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.

翻译：本文形式化了由符号二次驱动映射支配的系统冗余解构的微分拓扑。通过分析最小冗余情况，建立了定义非线性驱动零空间的连续纤维丛的全局拓扑。证明了与这些纤维正交的分布全局可积，并由精确对数势场支配。该势场对执行器空间进行分层，诱导出所有卦限的结构化分层，其组合大小遵循严格的二项式递进。在这些分层内部，相邻卦限通过称为倒易铰链的低维流形连续连接，而各层之间由作为纤维全局截面的边界超平面（即门户）分隔。这一划分严格区分了极值层与过渡层，二者呈现出根本不同的纤维拓扑与分层性质。通过该几何框架，经典伪线性静态分配策略被证明不可避免地与奇异边界超平面相交，引发无穷导数运动学奇点，并将任务空间分裂为指数级数量的奇点分隔扇区。相比之下，基于正交流形导出的分配器为过渡层生成仅含线性数量扇区的连续可微全局截面，甚至在两个极值层情况下可形成任务空间的单一全局微分同胚，从而完全避免了几何秩亏与边界穿越奇点。这些理论结果直接适用于螺旋桨驱动构型的控制分配，包括多旋翼无人机、海洋及水下航行器。