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1. Differential-Geometric-Control Formulation of Flapping Flight Multi-body Dynamics

   https://doi.org/10.1007/s00332-018-9520-8  

   Hassan, Ahmed M.; Taha, Haithem E. (November 2018, Journal of Nonlinear Science)

   Flapping flight dynamics is quite an intricate problem that is typically represented by a multi-body, multi-scale, nonlinear, time-varying dynamical system. The unduly simple modeling and analysis of such dynamics in the literature has long obstructed the discovery of some of the fascinating mechanisms that these flapping-wing creatures possess. Neglecting the wing inertial effects and directly averaging the dynamics over the flapping cycle are two major simplifying assumptions that have been extensively used in the literature of flapping flight balance and stability analysis. By relaxing these assumptions and formulating the multi-body dynamics of flapping-wing microair- vehicles in a differential-geometric-control framework, we reveal a vibrational stabilization mechanism that greatly contributes to the body pitch stabilization. The discovered vibrational stabilization mechanism is induced by the interaction between the fast oscillatory aerodynamic loads on the wings and the relatively slow body motion. This stabilizationmechanism provides an artificial stiffness (i.e., spring action) to the body rotation around its pitch axis. Such a spring action is similar to that of Kapitsa pendulum where the unstable inverted pendulum is stabilized through applying fast-enough periodic forcing. Such a phenomenon cannot be captured using the overly simplified modeling and analysis of flapping flight dynamics. 

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2. Aerodynamic-Dynamic Interactions and Multi-Body Formulation of Flapping Wing Dynamics: Part II - Trim and Stability Analysis

   Hassan, Ahmed; Taha, Haithem (January 2017, AIAA Guidance, Navigation, and Control Conference)

   Flapping ight dynamics constitutes a multi-body, nonlinear, time-varying system. The two major simplifying assumptions in the analysis of apping ight stability are neglecting the wing inertial eects and averaging the dynamics over the apping cycle. The challenges resulting from relaxing these assumptions naturally invoke the geometric control theory as an appropriate analysis tool. In this work, a reduced-order model (extracted from the full model derived in the rst part of this work) for the longitudinal apping ight dynamics near hover is considered and represented in a geometric control framework. Then, combining tools from geometric control theory and averaging, the full dynamic stability as well as balance analyses of hovering insects are performed. 

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3. Vibrational Control in Flapping-Wing Micro-Air-Vehicles

   Taha, H; Kiani, M; Navarro, J (June 2018, American Control Conference)

   Flapping-Wing Micro-Air-Vehicles (FWMAVs) are bio-inspired air vehicles that mimic insect and bird flight. The dynamic behavior of these systems is typically described by a multi-body, multi-time-scale, nonlinear, time-varying dynamical system. Interestingly, this rich dynamics lead to unconventional stabilization mechanisms whose study essentially necessitates a mathematically rigorous analysis. In this paper, we use higherorder averaging, which is based on chronological calculus, to show that insects and their man-made counterparts (FWMAVs) exploit vibrational control to stabilize their body pitching angle. Such an unconventional stabilization cannot be captured by direct averaging. We also experimentally demonstrate such a phenomenon by constructing an experimental setup that allows for two degrees of freedom for the body; forward motion and pitching motion. We measure the response of the body pitching angle using a digital camera and an image processing algorithm at different flapping frequencies. It is found that there is a flapping frequency threshold beyond which the body pitching response is naturally (without feedback) stabilized, which conforms with the vibrational control concept. Moreover, we also construct a replica of the experimental setup with the FWMAV being replaced by a propeller revolving at constant speed, which results in a constant aerodynamic force, leaving no room for vibrational control. The response of the propellersetup is unstable at all frequencies, which also corroborates the fact that the observed stabilization of the FWMAV-setup at high frequencies is a vibrational stabilization phenomenon. 

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4. Aerodynamic-Dynamic Interactions and Multi-Body Formulation of Flapping Wing Dynamics: Part I - Modeling

   Hassan, Ahmed; Taha, Haithem (January 2017, AIAA Guidance, Navigation, and Control Conference)

   Flapping ight dynamics constitutes a multi-body, nonlinear, time-varying system. The two major simplifying assumptions in the analysis of apping ight stability are neglecting the wing inertial eects and averaging the dynamics over the apping cycle. Relaxing the rst assumption invokes a multi-body formulation of the equations of motion. In this work, the full, multi-body, equations of motion governing the longitudinal apping ight dynamics near hover are considered. The aerodynamic loads are represented through a relatively simple analytical model that accounts for the dominant contributions; e.g., leading edge vortex and rotational eects. The dynamic and aerodynamic models are coupled together to account for mutual interactions. 

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5. Springs and Wings: A robotic study of the insect flight system

   Lynch, James (July 2023, escholarship.org)

   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. 

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