Rapid frequency modulation in a resonant system: aerial perturbation recovery in hawkmoths

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

Natural fliers with flapping wings face the dual challenges of energy efficiency and active control of wing motion for achieving diverse modes of flight. It is hypothesized that flapping-wing systems use resonance to improve muscle mechanical output energy efficiency, a principle often followed in bioinspired flapping-wing robots. However, resonance can limit the degree of active control, a trade-off rooted in the dynamics of wing motor systems and can be potentially reflected in muscle work loops. To systematically investigate how energy efficiency trades off with active control of wingbeat frequency and amplitude, here we developed a parsimonious model of the wing motor system with either synchronous or asynchronous power muscles. We then non-dimensionalized the model and performed simulations to examine model characteristics as functions of Weis-Fogh number and dimensionless flapping frequency. For synchronous power muscles, our model predicts that energy efficiency trades off with frequency control rather than amplitude control at high Weis-Fogh numbers; however, no such trade-off was found for models with asynchronous power muscles. The work loops alone are insufficient to fully capture wing motor characteristics, and therefore fail to directly reflect the trade-offs. Finally, using simulation results, we predict that natural fliers function at Weis-Fogh numbers close to 1.

Stability and agility trade-offs in spring-wing systems

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 withN) 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 from1 Collapse

Key Words

• dynamic scaling
• elasticity
• insect flight
• resonance
• robophysics

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MESH Headings

• Wings, Animal/physiology
• Animals
• Flight, Animal/physiology
• Models, Biological
• Computer Simulation
• Insecta/physiology
• Biomimetics/methods
• Biomimetics/instrumentation
• Elastic Modulus/physiology
• Equipment Design
• Energy Transfer/physiology

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Grants Collapse

Affiliation(s)

James Lynch Department of Mechanical & Aerospace Engineering, University of California, San Diego, CA, United States of America
Ethan S Wold School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA, United States of America
Jeff Gau Interdisciplinary Bioengineering Graduate Program and George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, United States of America
Simon Sponberg School of Biological Sciences, Georgia Institute of Technology, Atlanta, GA, United States of America School of Physics, Georgia Institute of Technology, Atlanta, GA, United States of America
Nick Gravish Department of Mechanical & Aerospace Engineering, University of California, San Diego, CA, United States of America

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The Weis-Fogh Number Describes Resonant Performance Tradeoffs in Flapping Insects

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.

Insect Flight: State of the Field and Future Directions

The evolution of flight in an early winged insect ancestral lineage is recognized as a key adaptation explaining the unparalleled success and diversification of insects. Subsequent transitions and modifications to flight machinery, including secondary reductions and losses, also play a central role in shaping the impacts of insects on broadscale geographic and ecological processes and patterns in the present and future. Given the importance of insect flight, there has been a centuries-long history of research and debate on the evolutionary origins and biological mechanisms of flight. Here, we revisit this history from an interdisciplinary perspective, discussing recent discoveries regarding the developmental origins, physiology, biomechanics, and neurobiology and sensory control of flight in a diverse set of insect models. We also identify major outstanding questions yet to be addressed and provide recommendations for overcoming current methodological challenges faced when studying insect flight, which will allow the field to continue to move forward in new and exciting directions. By integrating mechanistic work into ecological and evolutionary contexts, we hope that this synthesis promotes and stimulates new interdisciplinary research efforts necessary to close the many existing gaps about the causes and consequences of insect flight evolution.

Moth resonant mechanics are tuned to wingbeat frequency and energetic demands

An insect's wingbeat frequency is a critical determinant of its flight performance and varies by multiple orders of magnitude across Insecta. Despite potential energetic benefits for an insect that matches its wingbeat frequency to its resonant frequency, recent work has shown that moths may operate off their resonant peak. We hypothesized that across species, wingbeat frequency scales with resonance frequency to maintain favourable energetics, but with an offset in species that use frequency modulation as a means of flight control. The moth superfamily Bombycoidea is ideal for testing this hypothesis because their wingbeat frequencies vary across species by an order of magnitude, despite similar morphology and actuation. We used materials testing, high-speed videography and a model of resonant aerodynamics to determine how components of an insect's flight apparatus (stiffness, wing inertia, muscle strain and aerodynamics) vary with wingbeat frequency. We find that the resonant frequency of a moth correlates with wingbeat frequency, but resonance curve shape (described by the Weis-Fogh number) and peak location vary within the clade in a way that corresponds to frequency-dependent biomechanical demands. Our results demonstrate that a suite of adaptations in muscle, exoskeleton and wing drive variation in resonant mechanics, reflecting potential constraints on matching wingbeat and resonant frequencies.

The Functions of Phasic Wing-Tip Folding on Flapping-Wing Aerodynamics

Insects produce a variety of highly acrobatic maneuvers in flight owing to their ability to achieve various wing-stroke trajectories. Among them, beetles can quickly change their flight velocities and make agile turns. In this work, we report a newly discovered phasic wing-tip-folding phenomenon and its aerodynamic basis in beetles. The wings' flapping trajectories and aerodynamic forces of the tethered flying beetles were recorded simultaneously via motion capture cameras and a force sensor, respectively. The results verified that phasic active spanwise-folding and deployment (PASFD) can exist during flapping flight. The folding of the wing-tips of beetles significantly decreased aerodynamic forces without any changes in flapping frequency. Specifically, compared with no-folding-and-deployment wings, the lift and forward thrust generated by bilateral-folding-and-deployment wings reduced by 52.2% and 63.0%, respectively. Moreover, unilateral-folding-and-deployment flapping flight was found, which produced a lateral force (8.65 mN). Therefore, a micro-flapping-wing mechanism with PASFD was then designed, fabricated, and tested in a motion capture and force measurement system to validate its phasic folding functions and aerodynamic performance under different operating frequencies. The results successfully demonstrated a significant decrease in flight forces. This work provides valuable insights for the development of flapping-wing micro-air-vehicles with high maneuverability.

Bridging two insect flight modes in evolution, physiology and robophysics

Since taking flight, insects have undergone repeated evolutionary transitions between two seemingly distinct flight modes1-3. Some insects neurally activate their muscles synchronously with each wingstroke. However, many insects have achieved wingbeat frequencies beyond the speed limit of typical neuromuscular systems by evolving flight muscles that are asynchronous with neural activation and activate in response to mechanical stretch2-8. These modes reflect the two fundamental ways of generating rhythmic movement: time-periodic forcing versus emergent oscillations from self-excitation8-10. How repeated evolutionary transitions have occurred and what governs the switching between these distinct modes remain unknown. Here we find that, despite widespread asynchronous actuation in insects across the phylogeny3,6, asynchrony probably evolved only once at the order level, with many reversions to the ancestral, synchronous mode. A synchronous moth species, evolved from an asynchronous ancestor, still preserves the stretch-activated muscle physiology. Numerical and robophysical analyses of a unified biophysical framework reveal that rather than a dichotomy, these two modes are two regimes of the same dynamics. Insects can transition between flight modes across a bridge in physiological parameter space. Finally, we integrate these two actuation modes into an insect-scale robot11-13 that enables transitions between modes and unlocks a new self-excited wingstroke strategy for engineered flight. Together, this framework accounts for repeated transitions in insect flight evolution and shows how flight modes can flip with changes in physiological parameters.

Structural damping renders the hawkmoth exoskeleton mechanically insensitive to non-sinusoidal deformations

Muscles act through elastic and dissipative elements to mediate movement, which can introduce dissipation and filtering which are important for energetics and control. The high power requirements of flapping flight can be reduced by an insect's exoskeleton, which acts as a spring with frequency-independent material properties under purely sinusoidal deformation. However, this purely sinusoidal dynamic regime does not encompass the asymmetric wing strokes of many insects or non-periodic deformations induced by external perturbations. As such, it remains unknown whether a frequency-independent model applies broadly and what implications it has for control. We used a vibration testing system to measure the mechanical properties of isolated Manduca sexta thoraces under symmetric, asymmetric and band-limited white noise deformations. The asymmetric and white noise conditions represent two types of generalized, multi-frequency deformations that may be encountered during steady-state and perturbed flight. Power savings and dissipation were indistinguishable between symmetric and asymmetric conditions, demonstrating that no additional energy is required to deform the thorax non-sinusoidally. Under white noise conditions, stiffness and damping were invariant with frequency, suggesting that the thorax has no frequency-dependent filtering properties. A simple flat frequency response function fits our measured frequency response. This work demonstrates the potential of materials with frequency-independent damping to simplify motor control by eliminating any velocity-dependent filtering that viscoelastic elements usually impose between muscle and wing.

By a whisker: the sensory role of vibrissae in hovering flight in nectarivorous bats

Whiskers are important tactile structures widely used across mammals for a variety of sensory functions, but it is not known how bats-representing about a fifth of all extant mammal species-use them. Nectar-eating bats typically have long vibrissae (long, stiff hairs) arranged in a forward-facing brush-like formation that is not present in most non-nectarivorous bats. They also commonly use a unique flight strategy to access their food-hovering flight. Here we investigated whether these species use their vibrissae to optimize their feeding by assisting fine flight control. We used behavioural experiments to test if bats' flight trajectory into the flower changed after vibrissa removal, and phylogenetic comparative methods to test whether vibrissa length is related to nectarivory. We found that bat flight trajectory was altered after vibrissae removal and that nectarivorous bats possess longer vibrissae than non-nectivorous species, providing evidence of an additional source of information in bats' diverse sensory toolkit.

Dual spring force couples yield multifunctionality and ultrafast, precision rotation in tiny biomechanical systems

Small organisms use propulsive springs rather than muscles to repeatedly actuate high acceleration movements, even when constrained to tiny displacements and limited by inertial forces. Through integration of a large kinematic dataset, measurements of elastic recoil, energetic math modeling and dynamic math modeling, we tested how trap-jaw ants (Odontomachus brunneus) utilize multiple elastic structures to develop ultrafast and precise mandible rotations at small scales. We found that O. brunneus develops torque on each mandible using an intriguing configuration of two springs: their elastic head capsule recoils to push and the recoiling muscle-apodeme unit tugs on each mandible. Mandibles achieved precise, planar, circular trajectories up to 49,100 rad s-1 (470,000 rpm) when powered by spring propulsion. Once spring propulsion ended, the mandibles moved with unconstrained and oscillatory rotation. We term this mechanism a 'dual spring force couple', meaning that two springs deliver energy at two locations to develop torque. Dynamic modeling revealed that dual spring force couples reduce the need for joint constraints and thereby reduce dissipative joint losses, which is essential to the repeated use of ultrafast, small systems. Dual spring force couples enable multifunctionality: trap-jaw ants use the same mechanical system to produce ultrafast, planar strikes driven by propulsive springs and for generating slow, multi-degrees of freedom mandible manipulations using muscles, rather than springs, to directly actuate the movement. Dual spring force couples are found in other systems and are likely widespread in biology. These principles can be incorporated into microrobotics to improve multifunctionality, precision and longevity of ultrafast systems.

The hawkmoth wingbeat is not at resonance

Flying insects have elastic materials within their exoskeletons that could reduce the energetic cost of flight if their wingbeat frequency is matched to a mechanical resonance frequency. Flapping at resonance may be essential across flying insects because of the power demands of small-scale flapping flight. However, building up large-amplitude resonant wingbeats over many wingstrokes may be detrimental for control if the total mechanical energy in the spring-wing system exceeds the per-cycle work capacity of the flight musculature. While the mechanics of the insect flight apparatus can behave as a resonant system, the question of whether insects flap their wings at their resonant frequency remains unanswered. Using previous measurements of body stiffness in the hawkmoth, Manduca sexta, we develop a mechanical model of spring-wing resonance with aerodynamic damping and characterize the hawkmoth's resonant frequency. We find that the hawkmoth's wingbeat frequency is approximately 80% above resonance and remains so when accounting for uncertainty in model parameters. In this regime, hawkmoths may still benefit from elastic energy exchange while enabling control of aerodynamic forces via frequency modulation. We conclude that, while insects use resonant mechanics, tuning wingbeats to a simple resonance peak is not a necessary feature for all centimetre-scale flapping flyers.

Distinct forms of resonant optimality within insect indirect flight motors

Insect flight motors are extraordinary natural structures that operate efficiently at high frequencies. Structural resonance is thought to play a role in ensuring efficient motor operation, but the details of this role are elusive. While the efficiency benefits associated with resonance may be significant, a range of counterintuitive behaviours are observed. In particular, the relationship between insect wingbeat frequencies and thoracic natural frequencies is uncertain, with insects showing wingbeat frequency modulation over both short and long time scales. Here, we offer new explanations for this modulation. We show how, in linear and nonlinear models of an indirect flight motor, resonance is not a unitary state at a single frequency, but a complex cluster of distinct and mutually exclusive states, each representing a different form of resonant optimality. Additionally, by characterizing the relationship between resonance and the state of negative work absorption within the motor, we demonstrate how near-perfect resonant energetic optimality can be maintained over significant wingbeat frequency ranges. Our analysis leads to a new conceptual model of flight motor operation: one in which insects are not energetically restricted to a precise wingbeat frequency, but instead are robust to changes in thoracic and environmental properties-an illustration of the extraordinary robustness of these natural motors.

How and why do bees buzz? Implications for buzz pollination

Buzz pollination encompasses the evolutionary convergence of specialized floral morphologies and pollinator behaviour in which bees use vibrations (floral buzzes) to remove pollen. Floral buzzes are one of several types of vibrations produced by bees using their thoracic muscles. Here I review how bees can produce these different types of vibrations and discuss the implications of this mechanistic understanding for buzz pollination. I propose that bee buzzes can be categorized according to their mode of production and deployment into: (i) thermogenic, which generate heat with little mechanical vibration; (ii) flight buzzes which, combined with wing deployment and thoracic vibration, power flight; and (iii) non-flight buzzes in which the thorax vibrates but the wings remain mostly folded, and include floral, defence, mating, communication, and nest-building buzzes. I hypothesize that the characteristics of non-flight buzzes, including floral buzzes, can be modulated by bees via modification of the biomechanical properties of the thorax through activity of auxiliary muscles, changing the rate of activation of the indirect flight muscles, and modifying flower handling behaviours. Thus, bees should be able to fine-tune mechanical properties of their floral vibrations, including frequency and amplitude, depending on flower characteristics and pollen availability to optimize energy use and pollen collection.

Rapid frequency modulation in a resonant system: aerial perturbation recovery in hawkmoths

Centimetre-scale fliers must contend with the high power requirements of flapping flight. Insects have elastic elements in their thoraxes which may reduce the inertial costs of their flapping wings. Matching wingbeat frequency to a mechanical resonance can be energetically favourable, but also poses control challenges. Many insects use frequency modulation on long timescales, but wingstroke-to-wingstroke modulation of wingbeat frequencies in a resonant spring-wing system is potentially costly because muscles must work against the elastic flight system. Nonetheless, rapid frequency and amplitude modulation may be a useful control modality. The hawkmoth Manduca sexta has an elastic thorax capable of storing and returning significant energy. However, its nervous system also has the potential to modulate the driving frequency of flapping because its flight muscles are synchronous. We tested whether hovering hawkmoths rapidly alter frequency during perturbations with vortex rings. We observed both frequency modulation (32% around mean) and amplitude modulation (37%) occurring over several wingstrokes. Instantaneous phase analysis of wing kinematics revealed that more than 85% of perturbation responses required active changes in neurogenic driving frequency. Unlike their robotic counterparts that abdicate frequency modulation for energy efficiency, synchronous insects use wingstroke-to-wingstroke frequency modulation despite the power demands required for deviating from resonance.