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Stability and agility trade-offs in spring-wing systems

Bioinspiration & Biomimetics

December 2024
20(1)

DOI:10.1088/1748-3190/ad9535

Authors:

James Lynch

University of California, San Diego

Simon Sponberg

Georgia Institute of Technology

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Flying insects are thought to achieve energy-efficient flapping flight by storing and releasing elastic energy in their muscles, tendons, and thorax. However, ‘spring-wing’ flight systems consisting of elastic elements coupled to nonlinear, unsteady aerodynamic forces present possible challenges to generating stable and responsive wing motion. The energetic efficiency from resonance in insect flight is tied to the Weis-Fogh number (N), which is the ratio of peak inertial force to aerodynamic force. In this paper, we present experiments and modeling to study how resonance efficiency (which increases with N) influences the control responsiveness and perturbation resistance of flapping wingbeats. In our first experiments, we provide a step change in the input forcing amplitude to a series-elastic spring-wing system and observe the response time of the wing amplitude increase. In our second experiments we provide an external fluid flow directed at the flapping wing and study the perturbed steady-state wing motion. We evaluate both experiments across Weis-Fogh numbers from 1<N<10. The results indicate that spring-wing systems designed for maximum energetic efficiency also experience trade-offs in agility and stability as the Weis-Fogh number increases. Our results demonstrate that energetic efficiency and wing maneuverability are in conflict in resonant spring-wing systems, suggesting that mechanical resonance presents tradeoffs in insect flight control and stability.

Rise time in simulated linear spring-mass-damper. Time to full amplitude is proportional to Q. a and b show startup to 45° degree amplitude and an amplitude change from 45° to 90° for Q = 2 and Q = 8 respectively. The rise time is slower with higher Q, as shown in c for several values of full amplitude percentage, p.

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The series-elastic spring-wing system. (a) Conceptual diagram indicating the angle input, linear spring with structural damping, and rigid fixed-pitch wing. (b) Corresponding photo of the roboflapper indicating the ClearPath servo motor, silicone torsion spring, and acrylic wing in a large tank of water. (c) Diagram of the whole electromechanical system. Reproduced from [5]. CC BY 4.0.

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Responsiveness Experiments. (a) We drive the series-elastic system via servo (blue) and measure the emergent flapping kinematics (orange). We fit exponential curves to the flapping peaks during start up (yellow) and after an input step (purple) 15 cycles after start. The measured time (in wing strokes) to full amplitude is linearly related to N (b & (c). However, the effective value for N is less than prescribed after the step due to an increase in flapping amplitude (c).

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Description of the constant flow experiments. (a) Schematic of the orientation of the water jet relative to the wing, and conceptual representation of the effects of flow on time and phase domain plots. (b) Variation in limit cycle plots across N. Plots show 2 periods of steady oscillation with and without flow, at different values of N. (c) Plots of fit error for flow and no-flow cases across N, as well as fit lines for expected N⁻¹ function and a best fit curve. (d) Illustration of relative amplitude decrease across N, and mean at 84% of full amplitude.

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Amplitude increases due to a step increase in input amplitude lead to commensurate decreases in N and transient time. Data points are shown in color, and gray arrows indicate the movement due to increased control input. The arrows are all roughly aligned with the trendline.

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Bioinspir. Biomim. 20 (2025) 016024 https://doi.org/10.1088/1748-3190/ad9535

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20 November 2024

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23 December 2024

PAPER

Stability and agility trade-offs in spring-wing systems

James Lynch1, Ethan S Wold2, Jeff Gau3, Simon Sponberg2,4and Nick Gravish1,∗

1Department of Mechanical & Aerospace Engineering, University of California, San Diego, CA, United States of America

2School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA, United States of America

3Interdisciplinary Bioengineering Graduate Program and George W. Woodruff School of Mechanical Engineering, Georgia Institute of

Technology, Atlanta, GA, United States of America

4School of Physics, Georgia Institute of Technology, Atlanta, GA, United States of America

∗Author to whom any correspondence should be addressed.

E-mail: ngravish@ucsd.edu

Keywords: insect flight, resonance, dynamic scaling, elasticity, robophysics

Abstract

Flying insects are thought to achieve energy-efficient flapping flight by storing and releasing elastic

energy in their muscles, tendons, and thorax. However, ‘spring-wing’ flight systems consisting of

elastic elements coupled to nonlinear, unsteady aerodynamic forces present possible challenges to

generating stable and responsive wing motion. The energetic efficiency from resonance in insect

flight is tied to the Weis-Fogh number (N), which is the ratio of peak inertial force to aerodynamic

force. In this paper, we present experiments and modeling to study how resonance efficiency

(which increases with N) influences the control responsiveness and perturbation resistance of

flapping wingbeats. In our first experiments, we provide a step change in the input forcing

amplitude to a series-elastic spring-wing system and observe the response time of the wing

amplitude increase. In our second experiments we provide an external fluid flow directed at the

flapping wing and study the perturbed steady-state wing motion. We evaluate both experiments

across Weis-Fogh numbers from 1 <N<10. The results indicate that spring-wing systems

designed for maximum energetic efficiency also experience trade-offs in agility and stability as the

Weis-Fogh number increases. Our results demonstrate that energetic efficiency and wing

maneuverability are in conflict in resonant spring-wing systems, suggesting that mechanical

resonance presents tradeoffs in insect flight control and stability.

1. Introduction

Flapping flight is an extremely power-intensive mode

of locomotion, requiring both high frequency wing-

beats and large forces to produce lift and perform

agile maneuvers. Flying insects achieve efficient flight

through a combination of specialized flight muscles

[1] and elastic energy storage in the thorax [2–4]. The

insect flight system can thus be described as muscle

actuation of an elastic structure which oscillates wings

to generate aerodynamic forces. We call this combina-

tion of elastic, inertial, and aerodynamic mechanisms

a ‘spring-wing’ system [5]. While significant research

focus has been devoted to the aerodynamic force gen-

eration of flapping wings (see review in [6]), relat-

ively fewer studies have focused on understanding the

implications of elastic energy storage and return for

flight dynamics and control [3,4,7–10].

In the classic spring-mass-damper model, there

exists a particular actuation frequency which results

in the largest amplitude oscillation of a mass, the so-

called resonance frequency. In the performance con-

siderations for a ‘spring-wing’ system, there exist sev-

eral different resonant wingbeat frequencies at which

different forms of optimality (maximum amplitude,

lift, or efficiency, for example) are achieved [11].

Operating at a resonant frequency that maximizes

lift can enable significant performance advantage,

allowing insects to use smaller muscle force/power to

generate lift for flight. Indeed, roboticists designing

insect-scale flapping robots have found that incorpor-

ating elasticity and operating near resonance enables

Efficiency and control trade-offs and work loop characteristics of flapping-wing systems with synchronous and asynchronous muscles

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Flying animals resort to fast, large-degree-of-freedom motion of flapping wings, a key feature that distinguishes them from rotary or fixed-winged robotic fliers with limited motion of aerodynamic surfaces. However, flapping-wing aerodynamics are characterized by highly unsteady and three-dimensional flows difficult to model or control, and accurate aerodynamic force predictions often rely on expensive computational or experimental methods. Here, we developed a computationally efficient and data-driven state-space model to dynamically map wing kinematics to aerodynamic forces/moments. This model was trained and tested with a total of 548 different flapping-wing motions and surpassed the accuracy and generality of the existing quasi-steady models. This model used 12 states to capture the unsteady and nonlinear fluid effects pertinent to force generation without explicit information of fluid flows. We also provided a comprehensive assessment of the control authority of key wing kinematic variables and found that instantaneous aerodynamic forces/moments were largely predictable by the wing motion history within a half-stroke cycle. Furthermore, the angle of attack, normal acceleration and pitching motion had the strongest effects on the aerodynamic force/moment generation. Our results show that flapping flight inherently offers high force control authority and predictability, which can be key to developing agile and stable aerial fliers.

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Synopsis Dimensionless numbers have long been used in comparative biomechanics to quantify competing scaling relationships and connect morphology to animal performance. While common in aerodynamics, few relate the biomechanics of the organism to the forces produced on the environment during flight. We discuss the Weis-Fogh number, N, as a dimensionless number specific to flapping flight, which describes the resonant properties of an insect and resulting tradeoffs between energetics and control. Originally defined by Torkel Weis-Fogh in his seminal 1973 paper, N measures the ratio of peak inertial to aerodynamic torque generated by an insect over a wingbeat. In this perspectives piece, we define N for comparative biologists and describe its interpretations as a ratio of torques and as the width of an insect’s resonance curve. We then discuss the range of N realized by insects and explain the fundamental tradeoffs between an insect’s aerodynamic efficiency, stability, and responsiveness that arise as a consequence of variation in N, both across and within species. N is therefore an especially useful quantity for comparative approaches to the role of mechanics and aerodynamics in insect flight.

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Many insects utilize both amplitude and frequency modulation of their wingstrokes to control flight, a combination with potential advantages for flapping-wing micro-aerial vehicle (FWMAV) flight control as we attempt to mimic the agility of small biological fliers. However, frequency-modulated control is uncommon in insect-scale FWMAVs, and these vehicles have not yet demonstrated graceful, mid-wingstroke frequency transitions in flight. We propose a method to allow active frequency control as a primary control variable, and describe a method to achieve smooth frequency transitions in flight. We demonstrate that frequency, in combination with other flapping parameters, may be used to generate arbitrary, time-varying forces and torques. Additionally, we explore the advantages of frequency modulation in FWMAV flight, applying a frequency-based method to control the Harvard RoboBee during hovering flight.

Stability and agility trade-offs in spring-wing systems

Abstract and Figures

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