First published online August 3, 2006
Journal of Experimental Biology 209, 3114-3130 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02363
Flight control in the hawkmoth Manduca sexta: the inverse problem of hovering
T. L. Hedrick* and
T. L. Daniel
Department of Biology, University of Washington, Seattle, WA 98195,
USA
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Fig. 2. Two successive wingbeats run continuously together as long as none of the
defining parameters change from wingbeat to wingbeat (black line). However,
changing parameters between wingbeats ( in this instance) creates a
discontinuity at the boundary from one wingbeat to the next (cyan line). A
hyperbolic tangent combines the distinct second wingbeat with the prior
wingbeat (Eqn 4, broken red line) for a smooth transition.
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Fig. 3. (A) Outline of the steps that make up the microgenetic algorithm (µGA),
starting from an initial population. We terminated the µGA after 50
generations and used a simplex search algorithm follow the gradient from the
best µGA result to the local maxima. (B) How the combination of a µGA
and simplex search might operate in a two-dimensional parameter space defined
by the function z=f(x,y). The µGA searches broadly,
improving slightly with every generation, while the simplex algorithm proceeds
from the best µGA result to the local maximum. Note that although the
example here shows a search for a maximum for ease of illustration, the moth
simulation searches for a minimum using an otherwise identical procedure.
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Fig. 4. The measured position and orientation of a real hawkmoth hovering for 3.5 s
in front of an artificial flower. The moth's estimated centre of mass position
differed by an average of 0.29 cm from its overall mean location and the moth
maintained a body angle of 34.3±3.3°. Body position and orientation
were drawn for every 0.044 s of flight.
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Fig. 6. Kinematic parameter variation for 30 redundantly actuated trials, coded by
color. Each trial begins with the same initial conditions and all trials met
our definition of adequate hovering (remaining within a 4096 cm3
volume for the duration of the trial). Actual performance greatly exceeded
this definition of adequate; the mean distance from the center of the volume
to the moth was 2.6 cm. The Y-axes are identical to those in
Fig. 5.
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Fig. 7. Changes in the performance of the simulated moth in hovering flight as the
number of free parameters is reduced (and the number of fixed parameters
increased). Performance was quantified by (A) the number of wingbeats the
model executed without leaving a 4096 cm3 volume, up to a maximum
of 41 wingbeats (1.5 s of flight) and (B) the mean distance from the moth to
the centre of the volume. Note that the model requires 5 wingbeats to fall
from its initial position to a location outside the target volume. Trends are
shown as the mean ± 1 s.d., with the maximum and minimum values
indicated by diamonds and circles, respectively. N=4096 trials.
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Fig. 8. Results from an under-actuated hovering simulation with two free parameters
over a 15 s time span. (A) The location and orientation of the moth through
time with a stick-figure shown for every 5.6 wingbeats. The simulation does
not control X-axis position as precisely as in fully actuated cases.
(B) Variation in the two free parameters, the wing sweep angle phase and the
wing rotation angle amplitude, through time. The eight parameters not shown
were fixed at their average values (taken from the set of hovering trials with
all parameters free. The Y-axis scale for both variables reflects the
limits imposed by the model. (C) Variation in the simulated moth's three
degrees of freedom when restricted to two free kinematic parameters. There was
a correlation between the pitch and X-velocity.
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Fig. 9. Kinematic parameter variation for 20 forward flight trials plotted against
time. Each trial begins with the same initial conditions and all trials met
our definition of adequate forward flight (remaining within 20 cm of a target
traveling at exactly 3 m s-1 along the X-axis).
Y-axes are identical to those in Figs
5 and
6.
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Fig. 10. The 2-D projection of the paths of the right wingtip in the X-Z
plane with variation in kinematic parameters. Here we show the standard
wingbeat, the average kinematic parameter set adopted by the simulated moth
with all ten kinematic parameters free to vary (solid black line), and two
variations on this standard wingbeat. The variations show the kinematic effect
of changing two of the parameters most clearly associated with maintaining
hovering flight. The arrows indicate the direction of wing motion along the
wingtip path. Body angle and centre of gravity location were recorded from the
model. Artwork courtesy of Michael Tu.
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© The Company of Biologists Ltd 2006
p) angles. (B) A rear view of the moth in the body
coordinate system showing the wing elevation (
) and span axis rotation (
)
angles.
) and mean (bar over). ß is the abdominal angle.