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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. 

Synopsis 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. 

Abstract Since taking flight, insects have undergone repeated evolutionary transitions between two seemingly distinct flight modes^1–3. 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 stretch^2–8. These modes reflect the two fundamental ways of generating rhythmic movement: time-periodic forcing versus emergent oscillations from self-excitation^8–10. 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 phylogeny^3,6, 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 robot^11–13that 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. 

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 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. 

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 frequencydependent 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. 

In the last decade, roboticists have had significant success building centimeter-scale flapping wing micro aerial vehicles (FWMAVs) inspired by the flight of insects. Evidence suggests that insects store and release energy in the thoracic exoskeleton to improve energy efficiency by flapping at resonance. Insect-inspired micro flying robots have also leveraged resonance to improve efficiency, but they have discovered that operating at the resonant frequency leads to issues with flight control. This research seeks to investigate the roles that elasticity, aerodynamics, and muscle dynamics play in the emergent dynamics of flapping flight by studying elastic flapping spring-wing systems using dynamically-scaled robophysical models of springwings. Studying the dynamics of a robot with comparable features enables the validation of models from biology that are otherwise difficult to test in living insects, the generation of new hypotheses, and the development of novel FWMAV designs. In Chapter 1, the spring-wing system is characterized via a nonlinear spring-mass-damper model. A robophysical model validates that such systems gain energetic benefits from operating at resonance, but reveals that the benefit scales with an underappreciated dimensionless ratio of inertial to aerodynamic forces, the Weis-Fogh number. We show through dimensional analysis that any real system, living or robotic, must balance the mechanical advantage gained from operating at resonance with diminishing returns in efficiency. Chapter 2 further explores the impact of the Weis-Fogh number on flapping dynamics, showing that responsiveness to control inputs is reduced and resistance to environmental perturbations is increased as the dimensionless ratio increases. Together with calculations of Weis-Fogh number in insects, these studies illustrate tradeoffs that drive evolution of resonant flight in nature and guide development of future FWMAVs with elastic energy exchange. In the second half of the thesis, muscle dynamics are introduced in the form of a simplified model of self-excited asynchronous insect muscle. In Chapter 3, a form of velocity feedback, adapted from experiments on insect flight muscle, is developed and integrated with the springwing model, producing a system that generates steady flapping via limit-cycle oscillations despite the absence of periodic control inputs. The model is explored analytically, in simulation, and via implementation on the robotic spring-wing. Novel dynamic characteristics that enable adaptation to damage and passive response to wing collisions are described. Chapter 4 leverages the asynchronous feedback model as part of an interdisciplinary study of the evolution of asynchronous muscle. Phylogenetic analysis, direct measurement of insect muscle dynamics, and experiments on the robophysical system show that evolutionary transitions between periodicallyforced and self-excited insect muscle were likely made possible by a ”bridge” in the dynamic parameter space that could be traversed under specific conditions. The asynchronous spring-wing model provides new insight into the flight and evolution of some of the most agile insects in nature, and presents a novel adaptive control scheme for future FWMAVs. 

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. 

Mobile millimeter and centimeter scale robots often use smart composite manufacturing (SCM) for the construction of body components and mechanisms. The fabrication of SCM mechanisms requires laser machining and laminating flexible, adhesive, and structural materials into small-scale hinges, transmissions, and, ultimately, wings or legs. However, a fundamental limitation of SCM components is the plastic deformation and failure of flexures. In this work, we demonstrate that encasing SCM components in a soft silicone mold dramatically improves the durability of SCM flexure hinges and provides robustness to SCM components. We demonstrate this advance in the design of a flapping-wing robot that uses an underactuated compliant transmission fabricated with an inner SCM skeleton and exterior silicone mold. The transmission design is optimized to achieve desired wingstroke requirements and to allow for independent motion of each wing. We validate these design choices in bench-top tests, measuring transmission compliance, kinematics, and fatigue. We integrate the transmission with laminate wings and two types of actuation, demonstrating elastic energy exchange and limited lift-off capabilities. Lastly, we tested collision mitigation through flapping-wing experiments that obstructed the motion of a wing. These experiments demonstrate that an underactuated compliant transmission can provide resilience and robustness to flapping-wing robots. 

In most instances, flapping wing robots have emulated the “synchronous” actuation of insects in which the wingbeat timing is generated from a time-dependent, rhythmic signal. The internal dynamics of asynchronous insect flight muscle enable high-frequency, adaptive wingbeats with minimal direct neural control. In this paper, we investigate how the delayed stretch-activation (dSA) response of asynchronous insect flight muscle can be transformed into a feedback control law for flapping wing robots that results in stable limit cycle wingbeats. We first demonstrate - in theory and simulation - the mechanism by which asynchronous wingbeats self-excite. Then, we implement the feedback law on a dynamically-scaled robophysical model as well as on an insect-scale robotic flapping wing. Experiments on large- and small-scale robots demonstrate good agreement with the theory results and highlight how dSA parameters govern wingbeat amplitude and frequency. Lastly, we demonstrate that asynchronous actuation has several advantages over synchronous actuation schemes, including the ability to rapidly adapt or halt wingbeats in response to external loads or collisions through low-level feedback control.