Aero-Promptness: Drag-Aware Aerodynamic Manipulability for Propeller-driven Vehicles

Aero-Promptness: Drag-Aware Aerodynamic Manipulability for Propeller-driven Vehicles Antonio Franchi1,2 1Robotics and Mechatronics Department, Electrical Engineering, Mathematics, and Computer Science Faculty, University of Twente, The Netherlands. 2Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy. schol@r-franchi.euThe work was partially funded by the European Commission Horizon Europe Framework under project Autoassess (101120732) Abstract This work introduces the Drag-Aware Aerodynamic Manipulability (DAAM), a geometric framework for control allocation in redundant multirotors. By equipping the propeller spin-rate space with a Riemannian metric based on the remaining symmetric acceleration capacity of each motor, the formulation explicitly accounts for motor torque limits and aerodynamic drag. Mapping this metric through the nonlinear thrust law to the generalized force space yields a state-dependent manipulability volume. The log-determinant of this volume acts as a natural barrier function, strictly penalizing drag-induced saturation and low-spin thrust loss. Optimizing this volume along the allocation fibers provides a redundancy resolution strategy inherently invariant to arbitrary coordinate scaling in the generalized-force space. Analytically, we prove that the resulting optimal allocations locally form smooth embedded manifolds, and we geometrically characterize the global jump discontinuities that inevitably arise from physical actuator limits and spin-rate sign transitions. I Introduction The paradigm of aerial robotics has fundamentally shifted from underactuated navigation to fully actuated and redundantly actuated physically interactive multirotors [undef, undefa, undefb]. Early foundational work demonstrated that thrust-vectoring via tilt-rotor architectures enables decoupled position and attitude control for efficient force exertion and robust spatial navigation [undefc, undefd]. This evolved into the formalization of multi-directional and true omnidirectional wrench generation, achieved through both fixed-tilt configurations [undefe, undeff, undefg, undefh] and actively synchronized tilting mechanisms [undefi, undefj, undefk]. Pushing morphological boundaries further, multi-linked aerial platforms now dynamically alter their topologies to adapt to complex manipulation tasks [undefl]. As these platforms integrate into dynamic, human-centric environments, modern control frameworks have necessarily expanded to encompass energy-aware interaction policies [undefm] and admittance models for collaborative transportation among multiple agents [undefn]. To govern complex actuation redundancy, manipulability has served as a foundational concept in robotics for four decades. Following Yoshikawa’s seminal formulation of kinematic and dynamic manipulability [undefo, undefp], the differential-geometric literature rigorously established that manipulability indices depend intrinsically on the choice of Riemannian metrics defining the domain and codomain [undefq, undefr, undefs, undeft]. These geometric insights have successfully transitioned into real-time control stacks for grounded and legged robots via Jacobian-based task-priority resolution [undefu] and feasible wrench set computations [undefv, undefw]. Furthermore, biological systems inherently exploit forms of manipulability; human motor control dynamically trades metabolic efficiency for kinematic readiness via muscle co-contraction and impedance shaping [undefx, undefy, undefz, undefaa]. Despite this broad interdisciplinary adoption, manipulability in aerial robotics remains largely relegated to a secondary role. It is typically employed merely as an offline design proxy for arm-on-drone architectures [undef, undefa] or appended as a heuristic scalar penalty in redundancy allocation algorithms [undefab, undefac]. Crucially, the literature lacks a rigorous geometric formulation of multirotor manipulability that explicitly maps the mathematical concept of control authority to the fundamental physical constraint governing the system: highly nonlinear propeller aerodynamics. Both rotor thrust and aerodynamic drag torque arise from the acceleration of air particles. This shared physics dictates a quadratic dependence on the propeller spin rate, introducing two competing aerodynamic bottlenecks not present in standard robotic joint actuators. First, at low spin rates, the zero slope of the thrust curve creates a loss of control authority (a thrust-slope degeneracy). Second, at high spin rates, the aerodynamic drag limits the motor’s ability to symmetrically accelerate the propeller, shrinking the available control headroom [undefad, undefae]. TABLE I: Research Dossier: Executive Summary of Context, Methods, and Findings. [gray]0.9 Attribute Description & Specification I. CONTEXT & SCOPE Problem Domain Multirotor Control Allocation. Resolution of actuation redundancy in redundant multirotors mapping n actuator commands to m generalized forces. Core Framework Drag-Aware Aerodynamic Manipulability (DAAM). Shifting the allocation paradigm from heuristic energy minimization to maximizing instantaneous aerodynamic readiness and control authority. Core Question How does explicitly modeling physical limits (torque saturation) and aerodynamic realities (drag-induced acceleration loss) mathematically shape the optimal global solution landscape for redundancy resolution? I. FORMULATION Search Space The allocation fibers FwF_w within the feasible actuator space E¯⊂ℝn E ^n. Objective Function Log-Determinant of the DAAM Metric. Why? It acts as a natural, strictly concave barrier function. It directly penalizes rank deficiency, zero-spin thrust-slope degeneracy, and drag-induced actuator saturation while remaining invariant to task-space coordinate scaling. Assumptions 1. First-Principles Aerodynamics: Standard quadratic thrust and drag models, bidirectional propellers. 2. Rigid Body Dynamics: Standard Euler-Lagrange multirotor formulation. Justification: These models are universally validated in the multirotor literature. The optimization of intrinsic control readiness relies entirely on these steady-state maps; transient aerodynamic unmodeled effects are extrinsic disturbances for the feedback controller to reject. I. KEY DISCOVERIES Stratified Topologies Manifold Structure. The global set of optimal allocations is not a single smooth function but a stratified manifold. Local solutions form smooth embedded sheets αS_α governed by the DAAM metric. Topological Jumps Discontinuous Bifurcations. Continuous task-space demands w​(t)w(t) necessitate topological jumps in the actuator space when crossing fundamental physical boundaries: motor reversals (sign changes), capacity saturation, or authority loss. Antagonistic Tension Origin Avoidance. The DAAM framework mathematically proves that maximizing readiness requires antagonistic rotor tension. The optimizer actively avoids zero-spin states (v=0v=0) to prevent the loss of the aerodynamic thrust derivative. IV. METHODOLOGY Approach Differential Geometry & Mathematical Phenomenology. We analytically derive the local properties of the DAAM metric via the Implicit Function Theorem and numerically characterize the global topological phenomena via constrained fiberwise optimization. Scope Limitation Foundational Theory. This is a rigorous methodological and topological study proposing a universal framework applicable to any n-rotor configuration. Empirical hardware benchmarking or flight-test validation is explicitly out of scope. The universality of the geometric proofs renders instantiation on a specific hardware platform physically redundant for the scientific claims made herein. V. IMPACT & IMPLICATIONS Target Audience Robotic multirotor designers and developers, control theorists, and applied mathematicians focusing on geometric mechanics and allocation algorithms. Algorithmic Shift Replaces hand-tuned, pseudo-inverse heuristics with a physics-grounded, invariant, geometric optimization. Future Capabilities Lays the theoretical groundwork for next-generation highly redundant aerial vehicles requiring extreme agility, high-performance disturbance rejection, fault tolerance, and safe physical interaction, in situations where energy minimization is secondary to immediate control authority. No prior work has natively incorporated both this thrust nonlinearity and aerodynamic drag into a single manipulability framework. A recent preprint [undefaf] initiated this direction by isolating the thrust singularity, demonstrating that antagonistic rotor operation (akin to biological co-contraction) can enlarge dynamic manipulability at the cost of energy. The present work unites thrust nonlinearities and drag limits into a capacity-aware differential geometry framework. We introduce a state-dependent geometry on the actuator space based on the symmetric acceleration capacity (SAC) of the rotors. Directions with little remaining acceleration headroom are penalized, while directions with ample headroom remain inexpensive. By pushing this metric forward through the nonlinear allocation map, we obtain the drag-aware aerodynamic manipulability (DAAM) metric. Its associated log-volume acts as a smooth barrier function that penalizes both drag-induced saturation and thrust-slope degeneracy. We propose using this geometry to resolve multirotor actuation redundancy. For any desired generalized force, we perform a fiberwise selection: searching exclusively along the allocation fiber (the set of all redundant motor commands that produce the target generalized force) to find the configuration that maximizes the DAAM volume. Because this optimization relies on the physical SAC metric, the resulting optimal control allocation is intrinsic—it remains invariant to arbitrary metric or coordinate scaling choices in the generalized-force space. Unlike traditional control allocation schemes that primarily minimize instantaneous power consumption [undefag, undeff, undefah], the proposed objective prioritizes aerodynamic readiness. By maximizing the DAAM volume, the multirotor maintains the multi-directional control authority necessary to rapidly generate changes in the generalized force. This promptness is critical in highly dynamic scenarios, such as reacting to sudden wind gusts, absorbing unexpected impacts, lifting unknown payloads, or generating rapid forces during physical interaction. This metric serves as a complementary strategy to energy-efficient allocation, suited for flight phases demanding high reactivity. Future work will explore dynamically blending this capability-aware cost with traditional energy metrics. I-A Main Contributions This work bridges the gap between differential geometry and aerodynamic control allocation, making three primary contributions: • Drag-Aware Aerodynamic Manipulability (DAAM): We formulate a capacity-aware geometry on the actuator space derived from aerodynamic drag limits, propagating it through the nonlinear thrust law. The resulting DAAM log-volume serves as a unified scalar barrier function capturing drag saturation and thrust-slope degeneracy. • Intrinsic Fiberwise Optimization: We propose a redundancy resolution scheme that minimizes the DAAM log-volume strictly along allocation fibers. We mathematically prove that the resulting minimizers are intrinsic, depending only on vehicle aerodynamics and independent of arbitrary task-space (generalized-force) metrics. • Topological Stratification of Optimal Sections: Under standard rank and curvature conditions, we formally establish the existence and smoothness of local optimal sections mapping generalized forces to actuator commands. Furthermore, we characterize the global stratification of this solution set, revealing how physical boundaries (sign changes, saturation, rank loss) dictate branch continuations and jump bifurcations in the control allocation. The remainder of the paper is organized as follows. Section I formalizes the SAC metric and the induced DAAM geometry. Section I provides canonical visual examples. Section IV develops the fiberwise optimization, proves metric independence, and establishes the local/global topological structure. Section V presents numerical landscapes of the optimal sections in two and three dimensions. Finally, Section VI concludes with an outlook on limitations and future research directions opened by the proposed methodology. I Drag-Aware Aerodynamic Manipulability Consider a redundant propeller-driven robotic vehicle—a rigid body whose generalized forces are produced by rotating propellers (e.g., redundant multirotors). This system is modeled as a (possibly constrained) rigid body with m≤6m≤ 6 unconstrained degrees of freedom, endowed with n>mn>m rigidly anchored motor–propeller units capable of spinning at independent rates. Let v=(v1,…,vn)∈E=ℝnv=(v_1,…,v_n)∈ E=R^n denote the spin rates of the brushless motors driving the propellers about their respective axes, and |v|=(|v1|,…,|vn|)|v|=(|v_1|,…,|v_n|). Let B=ℝmB=R^m denote the coordinate space of the m generalized forces corresponding to the unconstrained degrees of freedom generated by the propeller thrust forces and drag moments acting on the platform. The nonlinear allocation map (see, e.g., [undefai, undefae]) is defined as f:E→B,f​(v)=A​(v⊙|v|),f:E→ B, f(v)=A(v |v|), (1) where A∈ℝm×nA ^m× n is the allocation matrix, and v⊙|v|v |v| denotes the entry-wise (Hadamard) product. We assume f is surjective (equivalently, A has full row rank). Surjectivity of f allows us to call E and B the total space and the base space of the map f, respectively. The total space E is partitioned by the fibers of the map f; given generalized force coordinates w∈Bw∈ B, the allocation fiber corresponding to w is the pre-image of w in E, i.e., Fw:=f−1​(w)=v∈E:f​(v)=w⊂E.F_w:=f^-1(w)=\\,v∈ E\;:\;f(v)=w\,\⊂ E. (2) Where defined, the Jacobian of f is Jf​(v)=∂f∂v=2​A​diag⁡(|v1|,…,|vn|).J_f(v)= ∂ f∂ v=2A\,diag(|v_1|,…,|v_n|). (3) The map f is a submersion almost everywhere in E; Jf​(v)J_f(v) is singular only at critical points where sufficient spin rates viv_i concurrently vanish (e.g., at the origin and other zero-measure subspaces in E). For each propeller–motor unit, consider the standard first-order motor dynamics [undefae]: mi​v˙i=−bi​vi​|vi|+τi,m_i\, v_i=-\,b_i\,v_i\,|v_i|\;+\; _i, (4) where mim_i is the lumped inertia of the motor–propeller unit, bi​vi​|vi|b_i\,v_i|v_i| is the dissipative aerodynamic drag torque due to the propeller spinning in air (with bi>0b_i>0 lumping aerodynamic effects), and τi _i is the motor torque input. Assume the torque input is symmetrically limited as τi∈[−τ¯i,τ¯i] _i∈[- τ_i, τ_i] with τ¯i>0 τ_i>0. For a given spin rate viv_i, we seek the largest zero-centered symmetric interval [−a¯i​(vi),a¯i​(vi)][- a_i(v_i), a_i(v_i)] of accelerations permitted by (4) under the torque limits. This leads to what we call the Symmetric Acceleration Capacity (SAC) a¯i​(vi):=max⁡(0,τ¯i−bi​vi2mi). a_i(v_i)\;:=\; \! (0,\; τ_i-b_i\,v_i^2m_i ). (5) Figure 1 illustrates the SAC as a function of viv_i for various values of mim_i, τ¯i τ_i, and bib_i. All curves attain a maximum value at vi=0v_i=0 equal to τ¯i/mi τ_i/m_i and decrease quadratically to zero at the critical spin rates vi=±τ¯i/biv_i=± τ_i/b_i. The SAC quantifies the amount of acceleration equally available in both directions at a given spin rate. An SAC equal to zero corresponds to the inability to accelerate the propeller in a zero-centered neighborhood, meaning acceleration is impossible in at least one of the two directions. −12-12−10-10−8-8−6-6−4-4−2-22244668810101212551010Spin rate viv_iSAC a¯i​(vi) a_i(v_i)mi=1m_i=1,τ¯i=10 τ_i=10, bi=0.1b_i=0.1mi=1m_i=1,τ¯i=10 τ_i=10, bi=0.2b_i=0.2mi=2m_i=2,τ¯i=10 τ_i=10, bi=0.1b_i=0.1 Figure 1: Symmetric Acceleration Capacity (SAC) of the i-th moto-propeller unit a¯i​(vi) a_i(v_i) as a function of the spin rate viv_i. The curves peak at τ¯i/mi τ_i/m_i when vi=0v_i=0 and drop to zero at the critical spin rate vi=±τ¯i/biv_i=± τ_i/b_i. We use the SAC to define a metric on E that encodes the symmetric acceleration authority remaining along d​vidv_i at each viv_i. Define the n×n× n diagonal matrix W​(v)=diag⁡(a¯1​(v1)2,…,a¯n​(vn)2). W(v)=diag ( a_1(v_1)^2,…, a_n(v_n)^2 ). (6) Let the open feasible region be E¯=v∈E||vi|<τ¯i/bi,∀i=1,…,n, E= \\,v∈ E\; |\;|v_i|< τ_i/b_i,\ ∀ i=1,…,n \, and equip E¯ E with the Riemannian metric111A Riemannian metric assigns a smoothly varying inner product to each tangent space, requiring G​(v)G(v) to be symmetric positive definite. Because a¯i​(vi)>0 a_i(v_i)>0 on E¯ E, W​(v)W(v) is positive definite and smooth; thus, its inverse G​(v)G(v) is also symmetric positive definite. Therefore, gE¯g_ E is a well-defined Riemannian metric on E¯ E. gE¯​(v)=G​(v)=W​(v)−1,g_ E(v)=G(v)=W(v)^-1, (7) which we call the SAC metric. Intuitively, this metric elongates directions with smaller acceleration headroom: as a¯i​(vi)→0+ a_i(v_i)→ 0^+ (approaching saturation), the metric stiffens along the coordinate direction of d​vidv_i. The use of the squared SACs, a¯i​(vi)2 a_i(v_i)^2, rather than the linear capacities, follows from the quadratic nature of the metric tensor. The squared line element is d​s2=d​v⊤​G​(v)​d​v=∑i=1nd​vi2a¯i​(vi)2.ds^2=dv G(v)\,dv= _i=1^n dv_i^2 a_i(v_i)^2. Thus, the metric naturally normalizes variations by the available capacity, mirroring weighted least-squares control allocation, where penalizing the squared normalized variation ∑i(d​vi/a¯i​(vi))2 _i (dv_i/ a_i(v_i) )^2 utilizes actuators proportionally to their instantaneous capability. To measure multidirectional capability in the generalized force (base) space B, the metric gE¯g_ E must be transferred from the actuator space E¯ E. However, a covariant tensor such as a metric does not admit a natural pushforward unless the mapping is a diffeomorphism. Because the allocation map f of a redundant system (n>mn>m) is many-to-one, pushing the metric forward directly is ill-defined: evaluating vectors in Tf​(v)​BT_f(v)B requires mapping them back via an inverse differential, but f−1​(w)f^-1(w) contains multiple preimages. To resolve this, we adopt a standard geometric approach from redundancy resolution. While vectors cannot be pulled back without an inverse, vectors can be pushed forward via the differential D​fvDf_v, and by duality, covectors can be pulled back unconditionally via the adjoint (D​fv)∗(Df_v)^*. This motivates operating on the cotangent spaces using the dual metric (co-metric), represented by the inverse matrix222The metric induces the musical isomorphism ♭:Tv​E¯→Tv∗​E¯,x↦gE¯​(v)​(x,⋅)\, :T_v E→ T_v^* E,\;x g_ E(v)(x,·), whose inverse ♯:Tv∗​E¯→Tv​E¯\, :T_v^* E→ T_v E is represented by W​(v)W(v) and acts as a co-metric in the cotangent space.: W​(v)=gE¯​(v)−1.W(v)=g_ E(v)^-1. The differential D​fvDf_v (algebraically, the Jacobian Jf​(v)J_f(v)) induces the pullback of 1-forms (D​fv)∗:Tf​(v)∗​B⟶Tv∗​E¯,(D​fv)∗≡Jf​(v)⊤.(Df_v)^*:T_f(v)^*B T_v^* E, (Df_v)^*≡ J_f(v) . We define a dual metric on the task space by pulling two covectors in Tf​(v)∗​BT_f(v)^*B back to Tv∗​E¯T_v^* E and measuring them with W​(v)W(v). This yields the Drag-Aware Aerodynamic Manipulability (DAAM) matrix ​(v)=Jf​(v)​W​(v)​Jf​(v)⊤,D(v)=J_f(v)\,W(v)\,J_f(v) , (8) which is a co-metric on B (i.e., a positive semidefinite inner product on Tf​(v)∗​BT_f(v)^*B). On the regular set E¯reg=v∈E¯|rank⁡Jf​(v)=m, E_reg=\\,v∈ E\;|\;rankJ_f(v)=m\,\, ​(v)D(v) is symmetric positive definite;333At singular points where rank⁡Jf​(v)<mrankJ_f(v)<m, ​(v)D(v) is only semidefinite. One may then work on the image subspace or use Moore–Penrose pseudoinverses, but this paper focuses on E¯reg E_reg. consequently, its inverse ​(v)−1D(v)^-1 is a well-defined covariant Riemannian metric on Tf​(v)​BT_f(v)B. Geometrically, the induced metric measures the effort required to generate coordinate generalized force variations d​wdw: among all actuator spin rate variations d​vdv satisfying Jf​(v)​d​v=d​wJ_f(v)\,dv=dw, it implicitly selects the minimizer of the actuator-space metric gE¯g_ E, i.e., mind​v:Jf​(v)​d​v=d​w⁡d​v⊤​gE¯​(v)​d​v=d​w⊤​(v)−1​d​w, _dv:\,J_f(v)dv=dw\;dv g_ E(v)\,dv=dw D(v)^-1dw, and the corresponding optimal d​vdv is orthogonal to the null space with respect to gE¯​(v)g_ E(v) [undefaj, undefak]. The unit ball under this metric in B forms what we call the DAAM ellipsoid in each tangent space at w=f​(v)w=f(v), ℰB​(v)=w˙∈Tf​(v)​B|w˙⊤​(v)−1​w˙≤1.E_B(v)= \\, w∈ T_f(v)B\ |\ w D(v)^-1\, w≤ 1\, \. Remark 1. The Drag-Aware Aerodynamic Manipulability (DAAM) ellipsoids describe the multidirectional generalized force‑rate authority under the Symmetric Acceleration Capacity (SAC) metric. While the mathematical structure parallels Yoshikawa’s dynamic manipulability (where W typically derives from the inverse mass matrix), DAAM represents a distinct physical concept. Its geometry is governed by actuator limits and aerodynamic drag—rather than sole inertial properties—and is critically shaped by the thrust nonlinearity ui​(vi)=vi​|vi|u_i(v_i)=v_i|v_i|. The zero slope at vi=0v_i=0 produces an aerodynamic degeneracy of Jf​(v)J_f(v) at low spin rates. The volume of ℰB​(v)E_B(v) is, up to the constant volume of the Euclidean unit ball in ℝmR^m, δ​(v)=1/det(​(v)−1)=det(​(v)),δ(v)=1/ \! (D(v)^-1 )= \! (D(v) )\,, (9) which we call the DAAM volume. To exploit the system’s redundancy, we wish to maximize δ​(v)δ(v). Equivalently, we define the DAAM log‑det cost as the negative logarithm of δ​(v)δ(v), ℒ​(v)=−ln⁡δ​(v)=−12​ln​det(​(v)).L(v)=- δ(v)=- 12\, \! (D(v) ). (10) This formulation unifies two distinct physical failure modes. The DAAM volume δ​(v)δ(v) collapses to zero—diverging the cost ℒ​(v)L(v) to +∞+∞—under either of the following conditions: 1. When v approaches the boundary of the feasible actuator space E¯ E such that enough actuators saturate to drop the effective rank of ​(v)D(v) below m. Due to redundancy, a single vanishing capacity in W​(v)W(v) does not necessarily collapse the DAAM volume; divergence occurs strictly when the remaining unsaturated actuators can no longer span the task space. This constitutes a failure induced by the SAC metric (drag‑limited actuation). 2. At any aerodynamically induced singularity where the allocation Jacobian Jf​(v)J_f(v) loses rank. This occurs due to the zero thrust slope at vi=0v_i=0 (from vi​|vi|v_i|v_i|) combined with the specific anchoring and orientation of the propellers (encoded by A), causing a loss of authority to generate generalized forces in certain directions (thrust-rate vanishing). Thus, minimizing ℒ​(v)L(v) steers the redundant multirotor system away from both aerodynamic-drag-induced motor saturation and aerodynamic-thrust-induced singularities, increasing the capacity to react promptly to desired generalized force changes, thus the concept of aero-promptness. I An Illustrative Example This section provides a low-dimensional example to visualize the DAAM geometry in both the spin-rate (total) and generalized force (base) spaces, illustrating the interplay between actuator saturation and task authority at the boundary of the feasible set. Consider a system of two collinear rotors (n=2n=2, E=ℝ2E=R^2) generating a single generalized force (m=1m=1, B=ℝB=R), governed by the allocation map f​(v)=a1​v1​|v1|+a2​v2​|v2|f(v)=a_1v_1|v_1|+a_2v_2|v_2|. For visualization purposes, we apply the following numerical parameters: a=(a1,a2)=(1.0, 1.5)a=(a_1,a_2)=(1.0,\,1.5), b=(b1,b2)=(0.20, 0.40)b=(b_1,b_2)=(0.20,\,0.40), τ¯=(τ¯1,τ¯2)=(10.0, 15.0) τ=( τ_1, τ_2)=(10.0,\,15.0), and m=(m1,m2)=(1.0, 1.0)m=(m_1,m_2)=(1.0,\,1.0). The corresponding feasible spin rates are bounded by |v1|<τ¯1/b1=50≈7.07|v_1|< τ_1/b_1= 50≈ 7.07 and |v2|<τ¯2/b2=37.5≈6.12|v_2|< τ_2/b_2= 37.5≈ 6.12. We present two complementary actuator-space views on the (v1,v2)(v_1,v_2) plane. I-1 SAC Metric and Allocation Fibers Fig. 2 shows the feasible region E¯ E, the local unit balls of the actuator-space SAC metric gE¯​(v)=W​(v)−1g_ E(v)=W(v)^-1 rendered as axis-aligned ellipses with semi-axes a¯i​(vi) a_i(v_i), and the allocation fibers FwF_w as level sets of w∈Bw∈ B. The ellipses collapse along any direction where the SAC a¯i​(vi)→0+ a_i(v_i)→ 0^+, visually encoding the stiffening of gE¯g_ E near actuator saturation. The ellipses are scaled for legibility; their shape reflects local anisotropy, while the color gradient encodes overall acceleration capacity. Figure 2: Spin-rate space (v1,v2)(v_1,v_2) with feasible region E¯ E (light), SAC metric unit balls of gE¯​(v)=W​(v)−1g_ E(v)=W(v)^-1 (axis-aligned ellipses), and fibers FwF_w (contours of w). Ellipses collapse toward ∂E¯∂ E along directions where a¯i​(vi)→0+ a_i(v_i)→ 0^+, encoding the stiffening of the spin-rate space metric near saturation. (Ellipses are scaled for readability; shape reflects anisotropy, color encodes capacity.) I-2 DAAM Volume and Fibers Fig. 3 overlays the DAAM volume δ​(v)=det​(v)δ(v)= (v) as filled contours, with the fibers FwF_w superimposed. In this one-dimensional task space (m=1m=1), the determinant simplifies to: det​(v)=​(v)=4​(a12​v12​(a¯1​(v1))2+a22​v22​(a¯2​(v2))2). (v)=D(v)=4 (a_1^2v_1^2\, ( a_1(v_1) )^2+a_2^2v_2^2\, ( a_2(v_2) )^2 ). (11) Two critical properties distinguish this mapping: 1. In the spin-rate (total) space E, the SAC metric unit ball d​v:∑id​vi2/a¯i​(vi)2≤1\dv: _idv_i^2/ a_i(v_i)^2≤ 1\ strictly collapses along saturated directions (Fig. 2). 2. In the generalized force (base) space B, δ​(v)δ(v) does not identically vanish along generic portions of ∂E¯∂ E. If one actuator saturates while the other retains headroom, ​(v)>0D(v)>0, and the DAAM co-metric continues to provide tangential generalized force-rate authority. Consequently, the DAAM log-det cost −12​ln​det​(v)- 12 (v) acts as a barrier strictly at loci of complete authority loss (e.g., the center, side mid-points, and corners in this n=2n=2 example), allowing the system to operate safely along benign segments of the boundary. The collapse of metric ellipsoids in E therefore does not imply a total loss of control authority in B. Figure 3: Spin-rate space (v1,v2)(v_1,v_2) with the DAAM volume δ​(v)δ(v) (filled contours), masked fibers FwF_w, and a family of straight lines (solid) drawn orthogonally to the thrust-space zero-generalized force fiber a1​v1​|v1|+a2​v2​|v2|=0a_1v_1|v_1|+a_2v_2|v_2|=0. Dark regions indicate small δ​(v)δ(v) at the center, the four corners, and the side mid-points, highlighting where the effective allocation loses authority. Figure 4: Curves δ​(v)δ(v) versus w​(v)w(v) traced along the straight lines of Fig. 3. The upper envelope of these curves represents the theoretical maximum performance of a fiberwise DAAM selection strategy. I-3 Parametric Slices Orthogonal to the Zero-Generalized Force Fiber To analyze the organization of fiberwise minima in the DAAM landscape, we evaluate a family of straight lines in the actuator plane orthogonal to the zero-generalized force fiber in thrust space, a1​v1​|v1|+a2​v2​|v2|=0a_1v_1|v_1|+a_2v_2|v_2|=0. Taking the normal direction n=(a1,a2)n=(a_1,a_2), we generate parallel line segments in the (v1,v2)(v_1,v_2) plane along n, evenly offset in the transversal direction t=(a2,−a1)t=(a_2,-a_1). Fig. 3 overlays these lines on the δ​(v)δ(v) scalar field and the masked fibers FwF_w. Fig. 4 plots the resulting δ​(v)δ(v) versus w​(v)w(v) profiles. The upper envelope of these nonconvex profiles illustrates the maximum capability attained by a fiberwise optimizer that selects the point of largest δ​(v)δ(v) along each corresponding fiber. IV Fiberwise DAAM Optimal Allocation Standard control architectures map desired generalized forces w∈Bw∈ B into physical actuator commands v∈Fw=f−1​(w)⊂Ev∈ F_w=f^-1(w)⊂ E via low-level allocation [undefal]. Traditional allocators exploit this redundancy to minimize instantaneous power [undefah, undefag, undeff] or mechanical manipulability [undefo, undefp]. As a complementary strategy for flight phases demanding high aerodynamic readiness (e.g., severe wind gusts or unmodeled interactions), we propose leveraging this redundancy to maximize the intrinsic aerodynamic capability encoded by the DAAM volume. We pose this as a fiberwise parametric optimization problem. Problem 1 (DAAM-Optimal allocation). For a given generalized force w∈Bw∈ B, we seek an allocation v on the fiber FwF_w that minimizes the DAAM log-det cost: minv∈E¯⁡ℒ​(v)s.t.f​(v)=w, _v∈ E\ L(v) .t. f(v)=w, (12) and we call any solution v∗​(w)v^*(w) a DAAM optimal allocation at w. IV-A Existence and Localization of Solutions On E¯reg E_reg, ​(v)D(v) is symmetric positive definite and ℒ​(v)=−12​ln​det​(v)L(v)=- 12 (v) is smooth. Define the effective actuator set I+​(v)=i∈1,…,n:|vi|>0​and​a¯i​(vi)>0.I_+(v)= \\,i∈\1,…,n\\;:\;|v_i|>0\ and\ a_i(v_i)>0\, \. Recalling (8), det​(v)>0 (v)>0 if and only if the columns of the allocation matrix indexed by I+​(v)I_+(v) span the task space ℝmR^m. It follows that det​(v)↓0 (v) 0 (and ℒ​(v)→+∞L(v)→+∞) precisely when the effective allocation loses rank. Consequently, ℒ​(v)L(v) acts as an interior-point barrier function for the safe operational space. Lemma 1 (Barrier at singularities). Let w∈Bw∈ B and assume the fiber f−1​(w)∩E¯f^-1(w)∩ E is nonempty. Along any sequence vk⊂f−1​(w)∩E¯\v^k\⊂ f^-1(w)∩ E where the effective set I+​(vk)I_+(v^k) fails to span ℝmR^m in the limit, ℒ​(vk)→+∞L(v^k)→+∞. Any minimizing sequence of (12) is strictly repelled from these authority-loss loci. Remark 2 (Boundary Sliding). As established in Section I, ℒ​(v)L(v) diverges only at total authority-loss loci. If, on the fiber FwF_w, one actuator saturates while the remaining effective actuators still span B, det​(v)>0 (v)>0. The cost thereby permits operation along benign boundary segments where the system mathematically retains full task-space authority. Because the feasible set E¯ E is bounded and the fiber Fw∩E¯F_w∩ E is closed in the relative topology, any minimizing sequence of (12) admits accumulation points in the closure E¯ E. By Lemma 1 and Remark 2, a global minimizer v∗​(w)v^*(w) is guaranteed to exist. It is localized either strictly within the regular interior E¯reg E_reg or on a benign boundary segment. IV-B First-order Optimality (KKT) on the Fiber On E¯reg E_reg, the Lagrangian of Problem 1 is ​(v,λ)=ℒ​(v)+λ⊤​(f​(v)−w)J(v,λ)=L(v)+λ (f(v)-w). The KKT conditions for (12) are ∇ℒ​(v∗)+Jf​(v∗)⊤​λ∗=0,f​(v∗)=w. (v^*)+J_f(v^*) λ^*=0, f(v^*)=w. (13) We evaluate the gradient using the differential d​ℒ=−12​tr⁡(S​(v)​d​)dL=- 12tr\! (S(v)\,dD ), where S​(v):=​(v)−1S(v):=D(v)^-1. Dropping the dependency on v and the subscript f for readability, the partial derivative of the D by the product rule is ∂vi=(∂viJ)​W​J⊤+J​(∂viW)​J⊤+[(∂viJ)​W​J⊤]⊤. ∂ v_i=( _v_iJ)\,W\,J \;+\;J\,( _v_iW)\,J \;+\; [( _v_iJ)\,W\,J ] . Let σi:=sign⁡(vi) _i:=sign(v_i). Applying ∂viJ=2​σi​A∙i​ei⊤ _v_iJ=2\, _i\,A_ ie_i and ∂viW=2​a¯i​(−2​bi​vimi)​ei​ei⊤ _v_iW=2\, a_i(- 2b_iv_im_i)e_ie_i , and summing the trace contributions yields: ∂vi=8​σi​|vi|​a¯i​A∙i​A∙i⊤​(a¯i−2​|vi|2​bimi). ∂ v_i=8\, _i\,|v_i|\, a_i\,A_ iA_ i ( a_i-2\,|v_i|^2\, b_im_i ). Substituting a¯i​(vi)=τ¯i−bi​|vi|2mi a_i(v_i)= τ_i-b_i|v_i|^2m_i into the parenthesis and applying the cyclic property of the trace to ∂ℒ∂vi=−12​tr⁡(S​∂vi) ∂ v_i=- 12tr (S\, ∂ v_i ), one obtains the componentwise gradient: ∂ℒ∂vi​(v)=−4​σi​|vi|​a¯i​(vi)​τ¯i−3​bi​|vi|2mi​(A∙i⊤​S​(v)​A∙i), ∂ v_i(v)=-4\, _i\,|v_i|\, a_i(v_i)\, τ_i-3b_i|v_i|^2m_i\, (A_ i \,S(v)\,A_ i ), (14) valid on E¯ E where a¯i​(vi)>0 a_i(v_i)>0 and vi≠0v_i≠ 0. By substituting (14) and the componentwise Jacobian into (13), the KKT stationarity condition for |vi∗|>0|v_i^*|>0 becomes: 2​|vi∗|​A∙i⊤​λ∗=4​σi∗​a¯i​(vi∗)​τ¯i−3​bi​|vi∗|2mi​(A∙i⊤​S​(v∗)​A∙i).2\,|v_i^*|\,A_ i λ^*=4\, _i^*\, a_i(v_i^*)\, τ_i-3b_i|v_i^*|^2m_i\, (A_ i S(v^*)A_ i ). (15) This analytical formulation provides direct insight into the optimal load distribution. The Lagrange multiplier λ∗λ^* acts as a base-space price for generating the required generalized force w, which (15) balances against the local aerodynamic penalties of each propeller. The term (τ¯i−3​bi​|vi∗|2)( τ_i-3b_i|v_i^*|^2) reveals an intrinsic threshold: the gradient contribution of the i-th propeller changes sign when its local drag torque exceeds one-third of the motor’s peak capability. Furthermore, this exact algebraic expression circumvents the need for finite differences or auto-differentiation, enabling the deployment of real-time nonlinear programming (NLP) solvers directly on embedded flight controllers. IV-C Intrinsic Optimality and Task-Space Metric Invariance Classical kinematic and dynamic manipulability indices often depend on the choice of task-space coordinates. In spaces with mixed physical units (e.g., forces and moments), changing the scaling or inner product of the base space can shift the optimal configuration. However, because the DAAM optimization is strictly constrained to the fiber FwF_w, any linear reparameterization or metric change in the base space B acts merely as a uniform volumetric scaling across the fiber. This factors out of the log-det objective as an additive constant. Consequently, the minimizer location and the authority-loss singularities are intrinsic properties governed exclusively by the SAC metric W​(v)W(v) and allocation map f, decoupled from arbitrary user choices in the task space. Formally, for any fixed generalized force w∈Bw∈ B, the fiberwise problem (12) exhibits the following invariance properties: 1) Coordinate change on B For any invertible linear reparametrization represented by a constant matrix C∈GL​(m)C (m), the congruently transformed co-metric is ~​(v)=C−T​(v)​C−1 D(v)=C^-TD(v)C^-1. Its determinant scales uniformly as det(~​(v))=(detC)−2​det(​(v)) ( D(v) )=( C)^-2 (D(v) ). The transformed cost ℒ~​(v)=−12​ln​det(~​(v))=ℒ​(v)+ln⁡|detC| L(v)=- 12 ( D(v) )=L(v)+ | C| differs from ℒ​(v)L(v) only by a constant. 2) Base inner product on B For any fixed positive-definite inner product on B represented by M∈ℝm×mM ^m× m, the induced co-metric is M​(v)=M−1/2​(v)​M−1/2D_M(v)=M^-1/2D(v)M^-1/2. This yields: ℒM​(v)=−12​ln​det(M​(v))=ℒ​(v)+12​ln​detM.L_M(v)=- 12 (D_M(v) )=L(v)+ 12 M. Thus, the argmin set of (12) on FwF_w remains invariant under coordinate and inner-product transformations. 3) Singular set For any invertible C and positive-definite M, ​(v)D(v) is singular if and only if C−T​(v)​C−1C^-TD(v)C^-1 (equivalently, M−1/2​(v)​M−1/2M^-1/2D(v)M^-1/2) is singular. The locus v∈E¯:det​(v)=0\\,v∈ E: (v)=0\,\ and its intersection with each fiber FwF_w are strictly intrinsic. IV-D Local Landscape on a Fiber From a control perspective, it is critical to determine when a continuously varying target generalized force w​(t)∈Bw(t)∈ B yields smoothly varying optimal actuator commands v∗​(w)v^*(w). We analyze the local geometry of the solution space near a reference optimal pair (w,v∗)(w,v^*) where v∗v^* is strictly within the regular interior E¯reg E_reg. Consequently, ​(v∗)D(v^*) is positive definite, the Linear Independence Constraint Qualification (LICQ) holds, and ℒL is locally smooth. Let λ∗λ^* denote the associated Lagrange multiplier. Assume the Second-Order Sufficient Condition (SOSC) holds at v∗v^*, meaning the reduced Hessian of the Lagrangian ​(v,λ)J(v,λ) is strictly positive definite on the tangent subspace ker⁡Jf​(v∗) J_f(v^*): d⊤​∇v​v2​(v∗,λ∗)​d> 0for all ​d∈ker⁡Jf​(v∗)∖0.d ∇^2_vJ(v^*,λ^*)\,d\;>\;0 all d∈ J_f(v^*) \0\. Theorem 1 (Local Optimal Section). Under the stated regularity and SOSC assumptions, there exist open neighborhoods U⊂BU⊂ B containing w and V⊂EV⊂ E containing v∗v^*, along with unique C1C^1 maps v∗:U→Vv^*:U→ V and λ∗:U→ℝmλ^*:U ^m, such that v∗​(w)v^*(w) uniquely solves (12) for all w∈Uw∈ U. Furthermore, the image =v∗​(w):w∈US=\v^*(w):w∈ U\ is a smoothly embedded m-dimensional manifold in E, and the restriction f|:→Uf|_S:S→ U is a local diffeomorphism. Proof. The KKT conditions for (12) define a system of n+mn+m nonlinear equations L​(v,λ,w)=0L(v,λ,w)=0: L​(v,λ,w)=[∇vℒ​(v)+Jf​(v)⊤​λf​(v)−w]=[00].L(v,λ,w)= bmatrix _vL(v)+J_f(v) λ\\ f(v)-w bmatrix= bmatrix0\\ 0 bmatrix. To apply the Implicit Function Theorem at (v∗,λ∗,w)(v^*,λ^*,w), we evaluate the Jacobian of L with respect to the primal-dual variables (v,λ)(v,λ): ∇(v,λ)L​(v∗,λ∗)=[∇v​v2​(v∗,λ∗)Jf​(v∗)⊤Jf​(v∗)0]. _(v,λ)L(v^*,λ^*)= bmatrix∇^2_vJ(v^*,λ^*)&J_f(v^*) \\ J_f(v^*)&0 bmatrix. Since v∗∈E¯regv^*∈ E_reg, Jf​(v∗)J_f(v^*) has full row rank m. Combined with the SOSC assumption that ∇v​v2∇^2_vJ is positive definite on the null space of Jf​(v∗)J_f(v^*), standard saddle-point matrix theory guarantees this block KKT matrix is non-singular. By the Implicit Function Theorem, the KKT system defines unique, continuously differentiable local solution mappings w↦v∗​(w)w v^*(w) and w↦λ∗​(w)w λ^*(w). Because w↦v∗​(w)w v^*(w) is a smooth immersion parameterized by the m-dimensional base space, its image S is an embedded m-dimensional manifold in E. By definition, f​(v∗​(w))=wf(v^*(w))=w, meaning f|f|_S smoothly continuously inverts the parameterization, establishing a local diffeomorphism. ∎ A direct corollary of Theorem 1 is that, locally on each fiber, the set of minimizers has the exact dimension of the task space B. Near any regular interior minimizer, the DAAM-optimal solution map generates a smooth, local section of the control allocation bundle, guaranteeing chattering-free actuator commands for continuous, small-signal task-space trajectories. However, because global continuous sections generally do not exist for complex allocation manifolds, discontinuities will eventually occur when a large-signal trajectory forces the system into topological obstructions or saturation boundaries. IV-E Global Landscape and Topological Transitions While Theorem 1 guarantees smooth actuator commands locally, the global solution set over the task space B consists of multiple smooth sheets (or branches) αS_α defined over open regions Uα⊂BU_α⊂ B, where the restriction f|α:α→Uαf|_S_α:S_α→ U_α acts as a local diffeomorphism. Transitions between these sheets are governed by physical and geometric boundaries, causing the continuous section to break down at three fundamental events: • Motor Reversals (vi=0v_i=0): Aerodynamic drag and thrust models switch signs across these hyperplanes, which partition the total space E into distinct orthants. Crossing a zero-spin boundary can create or destroy optimal branches, inducing fold or cusp singularities in the minimizer locus. • Actuator Saturation (a¯i​(vi)=0 a_i(v_i)=0): Reaching maximum thrust corresponds to the boundary ∂E¯∂ E, collapsing the actuator-space SAC unit ball along the i-th direction. As established in Remark 2, branches may continue by sliding along these boundary segments until system-wide authority is compromised. • Authority Loss (det​(v)=0 (v)=0): Rank loss of the allocation Jacobian Jf​(v)J_f(v) induces intrinsic aerodynamic singularities. At these loci, the log-det cost diverges to +∞+∞. Optimal branches either bifurcate discontinuously or terminate entirely to respect this barrier. This geometric framework underpins the subsequent analysis of specific configurations in Section V, which will illustrate typical local sections, branch continuations, and the mechanics of jump bifurcations across these stratification interfaces. V Optimal DAAM Landscapes for Three Illustrative Cases: A Qualitative Discussion To contextualize the mathematical analysis, we numerically explore the DAAM-optimal allocation landscape using three representative multirotor configurations. For a given generalized force w∈Bw∈ B, we minimize the DAAM log-det cost ℒ​(v)L(v) along the allocation fiber Fw=f−1​(w)F_w=f^-1(w) within the feasible spin-rate set E¯ E, where w∈Bw∈ B are the generalized forces. This yields the minimizing allocations v∗​(w)v^*(w), illustrating how physical aerodynamics and actuator limits organize the global solution space. V-A Two Propellers, One Generalized Force (n=2n=2, m=1m=1) (a) a1=0.7a_1=0.7 (b) baseline (balanced) (c) a2=0.7a_2=0.7 (d) m1=0.005m_1=0.005 (e) τ¯1=0.1 τ_1=0.1 (f) b1=0.01b_1=0.01 Figure 5: Case n=2n=2, m=1m=1: Six fiberwise DAAM landscapes varying single parameters from the baseline (a1=a2=1.0a_1=a_2=1.0, b1=b2=0.1b_1=b_2=0.1, m1=m2=0.05m_1=m_2=0.05, τ¯1=τ¯2=1.0 τ_1= τ_2=1.0). Axes represent spin rates v1v_1 and v2v_2. Panels display the cost field (contours), allocation fibers FwF_w (thin red curves), and global optimal allocation paths v∗​(w)v^*(w) (thick, dotted yellow curves). Consider the configuration from Section I (n=2n=2, m=1m=1), governed by f​(v)=a1​v1​|v1|+a2​v2​|v2|f(v)=a_1v_1|v_1|+a_2v_2|v_2|. Fig. 5 overlays the cost landscape ℒ​(v)L(v) (filled contours), generalized force fibers FwF_w, and numerically computed optimal allocations v∗​(w)v^*(w). The symmetric acceleration capacities define the rectangular feasible box E¯ E. Table I summarizes the single-parameter variations evaluated against a balanced baseline. TABLE I: Parameters for the n=2,m=1n=2,m=1 case (Fig. 5). Each case sweeps a single parameter from the balanced baseline (a1=a2=1.0a_1=a_2=1.0, b1=b2=0.1b_1=b_2=0.1, m1=m2=0.05m_1=m_2=0.05, τ¯1=τ¯2=1.0 τ_1= τ_2=1.0). Panel Case name Parameter change w.r.t. baseline 5(a) smaller a1a_1 a1=0.7a_1=0.7 5(b) balanced none 5(c) smaller a2a_2 a2=0.7a_2=0.7 5(d) tiny m1m_1 m1=0.005m_1=0.005 5(e) tiny τ¯1 τ_1 τ¯1=0.1 τ_1=0.1 5(f) tiny b1b_1 b1=0.01b_1=0.01 Balanced Baseline and Aerodynamic Gain Changes In the perfectly balanced baseline (Fig. 5(b)), the landscape exhibits total symmetry. As predicted in Section IV-D, the optimal allocations form continuous, smooth segments. Topological jumps occur precisely when crossing physical boundaries: the solution either passes through a zero-spin axis (changing the sign of a spin rate) or reaches the saturation boundary ∂E¯∂ E. During these jumps, the allocation instantly shifts the load between motors along a single target fiber FwF_w. In this symmetric baseline, these jumps act as bifurcations where a single optimal solution branches into two symmetric minima. Crucially, Fig. 5(b) illustrates the distinction between benign boundary sliding and total authority loss. At the corners of E¯ E, the optimal policy slides along the boundary, relying on the remaining effective propeller to span the task direction. Conversely, at the midpoints of the boundary edges, the fibers FwF_w become nearly tangent to ∂E¯∂ E. This tangency geometrically manifests control loss: moving along the boundary produces virtually no change in the task space. The log-det cost naturally repels the optimal paths from these degenerate loci. Decreasing the aerodynamic gain for one propeller (Fig. 5(a) and 5(c), e.g., via altered blade pitch) breaks this symmetry. The jump mechanics remain identical, but the bifurcations vanish, yielding a single global optimum per fiber that strictly favors the more efficient rotor. Heterogeneous Inertia (Small Mass) Fig. 5(d) illustrates a highly asymmetric case where the first rotor’s inertia is reduced by an order of magnitude (m1=0.005m_1=0.005). The feasible box E¯ E and allocation fibers FwF_w remain visually identical to the balanced baseline, as they depend exclusively on torque limits τ¯ τ, drag b, and thrust gains a. However, the cost landscape ℒ​(v)L(v) transforms dramatically into vertical stripes, dominated by the dynamics of v1v_1. The optimal allocation exhibits a strictly decoupled strategy: to satisfy variations in w, the optimizer pins v1v_1 at a constant value inside a low-cost valley, absorbing all required force variations through the slower rotor v2v_2. This strategy reflects a core physical principle of the DAAM framework. Because m1m_1 is significantly smaller, its maximum acceleration capacity is vastly larger, dominating the objective function. To maximize total aerodynamic readiness, the optimization parks the highly agile rotor at its aerodynamic peak—the locus where the thrust slope is steep but drag has not yet eroded acceleration headroom. The sluggish rotor v2v_2 is sacrificed to trim the baseline force, ensuring the system’s most responsive actuator remains primed to reject sudden disturbances. Heterogeneous Torque Capacity (Small Maximum Torque) In Fig. 5(e), the maximum torque of the first motor is reduced by an order of magnitude (τ¯1=0.1 τ_1=0.1). The feasible region E¯ E collapses horizontally into a narrow rectangle, confirming that the maximum steady-state spin rate scales with τ¯i/bi τ_i/b_i. The allocation fibers FwF_w remain unchanged, as they depend strictly on the unmodified thrust gains aia_i. The resulting cost landscape ℒ​(v)L(v) represents a 90-degree topological rotation of the low-mass case, exhibiting almost zero dependence on v1v_1. Because E¯ E is narrow and the cost is dominated by the strong motor (v2v_2), the optimal section navigates the fibers in a strictly boundary-adhering manner. This reveals a fundamental role reversal compared to Fig. 5(d). In the low-mass case, the highly capable actuator was parked in an optimal valley while the sluggish one absorbed variations. Here, the weak motor (v1v_1) contributes negligibly to the system’s total manipulability volume. Consequently, to satisfy generalized force variations, the system deliberately pushes the weak actuator to its absolute saturation boundaries to act as a constant bias. This strategy preserves the strong actuator (v2v_2) near the center of its dynamic capacity, maintaining the system’s baseline force while maximizing reactive authority. Heterogeneous Efficiency (Small Aerodynamic Drag) Fig. 5(f) illustrates the impact of reducing the first rotor’s aerodynamic drag coefficient (b1=0.01b_1=0.01). The feasible boundary E¯ E expands horizontally, visually confirming the physical capacity boundary’s inverse-square-root scaling with drag. The cost landscape exhibits alternating vertical valleys and ridges. Because the first rotor is highly efficient at high speeds, the optimal policy leverages this extended capacity. The resulting optimal section is structurally intricate, weaving through the landscape via twelve distinct smooth segments that alternate between interior curves and boundary coasting. Consistent with the theoretical guarantees in Section IV-E, topological jumps strictly occur at two well-defined events: when transitioning to/from a saturation boundary, or when crossing a zero-spin axis. (a) View 1 (b) View 2 (c) View 3 Figure 6: Case n=3n=3, m=1m=1: fibers FwF_w in three dimensions with the fiberwise DAAM-optimal section. Translucent colored surfaces show distinct levels of the scalar generalized force w; yellow markers depict the optimal allocations v∗​(w)v^*(w). The wireframe box indicates the feasibility bounds |vi|<τ¯i/bi|v_i|< τ_i/b_i. Three complementary viewpoints are provided to visually disambiguate depth and surface intersections. Invariant Pattern at Origin Crossing Across all asymmetric configurations, a consistent topological invariant emerges: a discontinuous jump across the origin as the generalized force w crosses zero. In the balanced baseline, the system possesses two equally optimal symmetric solutions for small |w||w|. However, when symmetry is broken, the optimal policy must commit to a single path. At exactly w=0w=0, the fiber Fw=0F_w=0 passes through the origin (v1=0,v2=0v_1=0,v_2=0). Because the DAAM cost heavily penalizes low spin rates due to thrust-slope degeneracy, the optimal policy actively avoids the origin, maintaining control authority by spinning the motors antagonistically. To cross w=0w=0, the system must fundamentally swap which motor provides positive thrust and which provides negative drag, forcing a macroscopic jump between opposing quadrants. This invariant demonstrates how the DAAM framework intrinsically prefers antagonistic internal tension over zero-spin states. It automatically guarantees maximum aerodynamic readiness even when the required task-space effort is zero, yielding a behavior fundamentally distinct from traditional energy-minimizing strategies. V-B Three Propellers, One Generalized Force (n=3n=3, m=1m=1) Expanding to a three-propeller system generating a single generalized force, the allocation map is f​(v)=∑i=13ai​vi​|vi|f(v)= _i=1^3a_iv_i|v_i|. The allocation fibers FwF_w manifest as two-dimensional surfaces embedded within the three-dimensional actuator space. The manipulability measure is the determinant of the capacity-scaled metric tensor: det​(v)=​(v)=∑i=13(2​ai​|vi|)2​a¯i​(vi)2. (v)=D(v)= _i=1^3 (2a_i|v_i| )^2\, a_i(v_i)^2. Unlike the 2D cases, the scalar cost landscape ℒ​(v)L(v) cannot be plotted alongside the actuator axes. Figure 6 renders multiple fiber surfaces FwF_w as translucent isosurfaces in (v1,v2,v3)(v_1,v_2,v_3), colored by the force level w. For a symmetric scan of w around zero, the fiberwise minimizers v∗​(w)v^*(w) inside E¯ E form one-dimensional curve segments (yellow markers) piercing the fiber surfaces. We utilize perfectly symmetric parameters matching the 2D baseline: a=(1.0, 1.0, 1.0)a=(1.0,\,1.0,\,1.0), b=(0.1, 0.1, 0.1)b=(0.1,\,0.1,\,0.1), m=(0.05, 0.05, 0.05)m=(0.05,\,0.05,\,0.05), and τ¯=(1.0, 1.0, 1.0) τ=(1.0,\,1.0,\,1.0). As w sweeps from minimum to maximum feasible values, the optimal section traverses the feasible space from the fully negative corner to the fully positive opposite corner. For extreme negative w, the optimal allocation slides along the exterior boundary planes of E¯ E. As |w||w| decreases to intermediate negative values, the solution detaches from the boundaries, traveling smoothly along the main internal diagonal (v1≈v2≈v3<0v_1≈ v_2≈ v_3<0), equally distributing effort among identical rotors. Due to perfect symmetry, the solution exhibits complex bifurcations analogous to the 2D baseline. As w approaches zero from the negative side, the single global minimum on the main diagonal bifurcates into two distinct optimal paths in opposite octants. Near the origin, the optimizer actively avoids the thrust-slope degeneracy at v=0v=0 by maintaining antagonistic tension, causing the solutions to jump into a second pair of opposite octants. As w becomes positive, the solutions jump again into a third pair of octants. This sequence of symmetrical branching illustrates how the DAAM geometry organizes the allocation nullspace to preserve control authority. Finally, for intermediate-large positive w, these bifurcated branches merge into a unique solution ascending the positive main diagonal (v1≈v2≈v3>0v_1≈ v_2≈ v_3>0). Upon reaching physical capacity limits for maximum w, the path resumes sliding along the positive boundary planes. V-C Three Propellers, Two General. Forces (n=3n=3, m=2m=2) (a) View 1 (b) View 2 (c) View 3 Figure 7: Case n=3n=3, m=2m=2: fibers FwF_w in three dimensions with the fiberwise DAAM-optimal section. Colored curves depict a selection of 1D fibers in the actuator space (v1,v2,v3)(v_1,v_2,v_3); gray markers indicate minimizers v∗​(w1,w2)v^*(w_1,w_2). The wireframe box shows feasibility limits. Optimal allocations track DAAM cost valleys along the fibers, forming m=2m=2-dimensional optimal surfaces that actively avoid the high-cost origin. Consider a three-propeller system generating two generalized forces (n=3n=3, m=2m=2): f1​(v) f_1(v) =a1​v1​|v1|+a2​v2​|v2|+a3​v3​|v3|, =a_1\,v_1|v_1|+a_2\,v_2|v_2|+a_3\,v_3|v_3|, f2​(v) f_2(v) =c1​v1​|v1|−c2​v2​|v2|. =c_1\,v_1|v_1|-c_2\,v_2|v_2|. Each fiber f−1​(w1,w2)⊂E¯f^-1(w_1,w_2)⊂ E is a one-dimensional curve. Adopting the coordinate change ui=vi​|vi|u_i=v_i|v_i|, these fibers are parameterized analytically by t=u3t=u_3: u1​(t)=w1−a3​t+(a2/c2)​w2a1+(a2​c1/c2),u2​(t)=c1​u1​(t)−w2c2,u_1(t)= w_1-a_3t+(a_2/c_2)\,w_2a_1+(a_2c_1/c_2), u_2(t)= c_1u_1(t)-w_2c_2, with vi​(t)=sign⁡(ui​(t))​|ui​(t)|v_i(t)=sign(u_i(t)) |u_i(t)| and feasibility enforced by |ui​(t)|<τ¯i/bi|u_i(t)|< τ_i/b_i. To minimize ℒ​(v)L(v), we utilize the system Jacobian: Jf​(v)=2​[a1​|v1|a2​|v2|a3​|v3|c1​|v1|−c2​|v2|0].J_f(v)=2 bmatrixa_1|v_1|&a_2|v_2|&a_3|v_3|\\ c_1|v_1|&-c_2|v_2|&0 bmatrix. For a grid of pairs (w1,w2)(w_1,w_2), solving the 1D optimization along t yields the minimizers v∗​(w1,w2)v^*(w_1,w_2). Figure 7 visualizes these 1D fibers alongside the optimal points (gray markers) using symmetric parameters: a=(1.0,1.0,1.0)a=(1.0,1.0,1.0), b=(0.1,0.1,0.1)b=(0.1,0.1,0.1), m=(0.05,0.05,0.05)m=(0.05,0.05,0.05), τ¯=(1.0,1.0,1.0) τ=(1.0,1.0,1.0), and c=(1.0,1.0,0.0)c=(1.0,1.0,0.0). The optimal allocations track the valleys of the DAAM cost, collectively describing m=2m=2-dimensional surfaces (sections of the fiber bundle) within the 3D actuator space. Notably, these surfaces actively avoid the center of the actuator cube (v1=v2=v3=0v_1=v_2=v_3=0). Allowing spin rates to approach zero annihilates the aerodynamic thrust derivative, eroding reactive control authority. By avoiding the origin, the optimizer inherently preserves antagonistic tension among the rotors, ensuring the multirotor remains primed to inject rapid forces. Furthermore, the symmetric parameters induce distinct bifurcation phenomena. The optimal solution surfaces split into opposite octants of the actuator space, intersecting the exact same 1D fiber curves. This proves the simultaneous existence of multiple global minima for specific task-space pairs (w1,w2)(w_1,w_2). As required forces increase in magnitude, these bifurcated interior branches merge or terminate, smoothly transitioning into boundary-coasting segments along the benign faces of E¯ E where task authority is safely maintained. VI Conclusion and Future Research Directions This paper introduces a novel geometric framework for multirotor control allocation, shifting the paradigm from traditional energy minimization to maximizing instantaneous aerodynamic readiness via Drag-aware Aerodynamic Manipulability (DAAM). As demonstrated, DAAM optimization autonomously generates complex, context-aware actuation strategies. Unlike convex energy metrics, this Riemannian measure induces a stratified manifold of optima. Smooth strata are joined along physical transition interfaces, such as actuator saturation or zero-spin crossings. This architecture is strictly intrinsic: the locations of optimal sheets, boundaries, and jump bifurcations depend entirely on the physical allocation map f and actuator limits, remaining invariant to task-space (generalized forces) coordinate choices. While this intrinsic geometry guarantees maximum instantaneous readiness, it opens several critical avenues for future research: Multi-Objective Blending The DAAM framework prioritizes readiness, whereas traditional metrics prioritize power efficiency. A fundamental open challenge is formulating principled blending strategies that smoothly interpolate between energy conservation during steady flight and DAAM-optimal readiness during dynamic maneuvering, disturbance rejection, or physical interaction. Global Continuity Algorithms Because DAAM optima exist on disconnected strata, large-signal task-space trajectories w​(t)w(t) that force the system across transition interfaces can cause jump discontinuities or active-set topological changes. While local operation around a constant equilibrium (e.g., hovering) remains safely within a single stratum, global trajectories executing large-angle rotations or high-force manipulation require continuous transitions. A critical open problem is deriving continuous, suboptimal sections through the fiber bundle by traversing saddle points in the DAAM cost landscape. This requires deliberately sacrificing global optimality to ensure a continuous lifting of the task-space trajectory into the actuator space. Experimental Stress-Testing for Unmodeled Aerodynamics While the underlying first-principles aerodynamic model is well-established, experimental validation on real hardware is an essential next step. Rather than serving strictly as a proof of concept, real-world deployments should explicitly stress-test the DAAM framework against unmodeled aerodynamic phenomena—such as aerodynamic interference, ground effect, complex aerodynamic drag, or blade flapping. Extracting empirical insights from these boundary conditions will drive the next generation of methodological advances in geometric control allocation for propeller-driven vehicles. Acknowledgments The authors thank their colleagues, mentors, and mentees for inspiring discussions. The authors also acknowledge the LLM Gemini 3 (early 2026) for assistance in proofreading and optimizing the data processing and plotting scripts. The functionality of all code and the accuracy of the resulting outputs were manually verified by the authors. 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