Floquet stability analysis of the longitudinal dynamics of two hovering model insects

Because of the periodically varying aerodynamic and inertial forces of the flapping wings, a hovering or constant-speed flying insect is a cyclically forcing system, and, generally, the flight is not in a fixed-point equilibrium, but in a cyclic-motion equilibrium. Current stability theory of insect flight is based on the averaged model and treats the flight as a fixed-point equilibrium. In the present study, we treated the flight as a cyclic-motion equilibrium and used the Floquet theory to analyse the longitudinal stability of insect flight. Two hovering model insects were considered—a dronefly and a hawkmoth. The former had relatively high wingbeat frequency and small wing-mass to body-mass ratio, and hence very small amplitude of body oscillation; while the latter had relatively low wingbeat frequency and large wing-mass to body-mass ratio, and hence relatively large amplitude of body oscillation. For comparison, analysis using the averaged-model theory (fixed-point stability analysis) was also made. Results of both the cyclic-motion stability analysis and the fixed-point stability analysis were tested by numerical simulation using complete equations of motion coupled with the Navier–Stokes equations. The Floquet theory (cyclic-motion stability analysis) agreed well with the simulation for both the model dronefly and the model hawkmoth; but the averaged-model theory gave good results only for the dronefly. Thus, for an insect with relatively large body oscillation at wingbeat frequency, cyclic-motion stability analysis is required, and for their control analysis, the existing well-developed control theories for systems of fixed-point equilibrium are no longer applicable and new methods that take the cyclic variation of the flight dynamics into account are needed.     

Vortex wake,downwash distribution,aerodynamic performance and wingbeat kinematics in slow-flying pied flycatchers

Many small passerines regularly fly slowly when catching prey, flying in cluttered environments or landing on a perch or nest. While flying slowly, passerines generate most of the flight forces during the downstroke, and have a ‘feathered upstroke’ during which they make their wing inactive by retracting it close to the body and by spreading the primary wing feathers. How this flight mode relates aerodynamically to the cruising flight and so-called ‘normal hovering’ as used in hummingbirds is not yet known. Here, we present time-resolved fluid dynamics data in combination with wingbeat kinematics data for three pied flycatchers flying across a range of speeds from near hovering to their calculated minimum power speed. Flycatchers are adapted to low speed flight, which they habitually use when catching insects on the wing. From the wake dynamics data, we constructed average wingbeat wakes and determined the time-resolved flight forces, the time-resolved downwash distributions and the resulting lift-to-drag ratios, span efficiencies and flap efficiencies. During the downstroke, slow-flying flycatchers generate a single-vortex loop wake, which is much more similar to that generated by birds at cruising flight speeds than it is to the double loop vortex wake in hovering hummingbirds. This wake structure results in a relatively high downwash behind the body, which can be explained by the relatively active tail in flycatchers. As a result of this, slow-flying flycatchers have a span efficiency which is similar to that of the birds in cruising flight and which can be assumed to be higher than in hovering hummingbirds. During the upstroke, the wings of slowly flying flycatchers generated no significant forces, but the body–tail configuration added 23 per cent to weight support. This is strikingly similar to the 25 per cent weight support generated by the wing upstroke in hovering hummingbirds. Thus, for slow-flying passerines, the upstroke cannot be regarded as inactive, and the tail may be of importance for flight efficiency and possibly manoeuvrability.     

Stable hovering of a jellyfish-like flying machine

Ornithopters, or flapping-wing aircraft, offer an alternative to helicopters in achieving manoeuvrability at small scales, although stabilizing such aerial vehicles remains a key challenge. Here, we present a hovering machine that achieves self-righting flight using flapping wings alone, without relying on additional aerodynamic surfaces and without feedback control. We design, construct and test-fly a prototype that opens and closes four wings, resembling the motions of swimming jellyfish more so than any insect or bird. Measurements of lift show the benefits of wing flexing and the importance of selecting a wing size appropriate to the motor. Furthermore, we use high-speed video and motion tracking to show that the body orientation is stable during ascending, forward and hovering flight modes. Our experimental measurements are used to inform an aerodynamic model of stability that reveals the importance of centre-of-mass location and the coupling of body translation and rotation. These results show the promise of flapping-flight strategies beyond those that directly mimic the wing motions of flying animals.     

A new twist on gyroscopic sensing: body rotations lead to torsion in flapping,flexing insect wings

Insects perform fast rotational manoeuvres during flight. While two insect orders use flapping halteres (specialized organs evolved from wings) to detect body dynamics, it is unknown how other insects detect rotational motions. Like halteres, insect wings experience gyroscopic forces when they are flapped and rotated and recent evidence suggests that wings might indeed mediate reflexes to body rotations. But, can gyroscopic forces be detected using only changes in the structural dynamics of a flapping, flexing insect wing? We built computational and robotic models to rotate a flapping wing about an axis orthogonal to flapping. We recorded high-speed video of the model wing, which had a flexural stiffness similar to the wing of the Manduca sexta hawkmoth, while flapping it at the wingbeat frequency of Manduca (25 Hz). We compared the three-dimensional structural dynamics of the wing with and without a 3 Hz, 10° rotation about the yaw axis. Our computational model revealed that body rotation induces a new dynamic mode: torsion. We verified our result by measuring wing tip displacement, shear strain and normal strain of the robotic wing. The strains we observed could stimulate an insect''s mechanoreceptors and trigger reflexive responses to body rotations.     

Hovering performance of Anna's hummingbirds (Calypte anna) in ground effect

Aerodynamic performance and energetic savings for flight in ground effect are theoretically maximized during hovering, but have never been directly measured for flying animals. We evaluated flight kinematics, metabolic rates and induced flow velocities for Anna''s hummingbirds hovering at heights (relative to wing length R = 5.5 cm) of 0.7R, 0.9R, 1.1R, 1.7R, 2.2R and 8R above a solid surface. Flight at heights less than or equal to 1.1R resulted in significant reductions in the body angle, tail angle, anatomical stroke plane angle, wake-induced velocity, and mechanical and metabolic power expenditures when compared with flight at the control height of 8R. By contrast, stroke plane angle relative to horizontal, wingbeat amplitude and wingbeat frequency were unexpectedly independent of height from ground. Qualitative smoke visualizations suggest that each wing generates a vortex ring during both down- and upstroke. These rings expand upon reaching the ground and present a complex turbulent interaction below the bird''s body. Nonetheless, hovering near surfaces results in substantial energetic benefits for hummingbirds, and by inference for all volant taxa that either feed at flowers or otherwise fly close to plant or other surfaces.     

Vision-based flight control in the hawkmoth Hyles lineata

Vision is a key sensory modality for flying insects, playing an important role in guidance, navigation and control. Here, we use a virtual-reality flight simulator to measure the optomotor responses of the hawkmoth Hyles lineata, and use a published linear-time invariant model of the flight dynamics to interpret the function of the measured responses in flight stabilization and control. We recorded the forces and moments produced during oscillation of the visual field in roll, pitch and yaw, varying the temporal frequency, amplitude or spatial frequency of the stimulus. The moths’ responses were strongly dependent upon contrast frequency, as expected if the optomotor system uses correlation-type motion detectors to sense self-motion. The flight dynamics model predicts that roll angle feedback is needed to stabilize the lateral dynamics, and that a combination of pitch angle and pitch rate feedback is most effective in stabilizing the longitudinal dynamics. The moths’ responses to roll and pitch stimuli coincided qualitatively with these functional predictions. The moths produced coupled roll and yaw moments in response to yaw stimuli, which could help to reduce the energetic cost of correcting heading. Our results emphasize the close relationship between physics and physiology in the stabilization of insect flight.     

Wake structure and wingbeat kinematics of a house-martin Delichon urbica.

The wingbeat kinematics and wake structure of a trained house martin in free, steady flight in a wind tunnel have been studied over a range of flight speeds, and compared and contrasted with similar measurements for a thrush nightingale and a pair of robins. The house martin has a higher aspect ratio (more slender) wing, and is a more obviously agile and aerobatic flyer, catching insects on the wing. The wingbeat is notable for the presence at higher flight speeds of a characteristic pause in the upstroke. The essential characteristics of the wing motions can be reconstructed with a simple two-frequency model derived from Fourier analysis. At slow speeds, the distribution of wake vorticity is more simple than for the other previously measured birds, and the upstroke does not contribute to weight support. The upstroke becomes gradually more significant as the flight speed increases, and although the vortex wake shows a signature of the pause phase, the global circulation measurements are otherwise in good agreement with surprisingly simple aerodynamic models, and with predictions across the different species, implying quite similar aerodynamic performance of the wing sections. The local Reynolds numbers of the wing sections are sufficiently low that the well-known instabilities of attached laminar flows over lifting surfaces, which are known to occur at two to three times this value, may not develop.     

Scaling law and enhancement of lift generation of an insect-size hovering flexible wing

We report a comprehensive scaling law and novel lift generation mechanisms relevant to the aerodynamic functions of structural flexibility in insect flight. Using a Navier–Stokes equation solver, fully coupled to a structural dynamics solver, we consider the hovering motion of a wing of insect size, in which the dynamics of fluid–structure interaction leads to passive wing rotation. Lift generated on the flexible wing scales with the relative shape deformation parameter, whereas the optimal lift is obtained when the wing deformation synchronizes with the imposed translation, consistent with previously reported observations for fruit flies and honeybees. Systematic comparisons with rigid wings illustrate that the nonlinear response in wing motion results in a greater peak angle compared with a simple harmonic motion, yielding higher lift. Moreover, the compliant wing streamlines its shape via camber deformation to mitigate the nonlinear lift-degrading wing–wake interaction to further enhance lift. These bioinspired aeroelastic mechanisms can be used in the development of flapping wing micro-robots.     

Flight efficiency is a key to diverse wing morphologies in small insects

Insect wings are hybrid structures that are typically composed of veins and solid membranes. In some of the smallest flying insects, however, the wing membrane is replaced by hair-like bristles attached to a solid root. Bristles and membranous wing surfaces coexist in small but not in large insect species. There is no satisfying explanation for this finding as aerodynamic force production is always smaller in bristled than solid wings. This computational study suggests that the diversity of wing structure in small insects results from aerodynamic efficiency rather than from the requirements to produce elevated forces for flight. The tested wings vary from fully membranous to sparsely bristled and were flapped around a wing root with lift- and drag-based wing kinematic patterns and at different Reynolds numbers (Re). The results show that the decrease in aerodynamic efficiency with decreasing surface solidity is significantly smaller at Re = 4 than Re = 57. A replacement of wing membrane by bristles thus causes less change in energetic costs for flight in small compared to large insects. As a consequence, small insects may fly with bristled and solid wing surfaces at similar efficacy, while larger insects must use membranous wings for an efficient production of flight forces. The above findings are significant for the biological fitness and dispersal of insects that fly at elevated energy expenditures.     

Force and flow structures of a wing performing flapping motion at low Reynolds number

Summary Aerodynamic force and flow structures of a wing performing a simplified flapping motion that emulates the wing motion of small insects in normal hovering flight are studied, using the method of numerically solving the Navier-Stokes equations. For a typical case (wing rotation-axis is at 0.25 chord position and wing rotation is symmetrical with respect to stroke reversal), large peaks inC _L andC _D are produced near the end of a stroke by the wing-rotation, but the wing-rotation also generates a vortical structure which induces strong downwash velocity, reducing the lift production in the early part of the following stroke; averaging over one flapping cycle,C _L is 28% larger than the steady-state value and is about that needed to support the weight of a small insect. The timing of the wing-rotation at stroke reversal can change the size of the peaks inC _L andC _D and their averages; e.g. in the case of shifting the wing-rotation forward in time by 7.5% of a stroke period, the averageC _L becomes 63% larger than the steady-state value, which is larger than that needed for weight supporting (and might provide extra force for control or maneuvers). When the rotation-axis is moved rearward to the middlechord position, the lift and drag peaks due to the wing-rotation become smaller, and in this case, the wing-rotation generates a vortical structure that tends to prevent the relative motion between positive and negative vorticities, also reducing the lift production in the early part of the following stroke, resulting in a smaller averageC _L .     

Mathematical modeling of sway, roll and yaw motions: order-wise analysis to determine coupled characteristics and numerical simulation for restoring moment’s sensitivity analysis

This paper investigates sway, roll and yaw motions of a floating body with the aim to determine coupled motion characteristics based on the order-wise analysis of hydrodynamic coefficients. To compute the hydrodynamic coefficients and wave force exerted on the floating body, we employ speed-dependent strip theory. The governing equations are solved analytically for linear restoring moment. For nonlinear restoring moment which is expressed as an odd-order polynomial of fifth degree in roll angle, we apply the Runge–Kutta–Gill method to solve the coupled equations. To investigate the effect of initial disturbances on sway, roll, yaw and speed of the body, numerical experiments have been carried out for a Panamax Container ship under the action of a sinusoidal wave of periodicity 11.2 s with varying wave height and speed. For the linear restoring moment, we first derive associated motion equations for various cases based on the relative magnitude of the hydrodynamic coefficients. The order-wise analysis leads to the classification of coupled characteristics exhibiting the nature of coupling. For the nonlinear restoring moment, we notice that the initial disturbance plays an important role in the ship’s stability. The effects of forward speed and variation in wave heights are illustrated through typical numerical experiments.     

A semi-empirical model of the aerodynamics of manoeuvring insect flight

Blade element modelling provides a quick analytical method for estimating the aerodynamic forces produced during insect flight, but such models have yet to be tested rigorously using kinematic data recorded from free-flying insects. This is largely because of the paucity of detailed free-flight kinematic data, but also because analytical limitations in existing blade element models mean that they cannot incorporate the complex three-dimensional movements of the wings and body that occur during insect flight. Here, we present a blade element model with empirically fitted aerodynamic force coefficients that incorporates the full three-dimensional wing kinematics of manoeuvring Eristalis hoverflies, including torsional deformation of their wings. The two free parameters were fitted to a large free-flight dataset comprising N = 26 541 wingbeats, and the fitted model captured approximately 80% of the variation in the stroke-averaged forces in the sagittal plane. We tested the robustness of the model by subsampling the data, and found little variation in the parameter estimates across subsamples comprising 10% of the flight sequences. The simplicity and generality of the model that we present is such that it can be readily applied to kinematic datasets from other insects, and also used for the study of insect flight dynamics.     

The role of wingbeat frequency and amplitude in flight power

Body-mounted accelerometers provide a new prospect for estimating power use in flying birds, as the signal varies with the two major kinematic determinants of aerodynamic power: wingbeat frequency and amplitude. Yet wingbeat frequency is sometimes used as a proxy for power output in isolation. There is, therefore, a need to understand which kinematic parameter birds vary and whether this is predicted by flight mode (e.g. accelerating, ascending/descending flight), speed or morphology. We investigate this using high-frequency acceleration data from (i) 14 species flying in the wild, (ii) two species flying in controlled conditions in a wind tunnel and (iii) a review of experimental and field studies. While wingbeat frequency and amplitude were positively correlated, R² values were generally low, supporting the idea that parameters can vary independently. Indeed, birds were more likely to modulate wingbeat amplitude for more energy-demanding flight modes, including climbing and take-off. Nonetheless, the striking variability, even within species and flight types, highlights the complexity of describing the kinematic relationships, which appear sensitive to both the biological and physical context. Notwithstanding this, acceleration metrics that incorporate both kinematic parameters should be more robust proxies for power than wingbeat frequency alone.     

昆虫飞行信息自动化采集装置的设计

针对现有昆虫飞行信息采集方法中存在的采集数据量大、采集效率低和束缚实验对象等问题,设计了一种基于视觉检测技术的自动化采集装置。该装置主要包括障碍通道模块、运动触发采集模块、飞行轨迹信息模块和自动采集控制系统。首先,根据控制功能的需求设计了间隙控制器,采用激光测距传感器和步进电机等硬件实现了闭环控制;其次,采用STM32微控制器为控制终端,并结合运动方向检测算法触发Blackfly S USB3高速相机,实现昆虫在指定运动方向的序列图像采集;然后,利用开源计算机视觉库OpenCV分析采集的序列图像,获取昆虫的飞行轨迹信息;最后,通过嵌入式控制系统协调各模块之间的通信,以达到高效、稳定获取昆虫飞行信息的设计要求。为了验证该装置的适用性和准确率,选取中华蜜蜂进行穿越间隙的实验,实验结果表明：在静态环境中,该装置能获取完整、清晰的飞行序列图像,平均准确率达到73.24%,采集性能稳定,有效提升了采集效率。通过分析蜜蜂的飞行轨迹信息,推断出其识别间隙的机制与横向运动的幅度和速度存在联系,这为深度研究蜜蜂的飞行机制提供了支持。     

The complex aerodynamic footprint of desert locusts revealed by large-volume tomographic particle image velocimetry

Particle image velocimetry has been the preferred experimental technique with which to study the aerodynamics of animal flight for over a decade. In that time, hardware has become more accessible and the software has progressed from the acquisition of planes through the flow field to the reconstruction of small volumetric measurements. Until now, it has not been possible to capture large volumes that incorporate the full wavelength of the aerodynamic track left behind during a complete wingbeat cycle. Here, we use a unique apparatus to acquire the first instantaneous wake volume of a flying animal''s entire wingbeat. We confirm the presence of wake deformation behind desert locusts and quantify the effect of that deformation on estimates of aerodynamic force and the efficiency of lift generation. We present previously undescribed vortex wake phenomena, including entrainment around the wing-tip vortices of a set of secondary vortices borne of Kelvin–Helmholtz instability in the shear layer behind the flapping wings.     

The wake of hovering flight in bats

Hovering means stationary flight at zero net forward speed, which can be achieved by animals through muscle powered flapping flight. Small bats capable of hovering typically do so with a downstroke in an inclined stroke plane, and with an aerodynamically active outer wing during the upstroke. The magnitude and time history of aerodynamic forces should be reflected by vorticity shed into the wake. We thus expect hovering bats to generate a characteristic wake, but this has until now never been studied. Here we trained nectar-feeding bats, Leptonycteris yerbabuenae, to hover at a feeder and using time-resolved stereoscopic particle image velocimetry in conjunction with high-speed kinematic analysis we show that hovering nectar-feeding bats produce a series of bilateral stacked vortex loops. Vortex visualizations suggest that the downstroke produces the majority of the weight support, but that the upstroke contributes positively to the lift production. However, the relative contributions from downstroke and upstroke could not be determined on the basis of the wake, because wake elements from down- and upstroke mix and interact. We also use a modified actuator disc model to estimate lift force, power and flap efficiency. Based on our quantitative wake-induced velocities, the model accounts for weight support well (108%). Estimates of aerodynamic efficiency suggest hovering flight is less efficient than forward flapping flight, while the overall energy conversion efficiency (mechanical power output/metabolic power) was estimated at 13%.     

Aerodynamic forces and flow structures of an airfoil in some unsteady motions at small Reynolds number

Summary The aerodynamic forces and flow structures of a NACA 0012 airfoil in some unsteady motions at small Reynolds number (Re=100) are studied by numerically solving the Navier-Stokes equations. These motions include airfoil acceleration and deceleration from one translational speed to another and rapidly pitching up in constant freestream (equivalent to pitching up during translational motion at constant speed). It is shown that at small Reynolds number (Re=100), when the airfoil is performing fast acceleration or deceleration from one speed to another, a large aerodynamic force can be generated during and for a time period after the acceleration or deceleration; a large aerodynamic force can also be generated when the airfoil is performing a fast pitching motion in a constant freestream. In these fast unsteady motions, an airfoil in low Re flow can produce a large aerodynamic force as effective as in large Re flow, or the effect of unsteady motion dominates the Reynolds number effect. During the fast unsteady motion of the airfoil, new layers of strong vorticity are formed near the upper and lower surfaces of the airfoil under the previously existing thick vorticity layers, and it is the generation and motion of the new vorticity layers that is mainly responsible for the generation of the large aerodynamic force; the large-scale structure and movement of the newly produced vorticity layers are similar to that of high Re flow.     

Insect-like flapping wing mechanism based on a double spherical Scotch yoke

We describe the rationale, concept, design and implementation of a fixed-motion (non-adjustable) mechanism for insect-like flapping wing micro air vehicles in hover, inspired by two-winged flies (Diptera). This spatial (as opposed to planar) mechanism is based on the novel idea of a double spherical Scotch yoke. The mechanism was constructed for two main purposes: (i) as a test bed for aeromechanical research on hover in flapping flight, and (ii) as a precursor design for a future flapping wing micro air vehicle. Insects fly by oscillating (plunging) and rotating (pitching) their wings through large angles, while sweeping them forwards and backwards. During this motion the wing tip approximately traces a "figure-of-eight" or a "banana" and the wing changes the angle of attack (pitching) significantly. The kinematic and aerodynamic data from free-flying insects are sparse and uncertain, and it is not clear what aerodynamic consequences different wing motions have. Since acquiring the necessary kinematic and dynamic data from biological experiments remains a challenge, a synthetic, controlled study of insect-like flapping is not only of engineering value, but also of biological relevance. Micro air vehicles are defined as flying vehicles approximately 150 mm in size (hand-held), weighing 50-100g, and are developed to reconnoitre in confined spaces (inside buildings, tunnels, etc.). For this application, insect-like flapping wings are an attractive solution and hence the need to realize the functionality of insect flight by engineering means. Since the semi-span of the insect wing is constant, the kinematics are spatial; in fact, an approximate figure-of-eight/banana is traced on a sphere. Hence a natural mechanism implementing such kinematics should be (i) spherical and (ii) generate mathematically convenient curves expressing the figure-of-eight/banana shape. The double spherical Scotch yoke design has property (i) by definition and achieves (ii) by tracing spherical Lissajous curves.     

A passerine spreads its tail to facilitate a rapid recovery of its body posture during hovering

We demonstrate experimentally that a passerine exploits tail spreading to intercept the downward flow induced by its wings to facilitate the recovery of its posture. The periodic spreading of its tail by the White-eye bird exhibits a phase correlation with both wingstroke motion and body oscillation during hovering flight. During a downstroke, a White-eye''s body undergoes a remarkable pitch-down motion, with the tail undergoing an upward swing. This pitch-down motion becomes appropriately suppressed at the end of the downstroke; the bird''s body posture then recovers gradually to its original status. Employing digital particle-image velocimetry, we show that the strong downward flow induced by downstroking the wings serves as an external jet flow impinging upon the tail, providing a depressing force on the tail to counteract the pitch-down motion of the bird''s body. Spreading of the tail enhances a rapid recovery of the body posture because increased forces are experienced. The maximum force experienced by a spread tail is approximately 2.6 times that of a non-spread tail.