Readiness Barrier Functions: Forward-Invariant Control Authority for Overactuated Multirotor Allocation

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.

We develop a task-relative framework for certifying residual wrench authority after hover and contact loading in multirotors with bounded actuators. Using convex geometry, we derive signed margins for prescribed convex reserves, including Euclidean balls and weighted ellipsoids. We obtain computable reserve certificates from actuator-interiority bounds to support slack-maximizing allocation. To preserve the required reserve, we propose command projection onto a tightened feasible set. We then connect the available reserve to structured gain synthesis and certify a local tracking region that respects actuator limits. Within this region, we establish nominal exponential convergence and robust ultimate boundedness. For the prescribed task and morphology family, we show that the optimized octarotor attains a larger margin than the optimized hexarotor at equal total thrust. We evaluate the proposed framework in closed-loop simulations of sustained rigid-wall contact using a fixed-geometry octarotor and a variable-tilt quadrotor with matched installed thrust. The quadrotor's local certificate admits a higher normalized push limit for the prescribed task family. In tests beyond the realizability boundaries, we observe the predicted loss of authority through rotor-thrust limits for the octarotor and tilt-servo limits for the quadrotor.

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.