Global Geometry of Orthogonal Foliations of Signed-Quadratic Systems

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.

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. An explicit alignment ratio states when tracking can stay exact, otherwise the wrench error is bounded by $\mathcal{O}(ρ^{-1/2})$, 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. Over $300$ randomized missions, greedy maximization violates the certified floor in up to $51\%$ of authority-scarce missions, while the proposed filter never leaves the certified set and, in the hardest moment reversal, commits a $79\times$ smaller wrench error. The robust variant holds the floor on all $150$ perturbed plants. A high-fidelity simulation with measured propeller and motor data shows that the floor holds under unmodeled aerodynamics, electrical limits, and delay, and where it stops applying.

Input--output feedback linearization fails at decoupling singularities, where the decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded. This paper develops a singularity-robust trajectory-tracking controller for square nonlinear control-affine systems that tracks through isolated decoupling singularities with bounded control. The method recasts tracking as real-time arc-length homotopy continuation, equivalently a continuous-time Newton/Davidenko flow, and replaces the inverse decoupling matrix by the least-norm Moore--Penrose solution of an augmented matrix $A=[Λ\mid b]$, where $b$ is the homotopy direction. A transversality condition $w^T b \ne 0$, with $w$ in the left null space of the decoupling matrix, keeps the augmented matrix full row rank through a generic rank-one loss. The resulting flow agrees with feedback linearization away from the singular set, tracks with $O(1/k)$ error, and re-locks after each crossing. The theory also characterizes the reflection-versus-branch-crossing dichotomy at Whitney folds and relates the reflection case to a Filippov sliding mode. Extensions cover dynamic relative-degree-one minimum-phase systems and arbitrary relative degree via filtered-error reduction. Simulations include a redundant 2-DOF manipulator, relative-degree-one and relative-degree-two plants, and a dual-active-bridge series-resonant DC/DC converter, where the method performs bounded inversion across buck/boost and resonance singularities while preserving zero-voltage soft switching.