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.
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Vibrational Control in Flapping-Wing Micro-Air-Vehicles
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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- Award ID(s):
- 1709746
- NSF-PAR ID:
- 10081739
- Date Published:
- Journal Name:
- American Control Conference
- Page Range / eLocation ID:
- 6445-6450
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.more » « less
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Abstract Bio-inspired flying robots (BIFRs) which fly by flapping their wings experience continuously oscillating aerodynamic forces. These oscillations in the driving force cause vibrations in the motion of the body around the mean trajectory. In other words, a hovering BIFR does not remain fixed in space; instead, it undergoes oscillatory motion in almost all directions around the stationary point. These oscillations affect the aerodynamic performance of the flier. Assessing the effect of these oscillations, particularly on thrust generation in two-winged and four-winged BIFRs, is the main objective of this work. To achieve such a goal, two experimental setups were considered to measure the average thrust for the two BIFRs. The average thrust is measured over the flapping cycle of the BIFRs. In the first experimental setup, the BIFR is installed at the end of a pendulum rod, in place of the pendulum mass. While flapping, the model creates a thrust force that raises the model along the circular trajectory of the pendulum mass to a certain angular position, which is an equilibrium point and is also stable. Measuring the weight of the BIFR and the equilibrium angle it obtains, it is straightforward to estimate the average thrust, by moment balance about the pendulum hinge. This pendulum setup allows the BIFR model to freely oscillate back and forth along the circular trajectory about the equilibrium position. As such, the estimated average thrust includes the effects of these self-induced vibrations. In contrast, we use another setup with a load cell to measure thrust where the model is completely fixed. The thrust measurement revealed that the load cell or the fixed test leads to a higher thrust than the pendulum or the oscillatory test for the two-winged model, showing the opposite behavior for the four-winged model. That is, self-induced vibrations have different effects on the two BIFR models. We felt that this observation is worth further investigation. It is important to mention that aerodynamic mechanisms for thrust generation in the two and four-winged models are different. A two-winged BIFR generates thrust through traditional flapping mechanisms whereas a four-winged model enjoys a clapping effect, which results from wing-wing interaction. In the present work, we use a motion capture system, aerodynamic modeling, and flow visualization to study the underlying physics of the observed different behaviors of the two flapping models. The study revealed that the interaction of the vortices with the flapping wing robots may play a role in the observed aerodynamic behavior of the two BIFRs.
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