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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Differential-Geometric-Control Formulation of Flapping Flight Multi-body Dynamics
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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- Award ID(s):
- 1709746
- NSF-PAR ID:
- 10081738
- Date Published:
- Journal Name:
- Journal of Nonlinear Science
- ISSN:
- 0938-8974
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00332-018-9520-8
