Differential Realizability of Static Control Allocation in Multirotors: An Impossibility under Nonredundant Full Actuation and a Pseudoinverse Obstruction under Redundant Actuation
Authors: Antonio Franchi, Mirko Mizzoni
Organizations: 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
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.
Figures & tables
Actuation and allocation
What can happen at the reversal
The multirotor is fully actuated but has no redundant actuation, regardless of the allocator used
One instantaneous generalized-force direction is necessarily lost.
The multirotor has redundant actuation and uses the standard pseudoinverse allocator
All instantaneous generalized-force directions may remain physically available, yet the commanded propeller rate can become unbounded.
TABLE I: Two limitations hidden by pointwise notions of full actuation at a single-propeller reversal.
Fig. 1: Spaces, differential structure, and section geometry. Panel (a) shows the continuous commutative diagram underlying the problem formulation in Section 2 ; the regularity of its physical lift is characterized in Section 4 . Panel (b) summarizes the structural rank loss caused by a zero actuator state without redundancy and the possible retention of surjectivity when rankA−i=m , as established in Section 3 . Panel (c) contrasts the pseudoinverse obstruction of Section 3 with the realizable crossing geometry of Section 4 , the local deformation of Section 7 , and the global fixed-orthant construction of Section 6 .
Fig. 2: Numerical comparison of the pseudoinverse, local, and fixed-orthant sections: physical rotor-speed command near reversal, peak finite-difference rotor-speed rate under temporal refinement, and normalized actuator-output effort.
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.
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
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.
Antonio Franchi, Barbara Bazzana
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.