多旋翼静态控制分配的可微可实现性：非冗余完全驱动下的不可能性与冗余驱动下的伪逆障碍
Differential Realizability of Static Control Allocation in Multirotors: An Impossibility under Nonredundant Full Actuation and a Pseudoinverse Obstruction under Redundant Actuation
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中文总结 AI 辅助
本文研究多旋翼静态控制分配的可微可实现性，证明非冗余完全驱动下不存在全局可微精确分配器，冗余驱动下伪逆存在速率奇异性，并提出零空间变形与固定象限构造方法。
中文摘要 AI 辅助
具有双向螺旋桨的多旋翼控制分配通常以带符号推力变量表示，此时力螺旋映射是线性的。然而，从物理转子转速到推力的带符号二次映射在零转速处不是局部微分同胚。本文推导出两个不同的结论。在非冗余完全驱动下，单个螺旋桨反转会移除一个瞬时任务方向；因此，在完整任务空间上不存在全局连续可微的精确静态分配器。在冗余驱动下，物理任务映射可能保持正则，但横向伪逆零交叉需要无界的转子转速导数。我们将微分可实现性定义为执行器输出截面的物理提升的正则性，推导出精确和一阶有效性条件，并区分结构秩损失与分配器引起的速率奇异性。局部零空间变形可修复孤立的伪逆反转，而全局固定象限构造则建立了正则截面的存在性，代价是持续的任务保持内部驱动。
英文摘要
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.
