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

Antonio Franchi

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.

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

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.
Paper Structure (18 sections, 2 theorems, 34 equations, 7 figures, 2 tables)

This paper contains 18 sections, 2 theorems, 34 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Drag-Aware Aerodynamic Manipulability
  4. An Illustrative Example
  5. SAC Metric and Allocation Fibers
  6. DAAM Volume and Fibers
  7. Parametric Slices Orthogonal to the Zero-Generalized Force Fiber
  8. Fiberwise DAAM Optimal Allocation
  9. Existence and Localization of Solutions
  10. First-order Optimality (KKT) on the Fiber
  11. Intrinsic Optimality and Task-Space Metric Invariance
  12. Local Landscape on a Fiber
  13. Global Landscape and Topological Transitions
  14. Optimal DAAM Landscapes for Three Illustrative Cases: A Qualitative Discussion
  15. Two Propellers, One Generalized Force ($n=2$, $m=1$)
  16. ...and 3 more sections

Key Result

Lemma 1

Let $w\in B$ and assume the fiber $f^{-1}(w)\cap\bar{E}$ is nonempty. Along any sequence $\{v^k\}\subset f^{-1}(w)\cap\bar{E}$ where the effective set $I_+(v^k)$ fails to span $\mathbb{R}^m$ in the limit, $\mathcal{L}(v^k)\to +\infty$. Any minimizing sequence of eq:opt is strictly repelled from thes

Figures (7)

  • Figure 1: Symmetric Acceleration Capacity (SAC) of the $i$-th moto-propeller unit $\bar{a}_i(v_i)$ as a function of the spin rate $v_i$. The curves peak at $\bar{\tau}_i/m_i$ when $v_i=0$ and drop to zero at the critical spin rate $v_i = \pm \sqrt{\bar{\tau}_i/b_i}$.
  • Figure 2: Spin-rate space $(v_1,v_2)$ with feasible region $\bar{E}$ (light), SAC metric unit balls of $g_{\bar{E}}(v)=W(v)^{-1}$ (axis-aligned ellipses), and fibers $F_w$ (contours of $w$). Ellipses collapse toward $\partial\bar{E}$ along directions where $\bar{a}_i(v_i)\to 0^+$, encoding the stiffening of the spin-rate space metric near saturation. (Ellipses are scaled for readability; shape reflects anisotropy, color encodes capacity.)
  • Figure 3: Spin-rate space $(v_1,v_2)$ with the DAAM volume $\delta(v)$ (filled contours), masked fibers $F_w$, and a family of straight lines (solid) drawn orthogonally to the thrust-space zero-generalized force fiber $a_1v_1|v_1|+a_2v_2|v_2|=0$. Dark regions indicate small $\delta(v)$ at the center, the four corners, and the side mid-points, highlighting where the effective allocation loses authority.
  • Figure 4: Curves $\delta(v)$ versus $w(v)$ traced along the straight lines of Fig. \ref{['fig:ex1_delta']}. The upper envelope of these curves represents the theoretical maximum performance of a fiberwise DAAM selection strategy.
  • Figure 5: Case $n=2$, $m=1$: Six fiberwise DAAM landscapes varying single parameters from the baseline ($a_1=a_2=1.0$, $b_1=b_2=0.1$, $m_1=m_2=0.05$, $\bar{\tau}_1=\bar{\tau}_2=1.0$). Axes represent spin rates $v_1$ and $v_2$. Panels display the cost field (contours), allocation fibers $F_w$ (thin red curves), and global optimal allocation paths $v^*(w)$ (thick, dotted yellow curves).
  • ...and 2 more figures

Theorems & Definitions (5)

  • Remark 1
  • Lemma 1: Barrier at singularities
  • Remark 2: Boundary Sliding
  • Theorem 1: Local Optimal Section
  • proof