Global Geometry of Orthogonal Foliations in the Control Allocation of Signed-Quadratic Systems

This work formalizes the differential topology of redundancy resolution for systems governed by signed-quadratic actuation maps. By analyzing the minimally redundant case, the global topology of the continuous fiber bundle defining the nonlinear actuation null-space is established. The distribution orthogonal to these fibers is proven to be globally integrable and governed by an exact logarithmic potential field. This field foliates the actuator space, inducing a structural stratification of all orthants into transverse layers whose combinatorial sizes follow a strictly binomial progression. Within these layers, adjacent orthants are continuously connected via lower-dimensional strata termed reciprocal hinges, while the layers themselves are separated by boundary hyperplanes, or portals, that act as global sections of the fibers. This partition formally distinguishes extremal and transitional layers, which exhibit fundamentally distinct fiber topologies and foliation properties. Through this geometric framework, classical pseudo-linear static allocation strategies are shown to inevitably intersect singular boundary hyperplanes, triggering infinite-derivative kinetic singularities and fragmenting the task space into an exponential number of singularity-separated sectors. In contrast, allocators derived from the orthogonal manifolds yield continuously differentiable global sections with only a linear number of sectors for transversal layers, or can even form a single global diffeomorphism to the task space in the case of the two extremal layers, thus completely avoiding geometric rank-loss and boundary-crossing singularities. These theoretical results directly apply to the control allocation of propeller-driven architectures, including multirotor UAVs, marine, and underwater vehicles.