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Vibrational control is an open loop stabilization technique via the application of highamplitude, high-frequency oscillatory inputs. The averaging theory has been the standard technique for designing vibrational control systems. However, it stipulates too high oscillation frequency that may not be practically feasible. Therefore, although vibrational control is very robust and elegant (stabilization without feedback), it is rarely used in practical applications. The only well-known example is the Kapitza pendulum; an inverted pendulum shose pivot is subject to vertical oscillation. the unstable equilibrium of the inverted pendulum gains asymptotic stability due to the high-frequency oscillation of the pivot. In this paper, we provide a new vibrational control system from Nature; flapping flight dynamics. Flapping flight is a rich dynamical system as a representative model will typically be nonlinear, time-varying, multi-body, multi-time-scale dynamical system. Over the last two decades, using direct averaging, there has been consensus in the flapping flight dynamics community that insects are unstable at the hovering equilibrium due to the lack of pitch stiffness. In this work, we perform higher-order averaging of the time-periodic dynamics of flapping flight to show a vibrational control mechanism due to the oscillation of the driving aerodynamic forces. We also experimentally demonstrate such a phenomenon on a flapping apparatus that has two degrees of freedom: forward translation and pitching motion. It is found that the time-periodic dynamics of the flapping micro-air-vehicle is naturally (without feedback) stabilized beyond a certain threshold. Moreover, if the averaged aerodynamic thrust force is produced by a propeller revolving at a constant speed while maintaining the wings stationary at their mean positions, no stabilization is observed. Hence, it is concluded that the observed stabilization in the flapping system at high frequencies is due to the oscillation of the driving aerodynamic force and, as such, flapping flight indeed enjoys vibrational stabilization. 

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. 

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. 

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. 

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. 

Because of the wing oscillatory motion with respect to the body, the ight dynam- ics of biological yers as well as their man-made mimetic vehicles, apping-wing micro- air-vehicles (FWMAVs), are typically represented by multi-body, nonlinear time-periodic (NLTP) system models whose balance and stability analyses are quite challenging. In this work, we consider a NLTP system model for a two-degree-of-freedom FWMAV that is conned to move along vertical rails. We combine tools from chronological calculus, geo- metric control, and averaging to provide a mathematically rigorous analysis for the balance of FWMAVs at hover; that is, relaxing the single-body and direct averaging assumptions that are commonly adopted in analyzing balance and stability of FWMAVs and insects. We also use optimized shooting to numerically capture the resulting periodic orbit and verify the obtained results. Finally, we provide a combined averaging-shooting approach for the balance and stability analysis of NLTP systems that (i) unlike typical shooting methods, does not require an initial guess; (ii) provides more accurate results than the analytical av- eraging approaches, hence relaxing the need for intractable high-order averaged dynamics; and (iii) allows a deeper scrutiny of the system dynamics, in contrast to numerical shooting methods.