Nonlinear flight dynamics and stability of hovering model insects

The effects of wing inertial forces and mean stroke angle on the pitch dynamics of hovering insects

This paper discusses the wing inertial effects and the important role of the mean stroke angle on the pitch dynamics of hovering insects. The paper also presents a dynamic model appropriate for averaging and discusses the pitch stability results derived from the model. The model is used to predict the body angle of five insect species during hover, which are in good agreement with the available experimental results from different literature. The results suggest that the wing inertial forces have a considerable effect on pitch dynamics of insect flight and should not be ignored in dynamic analysis of hovering insects. The results also suggest that the body of hovering insects can not be vibrationally stabilized in a non-vertical orientation. Instead, the pitch angle of a hovering insect's body is mainly due to a balance of the moment of the insect's weight and the aerodynamic moment due to flapping kinematics with a nonzero mean stroke angle. Experiments with a flapping wing device confirm this results. To clearly explain the used model and clarify the difference between vibrational and non-vibrational stabilization, first this paper discusses the vibrational control of a three-degree-of-freedom force-input pendulum with its pivot moving in a vertical plane.

Effect of Hindwings on the Aerodynamics and Passive Dynamic Stability of a Hovering Hawkmoth

Insects are able to fly stably in the complex environment of the various gusts that occur in nature. In addition, many insects suffer wing damage in their lives, but many species of insects are capable of flying without their hindwings. Here, we evaluated the effect of hindwings on aerodynamics using a Navier-Stokes-based numerical model, and then the passive dynamic stability was evaluated by coupling the equation of motion in three degrees of freedom with the aerodynamic forces estimated by the CFD solver under large and small perturbation conditions. In terms of aerodynamic effects, the presence of the hindwings slightly reduces the efficiency for lift generation but enhances the partial LEV circulation and increases the downwash around the wing root. In terms of thrust, increasing the wing area around the hindwing region increases the thrust, and the relationship is almost proportional at the cycle-averaged value. The passive dynamic stability was not clearly affected by the presence of the hindwings, but the stability was slightly improved depending on the perturbation direction. These results may be useful for the integrated design of wing geometry and flight control systems in the development of flapping-winged micro air vehicles.

Wing Kinematics-Based Flight Control Strategy in Insect-Inspired Flight Systems: Deep Reinforcement Learning Gives Solutions and Inspires Controller Design in Flapping MAVs

Flying insects exhibit outperforming stability and control via continuous wing flapping even under severe disturbances in various conditions of wind gust and turbulence. While conventional linear proportional derivative (PD)-based controllers are widely employed in insect-inspired flight systems, they usually fail to deal with large perturbation conditions in terms of the 6-DoF nonlinear control strategy. Here we propose a novel wing kinematics-based controller, which is optimized based on deep reinforcement learning (DRL) to stabilize bumblebee hovering under large perturbations. A high-fidelity Open AI Gym environment is established through coupling a CFD data-driven aerodynamic model and a 6-DoF flight dynamic model. The control policy with an action space of 4 is optimized using the off-policy Soft Actor-Critic (SAC) algorithm with automating entropy adjustment, which is verified to be of feasibility and robustness to achieve fast stabilization of the bumblebee hovering flight under full 6-DoF large disturbances. The 6-DoF wing kinematics-based DRL control strategy may provide an efficient autonomous controller design for bioinspired flapping-wing micro air vehicles.

Accommodating unobservability to control flight attitude with optic flow

Attitude control is an essential flight capability. Whereas flying robots commonly rely on accelerometers¹ for estimating attitude, flying insects lack an unambiguous sense of gravity^2,3. Despite the established role of several sense organs in attitude stabilization^3-5, the dependence of flying insects on an internal gravity direction estimate remains unclear. Here we show how attitude can be extracted from optic flow when combined with a motion model that relates attitude to acceleration direction. Although there are conditions such as hover in which the attitude is unobservable, we prove that the ensuing control system is still stable, continuously moving into and out of these conditions. Flying robot experiments confirm that accommodating unobservability in this manner leads to stable, but slightly oscillatory, attitude control. Moreover, experiments with a bio-inspired flapping-wing robot show that residual, high-frequency attitude oscillations from flapping motion improve observability. The presented approach holds a promise for robotics, with accelerometer-less autopilots paving the road for insect-scale autonomous flying robots⁶. Finally, it forms a hypothesis on insect attitude estimation and control, with the potential to provide further insight into known biological phenomena^5,7,8 and to generate new predictions such as reduced head and body attitude variance at higher flight speeds⁹.

Flexibility Effects of a Flapping Mechanism Inspired by Insect Musculoskeletal System on Flight Performance

Flying animals such as insects display great flight performances with high stability and maneuverability even under unpredictable disturbances in natural and man-made environments. Unlike man-made mechanical systems like a drone, insects can achieve various flapping motions through their flexible musculoskeletal systems. However, it remains poorly understood whether flexibility affects flight performances or not. Here, we conducted an experimental study on the effects of the flexibility associated with the flapping mechanisms on aerodynamic performance with a flexible flapping mechanism (FFM) inspired by the flexible musculoskeletal system of insects. Based on wing kinematic and force measurements, we found an appropriate combination of the flexible components could improve the aerodynamic efficiency by increasing the wingbeat amplitude. Results of the wind tunnel experiments suggested that, through some passive adjustment of the wing kinematics in concert with the flexible mechanism, the disturbance-induced effects could be suppressed. Therefore, the flight stability under the disturbances is improved. While the FFM with the most rigid spring was least efficient in the static experiments, the model was most robust against the wind within the range of the study. Our results, particularly regarding the trade-off between the efficiency and the robustness, point out the importance of the passive response of the flapping mechanisms, which may provide a functional biomimetic design for the flapping micro air vehicles (MAVs) capable of achieving high efficiency and stability.

A contralateral wing stabilizes a hovering hawkmoth under a lateral gust

Previous analysis on the lateral stability of hovering insects, which reported a destabilizing roll moment due to a lateral gust, has relied on the results of a single wing without considering a presence of the contralateral wing (wing-wing interaction). Here, we investigated the presence of the contralateral wing on the aerodynamic and flight dynamic characteristics of a hovering hawkmoth under a lateral gust. By employing a dynamically scaled-up mechanical model and a servo-driven towing system installed in a water tank, we found that the presence of the contralateral wing plays a significant role in the lateral static stability. The contralateral wing mitigated an excessive aerodynamic force on the wing at the leeward side, thereby providing a negative roll moment to the body. Digital particle image velocimetry revealed an attenuated vortical system of the leading-edge vortex. An excessive effective angle of attack in the single wing case, which was caused by the root vortex of previous half stroke, was reduced by a downwash of the contralateral wing. The contralateral wing also relocated a neutral point in close proximity to the wing hinge points above the actual center of gravity, providing a practical static margin to a hovering hawkmoth.

Flies land upside down on a ceiling using rapid visually mediated rotational maneuvers

Flies and other insects routinely land upside down on a ceiling. These inverted landing maneuvers are among the most remarkable aerobatic feats, yet the full range of these behaviors and their underlying sensorimotor processes remain largely unknown. Here, we report that successful inverted landing in flies involves a serial sequence of well-coordinated behavioral modules, consisting of an initial upward acceleration followed by rapid body rotation and leg extension, before terminating with a leg-assisted body swing pivoted around legs firmly attached to the ceiling. Statistical analyses suggest that rotational maneuvers are triggered when flies' relative retinal expansion velocity reaches a threshold. Also, flies exhibit highly variable pitch and roll rates, which are strongly correlated to and likely mediated by multiple sensory cues. When flying with higher forward or lower upward velocities, flies decrease the pitch rate but increase the degree of leg-assisted swing, thereby leveraging the transfer of body linear momentum.

Inertial Sensing and Encoding of Self-Motion: Structural and Functional Similarities across Metazoan Taxa

To properly orient and navigate, moving animals must obtain information about the position and motion of their bodies. Animals detect inertial signals resulting from body accelerations and rotations using a variety of sensory systems. In this review, we briefly summarize current knowledge on inertial sensing across widely disparate animal taxa with an emphasis on neuronal coding and sensory transduction. We outline systems built around mechanosensory hair cells, including the chordate vestibular complex and the statocysts seen in many marine invertebrates. We next compare these to schemes employed by flying insects for managing inherently unstable aspects of flapping flight, built around comparable mechanosensory cells but taking unique advantage of the physics of rotating systems to facilitate motion encoding. Finally, we highlight fundamental similarities across taxa with respect to the partnering of inertial senses with visual senses and conclude with a discussion of the functional utility of maintaining a multiplicity of encoding schemes for self-motion information.

Fuelling on the wing: sensory ecology of hawkmoth foraging

Hawkmoths (Lepidoptera, Sphingidae) comprise around 1500 species, most of which forage on nectar from flowers in their adult stage, usually while hovering in front of the flower. The majority of species have a nocturnal lifestyle and are important nocturnal pollinators, but some species have turned to a diurnal lifestyle. Hawkmoths use visual and olfactory cues including CO₂ and humidity to detect and recognise rewarding flowers; they find the nectary in the flowers by means of mechanoreceptors on the proboscis and vision, evaluate it with gustatory receptors on the proboscis, and control their hovering flight position using antennal mechanoreception and vision. Here, we review what is presently known about the sensory organs and sensory-guided behaviour that control feeding behaviour of this fascinating pollinator taxon. We also suggest that more experiments on hawkmoth behaviour in natural settings are needed to fully appreciate their sensory capabilities.

The roles of vision and antennal mechanoreception in hawkmoth flight control

Flying animals need continual sensory feedback about their body position and orientation for flight control. The visual system provides essential but slow feedback. In contrast, mechanosensory channels can provide feedback at much shorter timescales. How the contributions from these two senses are integrated remains an open question in most insect groups. In Diptera, fast mechanosensory feedback is provided by organs called halteres and is crucial for the control of rapid flight manoeuvres, while vision controls manoeuvres in lower temporal frequency bands. Here, we have investigated the visual-mechanosensory integration in the hawkmoth Macroglossum stellatarum. They represent a large group of insects that use Johnston’s organs in their antennae to provide mechanosensory feedback on perturbations in body position. Our experiments show that antennal mechanosensory feedback specifically mediates fast flight manoeuvres, but not slow ones. Moreover, we did not observe compensatory interactions between antennal and visual feedback.

Chordwise wing flexibility may passively stabilize hovering insects

Insect wings are flexible, and the dynamically deforming wing shape influences the resulting aerodynamics and power consumption. However, the influence of wing flexibility on the flight dynamics of insects is unknown. Most stability studies in the literature consider rigid wings and conclude that the hover equilibrium condition is unstable. The rigid wings possess an unstable oscillatory mode mainly due to their pitch sensitivity to horizontal velocity perturbations. Here, we show that a flapping wing flyer with flexible wings exhibits stable hover equilibria. The free-flight insect flight dynamics are simulated at the fruit fly scale in the longitudinal plane. The chordwise wing flexibility is modelled as a linear beam. The two-dimensional Navier-Stokes equations are solved in a tight fluid-structure integration scheme. For a range of wing flexibilities similar to live insects, all eigenvalues of the system matrix about the hover equilibrium have negative real parts. Flexible wings appear to stabilize the unstable mode by passively deforming their wing shape in the presence of perturbations, generating significantly more horizontal velocity damping and pitch rate damping. These results suggest that insects may passively stabilize their hover flight via wing flexibility, which can inform designs of synthetic flapping wing robots.

Design and evaluation of a deformable wing configuration for economical hovering flight of an insect-like tailless flying robot

Studies on wing kinematics indicate that flapping insect wings operate at higher angles of attack (AoAs) than conventional rotary wings. Thus, effectively flying an insect-like flapping-wing micro air vehicle (FW-MAV) requires appropriate wing design for achieving low power consumption and high force generation. Even though theoretical studies can be performed to identify appropriate geometric AoAs for a wing for achieving efficient hovering flight, designing an actual wing by implementing these angles into a real flying robot is challenging. In this work, we investigated the wing morphology of an insect-like tailless FW-MAV, which was named KUBeetle, for obtaining high vertical force/power ratio or power loading. Several deformable wing configurations with various vein structures were designed, and their characteristics of vertical force generation and power requirement were theoretically and experimentally investigated. The results of the theoretical study based on the unsteady blade element theory (UBET) were validated with reference data to prove the accuracy of power estimation. A good agreement between estimated and measured results indicated that the proposed UBET model can be used to effectively estimate the power requirement and force generation of an FW-MAV. Among the investigated wing configurations operating at flapping frequencies of 23 Hz to 29 Hz, estimated results showed that the wing with a suitable vein placed outboard exhibited an increase of approximately 23.7%  ±  0.5% in vertical force and approximately 10.2%  ±  1.0% in force/power ratio. The estimation was supported by experimental results, which showed that the suggested wing enhanced vertical force by approximately 21.8%  ±  3.6% and force/power ratio by 6.8%  ±  1.6%. In addition, wing kinematics during flapping motion was analyzed to determine the reason for the observed improvement.

Wing-wake interaction destabilizes hover equilibrium of a flapping insect-scale wing

Wing-wake interaction is a characteristic nonlinear flow feature that can enhance unsteady lift in flapping flight. However, the effects of wing-wake interaction on the flight dynamics of hover are inadequately understood. We use a well-validated 2D Navier-Stokes equation solver and a quasi-steady model to investigate the role of wing-wake interaction on the hover stability of a fruit fly scale flapping flyer. The Navier-Stokes equations capture wing-wake interaction, whereas the quasi-steady models do not. Both aerodynamic models are tightly coupled to a flight dynamic model, which includes the effects of wing mass. The flapping amplitude, stroke plane angle, and flapping offset angle are adjusted in free flight for various wing rotations to achieve hover equilibrium. We present stability results for 152 simulations which consider different kinematics involving the pitch amplitude and pitch axis as well as the duration and timing of pitch rotation. The stability of all studied motions was qualitatively similar, with an unstable oscillatory mode present in each case. Wing-wake interaction has a destabilizing effect on the longitudinal stability, which cannot be predicted by a quasi-steady model. Wing-wake interaction increases the tendency of the flapping flyer to pitch up in the presence of a horizontal velocity perturbation, which further destabilizes the unstable oscillatory mode of hovering flight dynamics.

Design and stable flight of a 21 g insect-like tailless flapping wing micro air vehicle with angular rates feedback control

An insect-like tailless flapping wing micro air vehicle (FW-MAV) without feedback control eventually becomes unstable after takeoff. Flying an insect-like tailless FW-MAV is more challenging than flying a bird-like tailed FW-MAV, due to the difference in control principles. This work introduces the design and controlled flight of an insect-like tailless FW-MAV, named KUBeetle. A combination of four-bar linkage and pulley-string mechanisms was used to develop a lightweight flapping mechanism that could achieve a high flapping amplitude of approximately 190°. Clap-and-flings at dorsal and ventral stroke reversals were implemented to enhance vertical force. In the absence of a control surface at the tail, adjustment of the location of the trailing edges at the wing roots to modulate the rotational angle of the wings was used to generate control moments for the attitude control. Measurements by a 6-axis load cell showed that the control mechanism produced reasonable pitch, roll and yaw moments according to the corresponding control inputs. The control mechanism was integrated with three sub-micro servos to realize the pitch, roll and yaw controls. A simple PD feedback controller was implemented for flight stability with an onboard microcontroller and a gyroscope that sensed the pitch, roll and yaw rates. Several flight tests demonstrated that the tailless KUBeetle could successfully perform a vertical climb, then hover and loiter within a 0.3 m ground radius with small variations in pitch and roll body angles.

The Function and Organization of the Motor System Controlling Flight Maneuvers in Flies

Animals face the daunting task of controlling their limbs using a small set of highly constrained actuators. This problem is particularly demanding for insects such as Drosophila, which must adjust wing motion for both quick voluntary maneuvers and slow compensatory reflexes using only a dozen pairs of muscles. To identify strategies by which animals execute precise actions using sparse motor networks, we imaged the activity of a complete ensemble of wing control muscles in intact, flying flies. Our experiments uncovered a remarkably efficient logic in which each of the four skeletal elements at the base of the wing are equipped with both large phasically active muscles capable of executing large changes and smaller tonically active muscles specialized for continuous fine-scaled adjustments. Based on the responses to a broad panel of visual motion stimuli, we have developed a model by which the motor array regulates aerodynamically functional features of wing motion. VIDEO ABSTRACT.

Aerodynamics, sensing and control of insect-scale flapping-wing flight

There are nearly a million known species of flying insects and 13 000 species of flying warm-blooded vertebrates, including mammals, birds and bats. While in flight, their wings not only move forward relative to the air, they also flap up and down, plunge and sweep, so that both lift and thrust can be generated and balanced, accommodate uncertain surrounding environment, with superior flight stability and dynamics with highly varied speeds and missions. As the size of a flyer is reduced, the wing-to-body mass ratio tends to decrease as well. Furthermore, these flyers use integrated system consisting of wings to generate aerodynamic forces, muscles to move the wings, and sensing and control systems to guide and manoeuvre. In this article, recent advances in insect-scale flapping-wing aerodynamics, flexible wing structures, unsteady flight environment, sensing, stability and control are reviewed with perspective offered. In particular, the special features of the low Reynolds number flyers associated with small sizes, thin and light structures, slow flight with comparable wind gust speeds, bioinspired fabrication of wing structures, neuron-based sensing and adaptive control are highlighted.

Hovering and forward flight of the hawkmoth Manduca sexta: trim search and 6-DOF dynamic stability characterization

We show that the forward flight speed affects the stability characteristics of the longitudinal and lateral dynamics of a flying hawkmoth; dynamic modal structures of both the planes of motion are altered due to variations in the stability derivatives. The forward flight speed u e is changed from 0.00 to 1.00 m s(-1) with an increment of 0.25 m s(-1). (The equivalent advance ratio is 0.00 to 0.38; the advance ratio is the ratio of the forward flight speed to the average wing tip speed.) As the flight speed increases, for the longitudinal dynamics, an unstable oscillatory mode becomes more unstable. Also, we show that the up/down (w(b)) dynamics become more significant at a faster flight speed due to the prominent increase in the stability derivative Z(u) (up/down force due to the forward/backward velocity). For the lateral dynamics, the decrease in the stability derivative L(v) (roll moment due to side slip velocity) at a faster flight speed affects a slightly damped stable oscillatory mode, causing it to become more stable; however, the t(half) (the time taken to reach half the amplitude) of this slightly damped stable oscillatory mode remains relatively long (∼12T at u(e) = 1 m s(-1); T is wingbeat period) compared to the other modes of motion, meaning that this mode represents the most vulnerable dynamics among the lateral dynamics at all flight speeds. To obtain the stability derivatives, trim conditions for linearization are numerically searched to find the exact trim trajectory and wing kinematics using an algorithm that uses the gradient information of a control effectiveness matrix and fully coupled six-degrees of freedom nonlinear multibody equations of motion. With this algorithm, trim conditions that consider the coupling between the dynamics and aerodynamics can be obtained. The body and wing morphology, and the wing kinematics used in this study are based on actual measurement data from the relevant literature. The aerodynamic model of the flapping wings of a hawkmoth is based on the blade element theory, and the necessary aerodynamic coefficients, including the lift, drag and wing pitching moment, are experimentally obtained from the results of previous work by the authors.

Controlling roll perturbations in fruit flies

Owing to aerodynamic instabilities, stable flapping flight requires ever-present fast corrective actions. Here, we investigate how flies control perturbations along their body roll angle, which is unstable and their most sensitive degree of freedom. We glue a magnet to each fly and apply a short magnetic pulse that rolls it in mid-air. Fast video shows flies correct perturbations up to 100° within 30 ± 7 ms by applying a stroke-amplitude asymmetry that is well described by a linear proportional-integral controller. For more aggressive perturbations, we show evidence for nonlinear and hierarchical control mechanisms. Flies respond to roll perturbations within 5 ms, making this correction reflex one of the fastest in the animal kingdom.

Feedback control as a framework for understanding tradeoffs in biology

Control theory arose from a need to control synthetic systems. From regulating steam engines to tuning radios to devices capable of autonomous movement, it provided a formal mathematical basis for understanding the role of feedback in the stability (or change) of dynamical systems. It provides a framework for understanding any system with regulation via feedback, including biological ones such as regulatory gene networks, cellular metabolic systems, sensorimotor dynamics of moving animals, and even ecological or evolutionary dynamics of organisms and populations. Here, we focus on four case studies of the sensorimotor dynamics of animals, each of which involves the application of principles from control theory to probe stability and feedback in an organism's response to perturbations. We use examples from aquatic (two behaviors performed by electric fish), terrestrial (following of walls by cockroaches), and aerial environments (flight control by moths) to highlight how one can use control theory to understand the way feedback mechanisms interact with the physical dynamics of animals to determine their stability and response to sensory inputs and perturbations. Each case study is cast as a control problem with sensory input, neural processing, and motor dynamics, the output of which feeds back to the sensory inputs. Collectively, the interaction of these systems in a closed loop determines the behavior of the entire system.

Dynamic flight stability of a hovering model dragonfly

The longitudinal dynamic flight stability of a model dragonfly at hovering flight is studied, using the method of computational fluid dynamics to compute the stability derivatives and the techniques of eigenvalue and eigenvector analysis for solving the equations of motion. Three natural modes of motion are identified for the hovering flight: one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. The flight is dynamically unstable owing to the unstable oscillatory mode. The instability is caused by a pitch-moment derivative with respect to horizontal velocity. The damping force and moment derivatives (with respect to horizontal and vertical velocities and pitch-rotational velocity, respectively) weaken the instability considerably. The aerodynamic interaction between the forewing and the hindwing does not have significant effect on the stability properties. The dragonfly has similar stability derivatives, hence stability properties, to that of a one-wing-pair insect at normal hovering, but there are differences in how the derivatives are produced because of the highly inclined stroke plane of the dragonfly.

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.