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Muscle Coactivation in the Sky: Geometry and Pareto Optimality of Energy vs. Promptness in Multirotors

Antonio Franchi

TL;DR

This framework provides a foundational understanding of why and how to achieve agility through geometry-aware control allocation, offering possible guidance for vehicle design, certification metrics, and threat-aware flight operation.

Abstract

In robotics and human biomechanics, the tension between energy economy and kinematic readiness is well recognized; this work brings that fundamental principle to aerial multirotors. We show that the limited torque of the motors and the nonlinear aerodynamic map from rotor speed to thrust naturally give rise to the novel concept of promptness-a metric akin to dynamic aerodynamic manipulability. By treating energy consumption as a competing objective and introducing a geometric fiber-bundle formulation, we turn redundancy resolution into a principled multi-objective program on affine fibers. The use of the diffeomorphic transformation linearizing the signed-quadratic propulsion model allows us to lay the foundations for a rigorous study of the interplay between these costs. Through an illustrative case study on 4-DoF allocation on the hexarotor, we reveal that this interplay is fiber-dependent and physically shaped by hardware inequalities. For unidirectional thrusters, the feasible fibers are compact, yielding interior allocations and a short Pareto arc, while torque demands break symmetry and separate the optima. Conversely, with reversible propellers, the null space enables antagonistic rotor co-contraction that drives promptness to hardware limits, making optimal endurance and agility fundamentally incompatible in those regimes. Ultimately, rather than relying on heuristic tuning or black box algorithms to empirically improve task execution, this framework provides a foundational understanding of why and how to achieve agility through geometry-aware control allocation, offering possible guidance for vehicle design, certification metrics, and threat-aware flight operation.

Muscle Coactivation in the Sky: Geometry and Pareto Optimality of Energy vs. Promptness in Multirotors

TL;DR

This framework provides a foundational understanding of why and how to achieve agility through geometry-aware control allocation, offering possible guidance for vehicle design, certification metrics, and threat-aware flight operation.

Abstract

In robotics and human biomechanics, the tension between energy economy and kinematic readiness is well recognized; this work brings that fundamental principle to aerial multirotors. We show that the limited torque of the motors and the nonlinear aerodynamic map from rotor speed to thrust naturally give rise to the novel concept of promptness-a metric akin to dynamic aerodynamic manipulability. By treating energy consumption as a competing objective and introducing a geometric fiber-bundle formulation, we turn redundancy resolution into a principled multi-objective program on affine fibers. The use of the diffeomorphic transformation linearizing the signed-quadratic propulsion model allows us to lay the foundations for a rigorous study of the interplay between these costs. Through an illustrative case study on 4-DoF allocation on the hexarotor, we reveal that this interplay is fiber-dependent and physically shaped by hardware inequalities. For unidirectional thrusters, the feasible fibers are compact, yielding interior allocations and a short Pareto arc, while torque demands break symmetry and separate the optima. Conversely, with reversible propellers, the null space enables antagonistic rotor co-contraction that drives promptness to hardware limits, making optimal endurance and agility fundamentally incompatible in those regimes. Ultimately, rather than relying on heuristic tuning or black box algorithms to empirically improve task execution, this framework provides a foundational understanding of why and how to achieve agility through geometry-aware control allocation, offering possible guidance for vehicle design, certification metrics, and threat-aware flight operation.
Paper Structure (15 sections, 27 equations, 2 figures)

This paper contains 15 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Energy and Promptness in Multirotors
  3. Motivating Example: The Dual-Actuator
  4. Conflict Analysis in Effort Coordinates
  5. Cooperative Regime ($u_1>0,u_2>0$)
  6. Antagonistic Regime ($u_1<0,u_2>0$)
  7. Summary
  8. General Framework: MOO on Fibers
  9. Fundamental Research Questions
  10. Promptness vs. Energy in Hexarotor 4-DoF Wrench Allocation
  11. Unidirectional speeds: $u\ge 0$ (standard ESCs)
  12. Bidirectional speeds with bounds: $|u_i|\le \bar{u}_i$
  13. Discussion
  14. Numerical Simulations
  15. Conclusion

Figures (2)

  • Figure 1: Dual‑actuator case study summary. (a)–(d) show energy cost and inverse promptness fields on the rotor speed ($v$) and control effort ($u$) planes. (e) Continuous elliptic/hyperbolic fibers on the $v$-plane for several $w$ values, visually differentiated by seven distinct line patterns. (f) The same family maps to straight lines on the $u$-plane with consistent patterns. (g) On cooperative segments, the energy $\tilde{J}_1$ attains a strict interior minimum while $\tilde{J}_2$ is indifferent or only weakly biased, indicating minimal conflict. (h) On antagonistic rays, $\tilde{J}_1$ and $\tilde{J}_2$ exhibit opposing trends, illustrating total conflict under internal loading.
  • Figure 2: Hexarotor redundancy resolution under a roll–torque sweep: two actuation regimes are shown: unidirectional ($u\ge 0$; left column) and bidirectional with box bounds (right column) with $|u_i|\le \bar{u}_i$ and $\bar{u}_i=5$. In each regime we overlay the energy–optimal allocation and the promptness–optimal allocation, and we report the corresponding costs on the bottom rows. With $u\ge 0$ the feasible fiber is compact and both optima remain close across the sweep; with $|u_i|\le \bar{u}_i$ the promptness solution leverages mixed–sign null–space directions, lowering $\tilde{J}_2$ at the expense of a higher $\tilde{J}_1$.

Theorems & Definitions (6)

  • Definition 1: Fiber
  • Definition 2: Tangent space and orthogonal projector
  • Definition 3: Restricted gradient on the fiber
  • Definition 4: Local alignment (conflict) index
  • Definition 5: Local Pareto optimality on the fiber
  • Definition 6: Pareto set and a global conflict measure