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

2026-04-02

AI Summary

This work addresses geometric singularities and task-space fragmentation arising from redundant parametrization in signed quadratic actuation systems. Leveraging differential topology and fiber bundle theory, it analyzes the global structure of the nonlinear actuation nullspace under minimal redundancy and establishes, for the first time, an intrinsic connection between the global integrability of the orthogonal foliation in actuation space and logarithmic potential fields. Building on this insight, the paper constructs a control allocator based on orthogonal manifolds that yields a continuously differentiable global section composed of only a linear number of sectors. In extreme cases, this approach achieves a global diffeomorphism of the task space, thereby completely avoiding the infinite-derivative singularities and boundary rank deficiencies inherent in conventional methods when crossing singular hyperplanes.

Abstract

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.

Problem

• control allocation
• signed-quadratic systems
• kinetic singularities
• task space fragmentation
• actuation redundancy

Innovation

• orthogonal foliations
• signed-quadratic systems
• global integrability
• control allocation
• geometric singularities