tracking cost.
Control allocation for multirotors with bidirectional propellers is commonly formulated in signed-thrust variables, where the wrench map is linear. The signed-quadratic map from physical rotor speed to thrust, however, is not a local diffeomorphism at zero speed. This work derives two distinct consequences. Under nonredundant full actuation, a single-propeller reversal removes one instantaneous task direction; hence, no global continuously differentiable exact static allocator exists over the complete task space. Under redundant actuation, the physical task map may remain regular, yet a transverse pseudoinverse zero crossing requires an unbounded rotor-speed derivative. We define differential realizability as regularity of the physical lift of an actuator-output section, derive exact and first-order validity conditions, and distinguish structural rank loss from an allocator- induced rate singularity. A local nullspace deformation repairs isolated pseudoinverse reversals, while a global fixed-orthant construction establishes existence of regular sections at the cost of persistent task-preserving internal actuation.
Antonio Franchi, Mirko Mizzoni
Robotics and Mechatronics Department, Electrical Engineering, Mathematics, and Computer Science (EEMCS) Faculty, University of Twente, 7500 AE Enschede, The Netherlands · Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy
Fully-actuated multirotor aerial vehicles must not only track nominal wrenches but retain the "readiness" to modulate them rapidly under disturbances. Classical effort-minimizing allocators ignore this dynamic limit, whereas maximizing readiness leads to topologically disconnected optimal sheets demanding physically impossible actuator rates. Enforcing a readiness safety floor on fixed-geometry symmetric platforms further encounters a zero-sum degeneracy: motor-speed redistribution cannot improve authority without conceding wrench tracking. This paper uses active morphology to break the degeneracy, treating servo tilt as a geometric resource supplying authority-recovery directions unavailable to static rotors. We construct a configuration-dependent, motor-only readiness certificate - the log-volume of the reachable wrench-rate set - that explicitly excludes servo capacity, preventing a "ghost capacity fallacy" in which the certificate would falsely credit slow mechanical kinematic limits instead of collapsing accurately at motor saturation. The certificate is enforced as a Control Barrier Function (CBF) within a Unified Physical-Command Quadratic Program acting on motor torques and servo setpoints. Closed-loop simulations of an articulated octorotor under severe gust disturbances show classical allocators diverging and uncertified articulated allocators violating the safety floor, while the proposed CBF filter bounds the system state and preserves vehicle authority.
Giuseppe Silano, Martin Saska
Department of Power Generation Technologies and Materials, Ricerca sul Sistema Energetico (RSE) S.p.A., 20134 Milan, Italy · Department of Cybernetics, Czech Technical University in Prague, 12135 Prague, Czech Republic