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

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.

Figures (7)

• Figure 1: The continuous fiber bundle defining the actuation null-space in the native kinetic space $\mathcal{V}$. The central fiber (red, thick) anchors the longitudinal flow. Adjacent fibers exhibit the expansion near the origin before asymptotically converging to and aligning with the central fiber.
• Figure 2: Level sets of the global logarithmic potential field defining the orthogonal manifolds. The top row utilizes the $n=2$ asymmetric allocation map from Fig. \ref{['fig:task_fibers']}b, showing the family of curves for $C \in \{-10, -4, 0, 3 5, 6.5\}$ in the extremal Quadrant I and $C \in \{-8, -4, -2, 1.5, 7\}$ in the transitional Quadrant II. The bottom row utilizes the $n=3$ standard allocation from Fig. \ref{['fig:task_fibers']}c, illustrating the surfaces for $C \in \{ -5,-1.5, 0.5, 2.5, 5\}$ in the transitional $(+,+,+)$ octant and $C \in \{ -4,-0.5, 1.5, 3, 4\}$ in the extremal $(+,-,+)$ octant. Thick colored boundaries denote the limit cases $C \to \pm\infty$, highlighting the difference between entry and exit portals in the transitional and extremal orthants. When present, background dashed lines indicate the fibers.
• Figure 3: Global orthogonal sections and layer partitioning in a low dimension $n=2$ kinetic space. The phase space is stratified into distinct topological layers ($\mathcal{L}_0, \mathcal{L}_1, \mathcal{L}_2$), indicated by the background shading. In $\mathcal{L}_1$ the fibers (dashed gray) flow strictly from the entry portals (orange) to the exit portals (purple). The global sections $\mathcal{M}^{(1)}_C$ act as complete orthogonal transversal barriers, seamlessly gluing the local orthant petals across the reciprocal hinge (the point $(0,0)$ for $n=2$) to form a continuous, $C^1$ smooth mapping of the task space.
• Figure 4: Global stratification and orthogonal sections of a 3D kinetic space ($n=3$). (Top Row) The sequential layer partition $\mathcal{L}_0 \to \mathcal{L}_3$. Exit portals are indicated in purple and entry portals in orange; folds are represented by blue semi-axes and hinges by gold semi-axes. The extremal layers are bounded exclusively by either exit portals (for the negative extremal orthant $\mathcal{L}_0 = \{\mathcal{O}^-\}$) or entry portals (for the positive extremal orthant $\mathcal{L}_3 = \{\mathcal{O}^+\}$). In both extremal cases, the bounding axes consist entirely of folds. The transitional layers $\mathcal{L}_1$ and $\mathcal{L}_2$ contain intermediate orthants geometrically glued by reciprocal hinges. The structural designations of the folds and hinges invert when transitioning from $\mathcal{L}_1$ to $\mathcal{L}_2$. Representative reconfiguration fibers (black curves) strictly traverse this sequence. Intersections with the spatial boundaries are rigidly categorized: fibers pierce entry portals (orange faces) at entry nodes (orange markers) and leave via exit portals (purple faces) at exit nodes (purple markers). (Bottom Row) The corresponding global orthogonal sections $\mathcal{M}^{(l)}_C$ foliating each layer, evaluated at level-set constants $C \in \{-3, 0, 3\}$ (translucent colored surfaces). Within the transitional layers $\mathcal{L}_1$ and $\mathcal{L}_2$, the localized orthogonal sections converge continuously at the reciprocal portals and bend toward the folds, which subsequently evolve into the hinges of the next layer. Across these transitional layers, the component orthogonal manifolds are structurally glued at the hinges to maintain $C^1$ smoothness, establishing the global section as a complete, smooth transversal barrier to the fiber flow.
• Figure 5: Sequential progression of the global orthogonal sections already displayed in Fig. \ref{['fig:3d_cases']} but this time displayed separately for better visualization. Displayed in reverse order from the positive external orthant ($\mathcal{L}_3$, top left) back to the negative extremal orthant ($\mathcal{L}_0$, bottom row). Passing through all the intermediate layers and the portals between them.

Theorems & Definitions (60)

• Definition 1: Kinetic space and state, Task, and Signed-quadratic Actuation Map
• Remark 1: Physical Interpretation
• Definition 2: Transformed State and Fiber Parameterization
• Remark 2
• Definition 3: Central Fiber and Extremal Orthants
• Theorem 1: Asymptotic Convergence of Fibers
• proof
• Lemma 1: Fiber Tangent Space
• proof
• Proposition 1: Asymptotic Tangent Alignment