Global Geometry of Orthogonal Foliations of Signed-Quadratic Systems

TL;DR

This paper develops a comprehensive topological framework for understanding the geometry of orthogonal foliations in signed-quadratic systems, ensuring smooth inverse mappings and singularity avoidance.

Contribution

It introduces a global topological classification of signed-quadratic actuation systems, proving the existence of smooth right-inverses that avoid singularities.

Findings

The orthogonal distribution is globally integrable and governed by a logarithmic potential.

The task space is stratified into layers with binomially distributed sizes.

Extremal orthants form a diffeomorphism to the task space, enabling singularity-free control.

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. Exploiting this geometric framework, we prove that the orthogonal manifolds within the extremal orthants form a global diffeomorphism to the entire unbounded task space. This establishes the theoretical existence of globally smooth right-inverses that permanently confine the system to a single orthant, guaranteeing the absolute avoidance of kinematic singularities. While motivated by the physical actuation maps of multirotor and marine vehicles, the results provide a strictly foundational topological classification of signed-quadratic surjective systems.