First published online October 7, 2004
Journal of Experimental Biology 207, 3813-3838 (2004)
Published by The Company of Biologists 2004
doi: 10.1242/jeb.01229
Neuromuscular control of aerodynamic forces and moments in the blowfly, Calliphora vicina
Claire N. Balint1,* and
Michael H. Dickinson2
1 Department of Integrative Biology, University of California, Berkeley, CA
94720, USA
2 Department of Bioengineering, California Institute of Technology,
Pasadena, CA 91125, USA
*
Author for correspondence at present address: ARL Division of Neurobiology, PO
Box 210077, University of Arizona, Tucson, AZ 85721, USA (e-mail:
cnbalint{at}cal.berkeley.edu)
Accepted 4 August 2004
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Summary
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Flies are among the most agile of flying insects, a capacity that
ultimately results from their nervous system's control over steering muscles
and aerodynamic forces during flight. In order to investigate the
relationships among neuromuscular control, musculo-skeletal mechanics and
flight forces, we captured high-speed, three-dimensional wing kinematics of
the blowfly, Calliphora vicina, while simultaneously recording
electromyogram signals from prominent steering muscles during visually induced
turns. We used the quantified kinematics to calculate the translational and
rotational components of aerodynamic forces and moments using a theoretical
quasi-steady model of force generation, confirmed using a dynamically scaled
mechanical model of a Calliphora wing. We identified three
independently controlled features of the wingbeat trajectory –
downstroke deviation, dorsal amplitude and mode. Modulation of each of these
kinematic features corresponded to both activity in a distinct steering muscle
group and a distinct manipulation of the aerodynamic force vector. This
functional specificity resulted from the independent control of downstroke and
upstroke forces rather than the independent control of separate aerodynamic
mechanisms. The predicted contributions of each kinematic feature to body
lift, thrust, roll, yaw and pitch are discussed.
Key words: insect flight, kinematics, aerodynamics, steering, motor control, Calliphora vicina
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Introduction
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The basis for much of an insect's flight abilities lies in the deceptively
simple back-and-forth flapping motion of its wings. The broad range of flight
maneuvers displayed by flies, from subtle course corrections to sudden darts
and saccades, suggests that wing motion can vary in complex ways. One of the
challenges in understanding the flight motor system is to identify the
components of wing motion that the fly can independently manipulate to produce
an array of behavioral outputs.
Previously, studies have correlated specific features of wing kinematics to
variation in aspects of free flight behavior such as forward velocity
(Dudley and Ellington, 1990a
;
Ennos, 1989;
Willmott and Ellington,
1997a). Other studies using tethered preparations have examined
the kinematic correlates of lift and thrust control
(Nachtigall and Roth, 1983;
Vogel, 1967;
Wortmann and Zarnack, 1993)
and responses to sensory manipulations such as visual or mechanical roll and
yaw (Faust, 1952;
Hengstenberg et al., 1986;
Lehmann and Dickinson, 1997;
Srinivasan, 1977;
Waldman and Zarnack, 1988;
Zanker, 1990;
Zarnack, 1988). However, the
functional relationship between variation in wing motion and behavioral output
has remained obscure due to two main complications. First, time-resolved,
three-dimensional measurements of wing kinematics are difficult to acquire,
especially over the duration of complete flight maneuvers. This difficulty
forces a trade-off between the number of kinematic parameters that may be
sensibly measured and the length of time over which they can be monitored.
Although, 50 years ago, Weis-Fogh and Jensen
(1956) emphasized the
importance of simultaneous measurements of wing speed and angle of attack in
particular for assessing the control of aerodynamic forces, such simultaneous
measurements have been rare. Second, even detailed analyses of conventional
kinematic parameters have been insufficient for predicting the resultant
forces due to the significant influence of unsteady mechanisms
(Cloupeau et al., 1979;
Wilkin and Williams, 1993;
Zanker and Gotz, 1990).
Fortunately, recent advances in high-speed video technology have greatly
facilitated the acquisition of detailed kinematic information
(Fry et al., 2003). Due to an
improved understanding of the contributions of delayed stall and rotational
forces to quasi-steady approximations (Sane and Dickinson,
2001,
2002), detailed kinematic
information, once obtained, can now be related to a reasonable approximation
of the resultant aerodynamic forces. This improved understanding of
aerodynamic mechanisms has both confirmed the importance of gross kinematic
features of wing motion (e.g. stroke amplitude, frequency) and emphasized the
need for measurement and analysis of finer-scale kinematic variation.
In the present study, we used high-speed videography to quantify the
changes in three-dimensional wing orientation with sufficient temporal
resolution to estimate the resultant force vector at various stages of the
wingbeat cycle and to confirm our estimate of force using a mechanical model.
However, in contrast to most previous studies that categorized wingbeat
kinematics (for review, see Taylor,
2001), as well as muscle activity
(Kutsch et al., 2003;
Spüler and Heide, 1978;
Thüring, 1986;
Waldman and Zarnack, 1988),
according to the amount of force or torque produced, we organized our analysis
based on particular features of wing motion we previously correlated with
patterns of steering muscle activity. These features were `downstroke
deviation', a correlate of basalare muscle activity, and `mode', a correlate
of activity in the pteralae III and pteralae I muscles
(Balint and Dickinson, 2001).
Using a bottom-up approach building upon these previous findings and
incorporating improved resolution of wing kinematics, we were able to bridge
three levels of analysis: the correlation between steering muscle activity and
wing kinematics, the mechanisms by which wing kinematics modify aerodynamic
forces, and the contribution of aerodynamic forces to body forces and moments.
The results of this approach suggest that it is the ability to manipulate the
coupling among aerodynamically relevant kinematic parameters, rather than the
ability to control these parameters independently, that allows Calliphora
vicina the flexibility of control observed in previous measurements of
its directional force and moment output
(Blondeau, 1981;
Schilstra and van Hateren,
1999).
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Materials and methods
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Tethering and filming procedure
Adult male blowflies, Calliphora vicina (R.-D.), were tethered and
implanted with extracellular electrodes as described previously
(Balint and Dickinson, 2001).
Male flies were selected from a laboratory colony, maintained at approximately
22°C with a 12 h:12 h light:dark cycle. The age of all individuals was
between one and two weeks post-eclosion at the time of tethering. Each tether
was composed of a modified #0 insect pin soldered onto a 1.5 mm-diameter brass
rod. Short lengths of 25 µm-diameter nickel chromium (NiChr) wire with
formvar insulation (A-M Systems, Sequim, WA, USA) were soldered to the
terminals of five pairs of 28-gauge wires, which were glued to the brass rod.
Each fly was anesthetized by placing it in a –4°C freezer for
3–4 min, then immediately attached to the end of the insect pin with a
mixture of collophonium and beeswax. We implanted the tips of a pair of the
NiChr wires into each of five steering muscles (b2, b1, III1, I1, III2-4;
nomenclature from Heide, 1968)
on the left side of the animal. Flies were allowed to recover for one day
following electrode implantation, and data were collected for 2–4
consecutive days following initial electrode implantation.
We secured the free end of the tether onto a piezoelectric crystal attached
to a rigid acrylic rod. The acrylic rod was then secured onto a metal
armature, so that the fly was held with its longitudinal body axis
approximately 15° relative to the ground. The mouth of a small open-throat
wind tunnel was positioned in front of the fly, 
5 cm from the front of
the head. A 7.0x0.8 cm black cylindrical brass rod pendulum was
suspended in front of the fly with the base of the rod level with the fly's
head.
Three Kodak MotionPro cameras were positioned above, behind and on the left
side of the fly (Fig. 1A). Each
camera was positioned so that their lines of sight were orthogonal to each
other and equidistant to the fly. We used identical 8.5 mm video lenses
(Computar, Torrance, CA, USA) on each camera. Small panels of infrared
light-emitting diodes (LEDs) placed opposite each camera acted as a backlight
against which the fly was imaged. The wings were sufficiently translucent,
such that the outline and venation were clearly visible in the camera image
(Fig. 1B). We filmed the flies
at a rate of 5000 frames s–1 and an electronic shutter speed
of 1/20 000.
Extracellular potentials from the implanted electrodes were amplified using
an AC amplifier (A-M Systems Model 1800) and digitized using a Digidata 1200
and Axoscope software (Axon Instruments, Union City, CA, USA). Oscillatory
signals from the piezoelectric crystal, which were in phase with the stroke
cycle, and frame-mark signals from the cameras were also recorded. All the
signals were digitized at 37 kHz in order to adequately discriminate the 5000
Hz frame-mark signals. To initiate each flight bout, the wind tunnel was
switched on and set to a wind speed of approximately 2 m s–1
at the mouth, and the pendulum rod was set into motion. When the fly reacted
to the pendulum motion with stereotyped modulations of wing motions and
steering muscle activity, we manually activated an external trigger to
initiate video capture and electrophysiological data acquisition. Data were
collected in this manner from seven animals.
Wing digitization
Captured images were directly downloaded to computer as bitmaps. The bitmap
images were then analyzed using a custom digitizing program in MATLAB
(Fry et al., 2003). For each
time sample, the program displayed the synchronously captured images from each
of the three cameras. Points were digitized simultaneously in all three
fields. We digitized the x-, y- and z-coordinates
of at least five points in each time sample: the anterior tip of the head,
posterior tip of the abdomen, left wing hinge, left wingtip and right wing
hinge. A sixth point, the right wingtip, was digitized when we chose to
include information about the position of the right wing. Because the body was
tethered and stationary, the head, tail and hinge coordinates were held
constant for each sequence.
The coordinates of each point were transformed such that the wing hinge was
the origin and the longitudinal body axis was tilted 50° relative to
horizontal (Fig. 1C). The
Cartesian coordinates of the wingtip were then converted to spherical
coordinates:
 | (1) |
 | (2) |
A wire-frame image of a Calliphora vicina wing was then fit to
match the hinge and tip coordinates and rotated about the hinge-to-tip axis
until the wire-frame and wing outline matched best, as judged by eye. The
digitized morphological wing angle, 
, was the angle between the
wire-frame plane and the vertical plane through the axis of rotation.
Using this procedure, we collected complete information about the wing
position and orientation for each time point. Although bending and torsion of
the wing were conspicuous during the upstroke and during wing rotations, these
kinematic changes were excluded from our analysis. The left wing was digitized
in a total of 19 523 time samples (569 wingbeat cycles), and the right wing
was digitized in a total of 10 078 time samples (294 wingbeat cycles).
Force measurements
We used the mechanical model from previous studies
(Fig. 1D;
Dickinson et al., 1999; Sane
and Dickinson, 2001,
2002) to measure the
aerodynamic forces resulting from the measured wing kinematics. An enlarged
planform of a Calliphora vicina wing was made by cutting a 2.3
mm-thick acrylic sheet into the shape of a wing isometrically scaled to 30 cm
length and 7.6 cm mean chord length. The proximal end of the wing was attached
to a two-dimensional force transducer and fixed to a gearbox driven in three
rotational degrees of freedom by three servo-motors. The wing, force
transducer and gearbox were immersed in mineral oil with a kinematic viscosity
of 11.5 cSt.
A series of manipulations were performed on the wing data before
replicating the kinematics on the dynamically scaled mechanical model. First,
each sequence was divided into sets of 40 wingbeat cycles or fewer. Second,
each of the three time series of wing angles (
, 
, )
describing the first wingbeat cycle in each sequence was distorted so that the
wing position at the beginning and end of the cycle was identical. This made
it possible to repeat this cycle indefinitely without producing any sudden
changes in position during transitions from one cycle to the next. This
distorted version of the first wingbeat cycle was copied and concatenated into
a series of four cycles and then added to the beginning of each data set. The
last wingbeat cycle was similarly distorted, concatenated and added to the end
of each data set. These sections of `junk kinematics' allowed the mechanical
model to reach speed and entrain the wake at the beginning of each sequence,
and to slow down gradually at the end of the sequence, without affecting the
kinematics of interest. Third, each of the three wing angle sequences was
smoothed using a B-spline algorithm (based on criteria from
Craven and Wahba, 1979) and
temporally re-sampled so that motion between time points was 1° or
less.
For each kinematic sequence, the mean wingbeat frequency of the mechanical
model was scaled such that the Reynolds number (as defined by
Ellington, 1984c) matched that
of each fly. The mean wingbeat frequency observed among flight sequences
ranged from 130 to 167 Hz. In order to match the Reynolds numbers for these
sequences, the wingbeat frequencies reproduced by the mechanical model ranged
from 0.125 to 0.145 Hz. Due to the large magnitude of the forces in this study
and the effects of backlash in the gears linking the motors to the wing, the
actual wing kinematics of the mechanical model differed depending on the
direction of motion. To ameliorate these effects, we ran each sequence twice:
once with the directional convention such that the wing moved from left to
right for the downstroke and right to left for the upstroke (`forward'), and a
second time such that the wing moved right to left for the downstroke and left
to right for the upstroke (`backward'). We were able to minimize the
directional bias due to backlash by using the `backward' measurements for the
downstroke and the `forward' measurements for the upstroke.
The calibrated two-dimensional force transducer measured forces parallel
and perpendicular to the wing. The voltage signals from the force transducer
were acquired at a rate of 200 Hz using a data acquisition board (National
Instruments, Austin, TX, USA) operated using a custom program written in
MATLAB (see Sane and Dickinson,
2001, for more details). The gravitational contribution to the
measured forces was subtracted, and the force signal was filtered offline
using a low-pass digital Butterworth filter with a zero phase delay and a
cut-off at 4 Hz. The resultant signal from the perpendicular channel was our
measure of the total aerodynamic force normal to the wing (measured
FN). Because fly wings are relatively flat and flap at
high angles of attack that separate flow, aerodynamic forces should be at all
times roughly normal to the surface of the wing
(Dickinson, 1996). Accordingly,
we confirmed that the forces measured from the parallel channel were
negligible.
The magnitude of forces measured using the mechanical model in oil are
related to those of a fly flying in air by a simple conversion factor, as
described previously (Fry et al.,
2003):
 | (3) |
where F is the force magnitude, 
is the fluid density, 
is
the kinematic viscosity of the fluid, and
is the
non-dimensional second moment of wing area
(Ellington, 1984a). We found
the conversion factor in our experiments to be 0.0018. Therefore, all measured
forces were multiplied by this factor in order to compare them with those
expected for an actual fly.
Force calculations
Theoretical calculations of the quasi-steady translational and rotational
components of aerodynamic force were made using the methods in Sane and
Dickinson (2002). The
translational force component normal to the wing surface was calculated as:
 | (4) |
where S is the projected surface area of the wing,
Ut is the wingtip velocity, and g is the
wing's geometrical angle of attack with respect to its path. The lift
(CLt) and drag coefficients (CDt) for
the model wing were measured at a comparable Reynolds number and fitted with
the following equations:
 | (5) |
and
 | (6) |
The rotational force normal to the wing surface was calculated from:
 | (7) |
where 
is the absolute rotational angular velocity of the wing,
is the mean chord length, R
is the wing length, 
is the
non-dimensional radial position along the wing, and
() is the
non-dimensional chord length (Ellington,
1984a). The rotational angular velocity, , is equivalent to
the temporal derivative of . Prior experiments have shown that the
rotational force coefficient, Crot, is dependent on the
value of non-dimensional rotational angular velocity
(
;
Sane and Dickinson, 2002):
 | (8) |
To estimate rotational forces, we used the relationship between
and Crot
measured in Sane and Dickinson
(2002) for model
Drosophila wings. Although this must introduce some error in our
estimates, these were deemed small relative to other sources of error based on
inspection of the data. For instantaneous values of
of less than 0.123,
Crot was 0, and for values of
greater than or equal to 0.374,
Crot was 1.55. For values of
between 0.123 and 0.374:
 | (9) |
Viscous forces that act parallel to the wing surface were ignored, a
reasonable assumption at the Reynolds numbers used in this study.
The total aerodynamic force normal to the wing, FN, was
approximated as the sum of the translatory (Ftrans) and
rotational (Frot) components normal to the wing. This
model neglects two additional terms: added mass forces and wake capture
forces, the latter resulting from the interaction between a wing and the shed
vorticity of the previous strokes
(Dickinson et al., 1999;
Sane and Dickinson, 2002).
However, using only translational and rotational components of the
quasi-steady model, we obtained reasonably accurate approximations of the
measured forces.
Rectangular components of force and moments relative to the body
The above measurements of the force normal to the wing surface were
combined with the three-dimensional wing orientation relative to the body in
order to calculate the directional components of force in the body's frame of
reference. The wing angles were transformed such that the fly's longitudinal
body axis was defined as the X-axis, its vertical axis was the
Y-axis and its cross-sectional axis was the Z-axis
(Fig. 2A). Note that this
converted the reference frame from the inclined body axis used for assessing
kinematic variation and reproducing the kinematics using the mechanical model
(Fig. 1C,D) to a horizontal
body axis (Fig. 2). The
three-dimensional angular orientation of the wing directs the aerodynamic
force into its rectangular components:
 | (10) |
 | (11) |
 | (12) |
where FN is the total aerodynamic force normal to the
wing, Fx is thrust, Fy is lift and
Fz is the radial or sideslip force.
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Fig. 2. Resultant forces and moments relative to the body. (A) Illustration of the
six degrees of body motion: the directional components (thrust, lift and
sideslip) and the moments (roll, yaw and pitch). The projection of the
aerodynamic force vector (FN) onto each of the three
rectangular axes constitutes its contribution to thrust
(Fx), lift (Fy) and sideslip
(Fz). The moment depends on the position vector r
and the force vector FN as described in the text. (B)
Simplified sideview of A. If the sideslip component of force is negligible,
lift and thrust depend primarily on the magnitude (FN) and
inclination (F) of the force vector as the wing moves
through the stroke (blue dots denote changes in position).
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The contribution of the force vector to the body moment, M, is
determined by:
 | (13) |
where r is the position vector between the body's center of mass and
the wing's center of pressure, and FN is the
three-dimensional aerodynamic force vector normal to the wing surface. We
estimated the center of mass as the point midway between the left and right
wing hinge and used the wing's center of area (0.54R or 4.9 mm from
wing base for a wing of 9 mm length) as an estimate of the wing's center of
pressure. Roll (Mx), yaw (My) and
pitch (Mz) moments were calculated from:
 | (14) |
 | (15) |
 | (16) |
Because the sideslip force (Fz) was small through our
dataset, we were able to summarize the direction of the force vector as one
angular measure, the force inclination (F), thereby reducing
the number of variables determining lift and thrust
(Fig. 2B). The relationship
between the force vector and the accompanying moment also simplifies, such
that roll is essentially a function of lift (Fy), and yaw
is essentially a function of thrust (Fx). Pitch remains a
function of the difference between lift and thrust. Whereas roll and yaw are
most sensitive to forces at mid-stroke when rz is maximal
(equations 14,
15), pitch is most greatly
influenced by forces generated during stroke reversal when
rx and ry are maximal
(equation 16).
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Results
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Analytical framework
The goal of our analysis is to quantify the relationship between the
kinematic adjustments correlated with steering muscle activity and the role of
these adjustments in controlling aerodynamic forces during steering maneuvers.
In order to study the relationship between muscle activity and body forces, we
must bridge several intermediate levels of analysis that have been described
previously. Therefore, the following discussion will introduce the known
aspects of these intermediate transformations that were used for the combined
analysis used in this study. First, we will describe the aspects of the
aerodynamic force vector relevant to body forces and moments. Second, we will
describe the kinematic variables relevant to control of aerodynamic forces.
Third, we will describe the kinematic adjustments correlated with steering
muscle activity. Finally, we will introduce the concerted nature of the
changes accompanying each kinematic adjustment.
The motion of each wing contributes to the body's six degrees of freedom by
varying the magnitude, direction and position of an aerodynamic force vector
(FN; Fig.
2A). We found that in our study on Calliphora, the
sideslip force generated by each wing was relatively small (maximum mean over
wingbeat cycle: sideslip force 1.0x10–4 N vs
lift and thrust forces 4.0x10–4 N). Therefore, the
magnitude (FN) and the inclination (F)
of the force vector were the primary output variables contributing to the
remaining five degrees of freedom (Fig.
2B). Due to the dependence of moments on the instantaneous
position of the wing, roll and yaw are most sensitive to forces at mid-stroke,
whereas pitch is most sensitive to forces during stroke reversals.
The repetitive pattern of wing motion is characterized by a roughly
harmonic back-and-forth motion, (t), during which the
morphological wing angle is relatively constant until the wing rotates at the
dorsal and ventral reversal points [(t);
Fig. 3A]. Variation in the wing
deviation is relatively small throughout the wingbeat cycle and follows a more
complicated waveform [(t);
Fig. 3A]. According to a recent
multi-component quasi-steady model (Sane and Dickinson,
2001,
2002), the primary kinematic
determinants of aerodynamic force production are the wingtip velocity
(Ut), the angle of attack (g) and the
rotational angular velocity (; Fig.
3B). The tip velocity and the angle of attack together determine
the translatory component of the force (Ftrans), which
reaches its peak during the middle of the stroke
(Fig. 3C). The tip velocity and
the rotational velocity together determine the rotational component of the
force (Frot), which acts from the end of one stroke to the
beginning of the next (Fig.
3C). The sum of quasi-steady translatory and rotational force
components is equal to the total calculated normal force. The time course of
the calculated forces was in reasonably close agreement with forces measured
by playing the kinematics on our dynamically scaled mechanical model
(Fig. 3C). The main source of
disagreement between the two traces was a positive transient in the measured
forces at the start of each stroke that was not captured by the two-component
quasi-steady model (Fig. 3C).
This is the same pattern observed by Sane and Dickinson
(2002) and is likely to be due
to a combination of acceleration reaction (added mass) forces and wake
capture.
Although a reasonably robust theory exists for predicting the forces
resulting from an arbitrary change in wing motion, the link between
aerodynamically relevant changes in wing kinematics and the activity of
specific steering muscles is less clear. Our previous study
(Balint and Dickinson, 2001)
indicated that activity in specific steering muscles is well correlated with
systematic and quantifiable distortions of the wingtip trajectory. In
particular, displacement of the downstroke trajectory along the roughly
anterio-posterior body axis, which we termed downstroke deviation, was a
robust correlate of cycle-by-cycle activity patterns in the basalare muscles.
However, changes in downstroke deviation were not isolated modulations of
deviation, (t), but were consistently coupled with modulation
of the ventral amplitude, the anterio-ventral maximum in elevation,
(t). The ventral amplitude accompanying changes in downstroke
deviation differed slightly depending on whether the muscles of pteralae III
were active (Mode 2) or those of pteralae I were active (Mode 1). In the
present study, our results concerning the correlation between muscle activity
and these features of the wingtip trajectory were consistent with the previous
findings (Fig. 4). However, our
use of three-dimensional high-speed video in the present study allowed us to
assess kinematic features related to changes in wing angle () in
addition to changes in wingtip elevation () and deviation (). We
found that changes in wing angle [(t)] and wing trajectory
[(t) and (t)], rather than being independent of
each other, were part of concerted kinematic programs. Therefore, downstroke
deviation was one component of a three-dimensional kinematic alteration. In
addition, the associated changes were not limited to the downstroke but
extended over the entire cycle. The shape of the wingbeat trajectory, or the
time course of (t) over the downstroke and following upstroke,
was closely associated with downstroke deviation
(Fig. 5A), as was the time
course of the wing angle [(t);
Fig. 5B]. In addition, the
ventral amplitude was correlated with downstroke deviation, except for the
subtle de-coupling between modes (Fig.
5C), as mentioned above. The dorsal amplitude – the
posterio-dorsal maximum in elevation – varied independently of
downstroke deviation and differed considerably between the two wings and
across individuals (Fig. 5D).
We also found that the wingbeat frequency was independent of downstroke
deviation (Fig. 5E) and all
other aspects of the wingbeat. The wingbeat frequency varied very little
overall, and all individuals fell roughly into one of two frequency groups.
However, the downstroke to upstroke ratio was correlated with downstroke
deviation within trials (Fig.
5F) and was correlated with dorsal amplitude across trials (see
Dorsal amplitude section below).
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Fig. 5. Variation of kinematic parameters in relation to downstroke deviation. (A)
Wingtip trajectories of downstrokes and accompanying upstrokes of a single
individual, color-coded according to downstroke deviation value (color bar is
shown on the left). Black circles indicate ventral and dorsal amplitudes. (B)
Examples of wing orientations accompanying high (red) and low (blue)
downstroke deviation, shown for the downstroke (above) and the upstroke
(below). (C) Relationship between downstroke deviation and ventral amplitude
within the experimental population (R2=0.78). A subtle
distinction exists between Mode 2 (gray points; blue line is a second order
regression) and Mode 1 (pink points; red line). No distinction between modes
was observed for the following parameters within the experimental population:
relationship between downstroke deviation and (D) dorsal amplitude
(R2=0.02), (E) instantaneous wingbeat frequency
(R2=0.01) and (F) downstroke/upstroke ratio
(R2=0.30). Black circles in C–F indicate values for
the individual shown in A.
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Given this combination of tightly and more loosely correlated features of
wing motion, the functional significance of downstroke deviation as a control
parameter is not directly evident in comparison with that of conventional
uni-dimensional parameters such as total stroke amplitude, which theoretically
correspond to a single aerodynamic variable, mean wingtip velocity
(Ut). Within the entire data set, we identified three
independently controlled features of the wingbeat trajectory: downstroke
deviation, mode and dorsal amplitude. Downstroke deviation and mode were
identified based on their robust match with patterns of muscle activity,
whereas dorsal amplitude was identified based on its considerable inter-wing
and inter-individual variability. For each of these components of the wingbeat
trajectory, the associated changes were multi-dimensional and specific to
different parts of the wingbeat cycle. We examined all changes in body forces
and moments caused by alteration of these three coordinated changes in wing
motion. This approach consists of correlating the translatory forces
(Ftrans), rotational forces (Frot) and
force inclinations (F) over each wingbeat cycle with each
kinematic parameter and then summarizing the consequences for mean lift,
thrust, roll, yaw and pitch. Through our analysis, we were able to confirm
that these three kinematic patterns are distinct with respect to both
behavioral function and neuromuscular control.
Downstroke deviation
As described in previous work (Balint
and Dickinson, 2001), downstroke deviation was correlated on a
cycle-by-cycle basis with the activity of the basalare muscles. However, the
more thorough three-dimensional analysis showed that downstroke deviation
accompanied a particular qualitative change in all three kinematic dimensions,
(t), (t) and (t), throughout
each cycle. In order to quantify the functional significance of these
coordinated changes for control of the aerodynamic force vector, we examined
the influence of downstroke deviation on translational
(Ftrans) and rotational (Frot)
mechanisms of force generation, as well as the inclination of these forces
(F). This combination of influences will be used to
demonstrate that the changes associated with downstroke deviation result in a
predicted modulation of body lift via control of the force generated
during the downstroke.
First, we investigated the changes relevant to control of the translatory
force. As a consequence of the complex of kinematic parameters involved,
changes in downstroke deviation were correlated with concerted changes of both
angle of attack and tip velocity. More importantly, changes in downstroke
deviation were not indicative of a mean change in these variables over the
wingbeat cycle but rather a more complex change in time course throughout the
stroke. Fig. 6A,D illustrates
the pattern of variation in angle of attack that accompanied changes in
downstroke deviation, and Fig.
6B,E illustrates the concomitant pattern of instantaneous tip
velocity. Although both angle of attack and tip velocity varied throughout the
cycle, the patterns of variation were quite distinct. During the downstroke,
the angle of attack (Fig. 6A)
and the tip velocity (Fig. 6B)
varied in a complementary way, so that the dependence of the resultant force
on downstroke deviation was relatively large
(Fig. 6C). By contrast, during
the upstroke, angle of attack (Fig.
6D) and tip velocity (Fig.
6E) varied inversely, such that the range of translatory force at
each time point remained relatively small
(Fig. 6F). Therefore, because
of the precise pattern of changes in angle of attack and tip velocity, changes
in downstroke deviation affected force during the downstroke but not the
upstroke. In order to confirm the pattern of force modulation described above,
we compared the relevant mid-stroke values for our experimental population.
For the downstroke, mid-stroke angle of attack, tip velocity and translatory
force were consistently correlated with downstroke deviation
(Fig. 7A), and the range of
variation was similar to that shown in Fig.
6A–C. By contrast, angle of attack and tip velocity measured
during the upstroke were much more variable across individuals
(Fig. 7B). However, no
inter-individual variation was evident in the upstroke translatory force
(Fig. 7B). A subtle correlation
existed between downstroke deviation and the upstroke translatory force, but
upstroke force was consistently less variable than downstroke force.
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Fig. 6. Temporal variation in angle of attack, tip velocity and translatory force
in a single individual, color-coded according to downstroke deviation as in
Fig. 5A. All data are shown
along a normalized time axis, where 0 is the beginning and 1 is the end of the
half-stroke. The slope and R2 regression statistics for
the correlation between each parameter and downstroke deviation are shown for
normalized time intervals of 0.05. Mid-stroke was defined as a normalized time
of 0.55, indicated by the vertical dotted lines. (A) Angles of attack over the
downstroke. (B) Tip velocities over the downstroke. (C) Translatory forces
over the downstroke. Regression statistics are shown for the individual (black
circles) and the population (gray circles). (D) Angles of attack over the
upstroke. (E) Tip velocities over the upstroke. (F) Translatory forces over
the upstroke. Regression statistics are shown for the individual (black
circles) and the population (gray circles).
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Fig. 7. Correlation between downstroke deviation and the mid-stroke angle of
attack, tip velocity and translatory force for the experimental population.
Black circles indicate values for the individual shown in
Fig. 6. (A) Correlation between
downstroke deviation and mid-downstroke angle of attack
(R2=0.41), tip velocity (R2=0.22) and
translatory force (R2=0.58). (B) Correlation between
downstroke deviation and mid-upstroke angle of attack
(R2=0.06), tip velocity (R2=0.05) and
translatory force (R2=0.23). Five separate regressions are
shown for five individuals.
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Second, we investigated the associated changes in the rotational force.
Although the mechanism for active control of rotation is not known
(Ennos, 1988), we found a
relatively strong correlation between downstroke deviation and the time course
of the ventral rotation (Fig.
8A,B). By contrast, the timing and magnitude of the dorsal
rotation was relatively constant. Whereas the ventral rotation elevates force
at the end of the downstroke, it also acts to diminish total force at the
start of the upstroke. As a consequence, ventral rotation contributed a small
force to the end of the downstroke (Fig.
8C) that was complementary to the concomitant translatory force,
so that both the rotational and translatory force components contributed to
the correlation of downstroke deviation with total force
(Fig. 8D). The ventral rotation
contributed a large negative force to the start of the upstroke
(Fig. 8F) due to the delay in
wing rotation relative to stroke reversal
(Fig. 8E), but addition of the
concomitant positive translatory force resulted in a smaller range of total
peak forces (Fig. 8G). The
contribution of the dorsal rotational force to total force at the end of the
upstroke was relatively large [mean rotational force peak,
5x10–4±1x10–4 N
(S.D.)], and its contribution to the start of the downstroke was
similar but more variable [mean rotational force peak,
–6x10–4±4x10–4 N
(S.D.)].
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Fig. 8. Correlation between downstroke deviation and rotational forces. (A) Time
course of rotational velocities and (B) time course of rotational forces for a
single individual, color-coded according to downstroke deviation. Data are
shown over a normalized time axis for the downstroke and upstroke. The filled
gray box indicates the ventral rotation, and the open gray box indicates the
dorsal rotation. For the ventral rotation, the pre-reversal portion occurs at
the end of the downstroke, and the post-reversal portion occurs at the
beginning of the upstroke. (C–G) The following data are for the
experimental population. Black circles indicate values for the individual
shown in A and B. Correlation between downstroke deviation and (C)
pre-reversal peak rotational force (R2=0.07), (D)
pre-reversal total force peak (R2=0.20), (E) post-reversal
normalized time delay of rotational force peak (R2=0.15),
(F) post-reversal peak rotational force (R2=0.60) and (G)
post-reversal total force peak (R2=0.31).
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Third, we investigated the relationship between downstroke deviation and
force inclination. The range and variability of force inclination differed
between downstrokes and upstrokes, as did force magnitude. The force
inclination over the downstroke was strongly correlated with downstroke
deviation (Fig. 9Ai). Although
the temporal pattern of force inclination was such that the sign of the
correlation with downstroke deviation changes at mid-stroke, the overall
variation was relatively small. The force was generally directed upward
relative to the body, between roughly 60 and 80° relative to horizontal at
the point of largest variation (Fig.
9Aii). By contrast, during the upstroke, force inclination was not
correlated with downstroke deviation (Fig.
9Bi). The total aerodynamic force was generally directed forward
relative to the body during the upstroke but varied over a wide range from
–30 to 40° relative to horizontal across the experimental population
(Fig. 9Bii). Therefore,
downstrokes and upstrokes differed not only in the general direction of the
force vector but also with respect to the degree of variation in force
inclination.
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Fig. 9. Correlation between downstroke deviation and force inclination. (Ai) Time
course of force inclination during the downstroke for a single individual,
color-coded according to downstroke deviation. The slope and
R2 regression statistics for the correlation between
downstroke deviation and force inclination are shown for normalized time
intervals of 0.05. Regression statistics are shown for the individual (black
circles) and the population (gray circles). The vertical dotted line indicates
a normalized time of 0.65. (Aii) Correlation between downstroke deviation and
the mid-downstroke (normalized time 0.65) force inclination for the
experimental population (R2=0.71). Black circles indicate
data for the individual in Ai. (Bi) Time course of force inclination during
the upstroke for a single individual, color-coded according to downstroke
deviation. The slope and R2 regression statistics for the
correlation between downstroke deviation and force inclination are shown at
the bottom, as in Ai. The vertical dotted line indicates a normalized time of
0.55. (Bii) Correlation between downstroke deviation and mid-upstroke
(normalized time 0.55) force inclination for the experimental population
(R2=0.03). Black circles indicate data for the individual
in Bi.
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Finally, we assessed the influence of downstroke deviation on mean
resultant forces and moments. The overall dichotomy between downstrokes and
upstrokes was that, during the downstroke, the force magnitude was variable
(Fig. 10A,B) while the force
inclination was relatively constant (Fig.
10A,C) whereas, during the upstroke, the force magnitude was
relatively constant (Fig.
10F,G) while the force inclination was variable
(Fig. 10F,H). The modulation
of force magnitude during the downstroke resulted mainly in modulation of lift
(Fig. 10D) and roll
(Fig. 10E). The small changes
in force magnitude during the upstroke resulted in a relatively constant
thrust (Fig. 10I) and yaw
(Fig. 10J). The uncorrelated
variation in upstroke force inclination had a greater effect on lift and roll
than on thrust (Fig. 10I,J).
The asymmetry between the variable downstroke lift and the less variable
upstroke thrust resulted in modulation of the mean pitch over each cycle that
was well correlated with the downstroke deviation within individuals
(Fig. 11A). However, the
uncorrelated lift component during the upstrokes
(Fig. 11B) resulted in
inter-individual variation in pitch (Fig.
11A).
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Fig. 10. Mean force vector (FN) and resultant directional forces
and moments. (A) Schematic illustrating the type of force vector modulation
occurring during the downstroke. As downstroke deviation changes, there is a
correlated modulation of force magnitude while force inclination remains
relatively constant. Correlation between downstroke deviation and downstroke
(B) mean force magnitude (R2=0.43), (C) mean force
inclination (R2=0.02), (D) mean lift (blue;
R2=0.54) and mean thrust (green;
R2=0.13), and (E) mean roll (blue;
R2=0.44) and mean yaw (green;
R2=0.05). (F) Schematic illustrating primary type of
variation in mean force vector during the upstroke. As downstroke deviation
changes, force magnitude is relatively constant, but there is unexplained
variation in force inclination. Correlation between downstroke deviation and
upstroke (G) mean force magnitude (R2=0.16), (H) mean
force inclination (R2=0.004), (I) mean thrust (green;
R2=0.22) and mean lift (blue;
R2=0.0001), and (J) mean yaw (green;
R2=0.17) and mean roll (blue;
R2=0.001). The direction of the roll and yaw moments in E
and J differ depending on whether they are generated by the left or right
wing. For the left wing, positive roll is a right side down roll, and positive
yaw is a yaw to the right. For the right wing, positive roll is a left side
down roll, and positive yaw is a yaw to the left.
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Fig. 11. Mean pitch. (A) Correlation between downstroke deviation and mean pitch
from each wingbeat cycle. Regression lines for five individuals super-imposed.
A relatively strong correlation exists within each individual
(R2=0.65 turquoise, 0.87 green, 0.97 orange, 0.83 pink,
0.76 purple), but the correlation differs across individuals (total
R2=0.49). (B) Upstroke lift from
Fig. 10I, with regression
lines for the same five individuals from A super-imposed.
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In conclusion, the primary role of changes in downstroke deviation and the
associated kinematic variables by the basalare muscles was to modulate the
lift force generated during the downstroke and thereby induce a roll moment as
well as some pitch. The accompanying kinematic changes also produced a more
subtle modulation of thrust and yaw during the upstroke.
Dorsal amplitude
Dorsal amplitude was a component of wing motion that remained relatively
constant as downstroke deviation varied. Because variation of dorsal amplitude
was small within individuals, we were unable to correlate differences with any
pattern of muscle activity. However, inter-wing and inter-individual variation
in dorsal amplitude was considerable. Therefore, we investigated the
relationship of dorsal amplitude to the inter-individual variation in the
upstroke parameters that were unexplained with respect to downstroke
deviation. We will demonstrate that the changes associated with dorsal
amplitude result in a predictable modulation of body lift via
inclination of the force vector during the upstroke.
Although dorsal amplitude was not associated with any significant
differences in the shape of the wingtip trajectory [(t);
Fig. 12A], it did accompany
differences in morphological wing angle during the upstroke
[(t); Fig.
12B]. Inter-individual differences in the downstroke to upstroke
ratio were also correlated with dorsal amplitude
(Fig. 12C). As a consequence
of the coupling of morphological wing angle and amplitude, changes in dorsal
amplitude resulted in concerted changes of both the geometrical angle of
attack and the tip velocity during the upstrokes.
Fig. 13Ai illustrates the
variation in the angle of attack through the upstroke for three sample
individuals differing in dorsal amplitude. Among these three individuals, as
well as across the experimental population, the mid-stroke angle of attack was
negatively correlated with dorsal amplitude
(Fig. 13Aii). Thus, the angle
of attack was lower in upstrokes that extended to a more dorsal position. At
the same time, the upstroke tip velocity of these individuals
(Fig. 13Bi), as well as across
all individuals (Fig. 13Bii),
was positively correlated with dorsal amplitude. Therefore, due to the inverse
relationship between angle of attack and tip velocity, translatory force
showed little variation with respect to dorsal amplitude
(Fig. 13C). We also found no
relationship between dorsal amplitude and rotational force (data not
shown).
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Fig. 12. Variation of kinematic parameters in relation to dorsal amplitude. (A)
Wingtip trajectories during the upstroke, shown for three individuals
differing in dorsal amplitude. Colored circles indicate dorsal amplitudes. (B)
Examples of wing orientations accompanying large (purple), intermediate
(green) and small (turquoise) dorsal amplitude during the upstroke. (C)
Relationship between dorsal amplitude and downstroke/upstroke ratio within the
experimental population (R2=0.45). Colored circles
indicate data for the three individuals shown in A.
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Fig. 13. Correlation between dorsal amplitude and upstroke translatory forces.
Colors indicate data from three individuals with large (purple), intermediate
(green) and small (turquoise) dorsal amplitudes. All time-course data are
shown along a normalized time axis. The slope and R2
regression statistics for the correlation between each parameter and dorsal
amplitude are shown for normalized time intervals of 0.05. Mid-upstroke was
defined as a normalized time of 0.55, indicated by the vertical dotted line.
(Ai) Time course of the angle of attack over the upstroke, shown for three
individuals. (Aii) Correlation between dorsal amplitude and mid-upstroke angle
of attack, for the experimental population (R2=0.71). (Bi)
Time course of the tip velocity over the upstroke, shown for three
individuals. (Bii) Correlation between dorsal amplitude and mid-upstroke tip
velocity, for the experimental population (R2=0.73). (Ci)
Time course of the translatory force over the upstroke, shown for three
individuals. (Cii) Correlation between dorsal amplitude and mid-upstroke
translatory force, for the experimental population
(R2=0.04).
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The associated variation in morphological wing angle resulted in alteration
of both the geometrical angle of attack and force inclination. Whereas dorsal
amplitude was negatively correlated with upstroke angle of attack, it was
positively correlated with force inclination. The correlation between dorsal
amplitude and force inclination was strong from the middle to the end of the
upstroke, across all individuals (Fig.
14A,B). In contrast to this variation in force inclination, the
kinematic changes associated with dorsal amplitude resulted in a constant
force magnitude (Fig. 15B,C).
As a result, the mean lift varied with dorsal amplitude more strongly than
mean thrust (Fig. 15D). As
expected, the variation in roll followed the variation in lift
(Fig. 15E). Therefore, the
inter-individual variation in upstroke lift and roll
(Fig. 10I,J) and mean pitch
(Fig. 11) that was
uncorrelated with downstroke deviation may be explained by independent,
inter-individual differences in dorsal amplitude (dorsal amplitude vs
mean pitch, R2=0.58).
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Fig. 14. Correlation between dorsal amplitude and upstroke force inclination. (A)
Time course of force inclination during the upstroke, shown for the three
individuals in Fig. 13. The
slope and R2 regression statistics for the correlation
between dorsal amplitude and force inclination in normalized time intervals of
0.05 for the experimental population are shown at the bottom. Mid-upstroke,
defined as a normalized time of 0.55, is indicated by the vertical dotted
line. (B) Correlation between dorsal amplitude and mid-upstroke force
inclination for the experimental population (R2=0.76).
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Fig. 15. Mean force vector and resultant directional forces and moments. (A)
Schematic illustrating primary type of variation in mean force vector during
the upstroke as in Fig. 10F.
Differences in force inclination were well correlated with dorsal amplitude.
Correlation between dorsal amplitude and upstroke (B) mean force magnitude
(R2=0.009), (C) mean force inclination
(R2=0.75), (D) mean lift (blue;
R2=0.79) and mean thrust (green;
R2=0.09), and (E) mean roll (blue;
R2=0.58) and mean yaw (green;
R2=0.18).
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In conclusion, the primary role of changes in dorsal amplitude and the
associated kinematic variables was to enhance the lift during the upstroke by
tilting the force vector and thereby contribute to variation in roll and pitch
moments.
Mode
The most obvious characteristic of wingtip trajectories that correlated
with changes in the activity of the muscles of pterale I and III was a shift
in the ventral amplitude accompanying changes in downstroke deviation. We
termed this qualitative alteration in stroke pattern a mode shift. Although no
other noticeable changes in the downstroke trajectory were associated with
differences in mode, the upstroke trajectory was slightly lower in deviation
during Mode 1 than during Mode 2 (Fig.
16A). In addition, the upstroke wing angles differed between modes
(Fig. 16B). Although we
defined a change in mode as a roughly binary shift in ventral amplitude, we
did observe graded, intra-mode variation in ventral amplitude accompanying
changes in downstroke deviation, as well as inter-individual variation in
dorsal amplitude. We examined the functional significance of mode shift by
comparing the changes associated with downstroke deviation and dorsal
amplitude within Mode 1 strokes, with changes associated with the same
parameters within Mode 2 strokes. We will demonstrate that the kinematic
changes specific to a mode shift result in a predicted modulation of body
thrust due to a change in the force generated during the upstroke.
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Fig. 16. Variation of kinematic parameters in relation to mode. (A) Comparison of
wingtip trajectories during Mode 2 (black) and Mode 1 (pink). Blue circles
indicate ventral amplitudes during Mode 2, and red circles indicate ventral
amplitudes during Mode 1. (B) Examples of wing orientations during the
upstroke for Mode 2 (black) and Mode 1 (pink).
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Fig. 17 compares the
temporal pattern of angle of attack, tip velocity and translatory force
associated with Mode 1 and 2 strokes. An equivalent range of downstroke
deviations is represented in each mode, and dorsal amplitude is constant. For
the downstroke, the angle of attack tended to be greater during Mode 1, most
dramatically at the beginning and end of the stroke
(Fig. 17A). By contrast, tip
velocities tended to be slightly lower during Mode 1 but overlapped with those
of Mode 2 (Fig. 17B). The
resultant range of translatory forces was equivalent within both modes,
although the force onset was slightly delayed in Mode 1 strokes
(Fig. 17C). For the upstroke,
the angle of attack was generally lower during Mode 1
(Fig. 17D), whereas the tip
velocities also tended to be lower during Mode 1 but overlapped with those
during Mode 2 (Fig. 17E).
However, because the changes in angle of attack and tip velocity were
complementary, the translatory force during the upstroke was much greater in
Mode 2 strokes than in Mode 1 strokes
(Fig. 17F).
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Fig. 17. Comparison of temporal variation in angle of attack, tip velocity and
translatory force between modes. Time course of each parameter over the
downstroke and upstroke shown along a normalized time axis. Mode 2 strokes are
shown in black, and Mode 1 strokes are shown in pink. Mid-stroke was defined
as a normalized time of 0.55, indicated by the vertical dotted lines. (A)
Angles of attack over the downstroke. (B) Tip velocities over the downstroke.
(C) Translatory forces over the downstroke. (D) Angles of attack over the
upstroke. (E) Tip velocities over the upstroke. (F) Translatory forces over
the upstroke.
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Comparing the mid-stroke values over our experimental population, we found
that, for the downstroke, the relationship of angle of attack and tip velocity
with downstroke deviation was similar within both modes, but with minor
differences. Whereas the angle of attack during Mode 1 strokes was
occasionally large, the accompanying tip velocity was comparatively small. Due
to a consistent relationship between angle of attack and tip velocity, the
correlation between downstroke deviation and translatory force remained nearly
identical for both modes (Fig.
18A). For the upstroke, we compared the relationship of angle of
attack and tip velocity with dorsal amplitude between modes. The angle of
attack was consistently lower during Mode 1 strokes than during Mode 2 strokes
(Fig. 18B). The tip velocities
were slightly lower during Mode 1 strokes but overlapped with those within
Mode 2. However, due to the consistently lower angle of attack, the
translatory force was consistently lower during Mode 1 than during Mode 2
strokes, even when the tip velocities overlapped
(Fig. 18B). The mid-upstroke
translatory forces were subtly correlated with downstroke deviation within
both modes (Fig. 18C).
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Fig. 18. Comparison of mid-stroke translatory forces between modes. (A) Relationship
between downstroke deviation and mid-downstroke angle of attack, tip velocity
and translatory force within Mode 2 (gray points) and Mode 1 (pink points).
The regression of downstroke deviation vs mid-downstroke translatory
force is shown for Mode 2 (black line; R2=0.58) and Mode 1
(red line; R2=0.63). (B) Relationship between dorsal
amplitude and mid-upstroke angle of attack, tip velocity and translatory force
within Mode 2 (gray points) and Mode 1 (pink points). The regression of dorsal
amplitude vs mid-upstroke translatory force is shown for Mode 2
(black line; R2=0.04) and Mode 1 (red line;
R2=0.05). (C) Relationship between downstroke deviation
and mid-upstroke translatory force within Mode 2 (gray points) and Mode 1
(pink points). The regression of downstroke deviation vs mid-upstroke
translatory force is shown for Mode 2 (black line;
R2=0.23) and Mode 1 (red line;
R2=0.43).
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Fig. 19A,B compares the
time course of rotation and rotational force during Mode 1 with that during
Mode 2. Mode 1 was associated with a delay in the rotational peak at the
beginning of the downstroke (Fig.
19C). This means that the dorsal flip was substantially delayed
during Mode 1 strokes. Although this delay was not correlated with a
consistent change in the magnitude of the rotational force peak at the
beginning of the downstroke (Fig.
19D), it was correlated with a decrease in the magnitude of the
rotational force peak at the end of the upstroke
(Fig. 19F). Although the
difference in rotational force at the end of the upstroke was small, it was
complementary to the difference in concomitant translatory forces, and
therefore the total force was substantially lower during Mode 1 than during
Mode 2 (Fig. 19G). We found no
significant differences in force inclination between modes (data not
shown).
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Fig. 19. Comparison of rotational forces between modes. (A) Time course of
rotational velocities and (B) time course of rotational forces within Mode 2
(black) and Mode 1 (pink). The filled gray box indicates the dorsal rotation,
and the open gray box indicates the ventral rotation. For the dorsal rotation,
the pre-reversal portion occurs at the end of the upstroke, and the
post-reversal portion occurs at the beginning of the downstroke. (C–G)
The following data are for the experimental population. Correlation between
downstroke deviation and (C) post-reversal normalized time delay of rotational
force peak, (D) post-reversal peak rotational force, (E) post-reversal total
force peak, (F) pre-reversal peak rotational force and (G) pre-reversal total
force peak for Mode 2 (gray points; black line regression) and Mode 1 (pink
points; red line regression).
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Finally, we compared the influence of mode on mean resultant forces and
moments. Due to the delay in dorsal rotation, the total mean force was
slightly lower within Mode 1 downstrokes than within Mode 2 downstrokes
(Fig. 20B). By comparison, due
to the decrease in both translatory and rotational forces during the upstroke,
the total mean force was substantially lower during Mode 1 upstrokes than
within Mode 2 upstrokes (Fig.
20G). The relatively small difference in force magnitudes within
the downstroke, as well as an equivalent range of force inclinations
(Fig. 20C), resulted in a
similar relationship between downstroke deviation and lift
(Fig. 20D) and roll
(Fig. 20E) for both modes. The
relatively large difference in force magnitudes between modes during the
upstroke resulted in an overall decrease in thrust
(Fig. 20I) and yaw
(Fig. 20J) during Mode 1
strokes relative to Mode 2. Although the force inclination during the upstroke
varied within Mode 1 as within Mode 2
(Fig. 20H), upstroke lift
during Mode 1 was small and did not vary considerably [mean upstroke lift,
5.5x10–5±3x10–5 N
(S.D.)]. Therefore, the upstroke roll was also small during Mode 1
[mean upstroke roll,
2x10–7±1x10–7 Nm
(S.D.)].
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Fig. 20. Mean force vector and resultant directional forces and moments. (A)
Schematic illustrating the difference in mean force vector during the
downstroke between modes. While force magnitude is similarly modulated within
each mode, the mean forces can be slightly lower during Mode 1. Correlation
between downstroke deviation and downstroke (B) mean force magnitude and (C)
mean force inclination for Mode 2 (gray points; black line regression) and
Mode 1 (pink points; red line). Correlation between downstroke deviation and
downstroke (D) mean lift and (E) mean roll for Mode 2 (blue points; black
line) and Mode 1 (pink points; red line). (F) Schematic illustrating the
difference in mean force vector during the upstroke between m | |
TOP
Summary
Introduction
