Neuromuscular control of aerodynamic forces and moments in the blowfly, Calliphora vicina
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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Key words: insect flight, kinematics, aerodynamics, steering, motor control, Calliphora vicina
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Introduction |
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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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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 s1 and an electronic shutter speed of 1/20 000.
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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 s1 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) |
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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):
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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:
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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:
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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) |
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The contribution of the force vector to the body moment, M, is
determined by:
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Results |
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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 1.0x104 N vs
lift and thrust forces 4.0x104 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.
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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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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. 6AC. 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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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,
5x104±1x104 N
(S.D.)], and its contribution to the start of the downstroke was
similar but more variable [mean rotational force peak,
6x104±4x104 N
(S.D.)].
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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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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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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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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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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. 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).
|
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).
|
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).
|
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, 5.5x105±3x105 N (S.D.)]. Therefore, the upstroke roll was also small during Mode 1 [mean upstroke roll, 2x107±1x107 Nm (S.D.)].
|
In conclusion, the primary role of a shift in mode was to change the thrust, and, as a consequence, yaw torque generated during the upstroke.
Comparison of calculated forces with measured forces
Although wingbeat frequency is an important kinematic parameter that
affects aerodynamic forces through changes in wingtip velocity, we found it
remained relatively constant relative to the other observed changes in
wingbeat trajectory (Fig. 4).
Within the experimental population, flies fell into roughly two frequency
groups. Five individuals flew with a mean wingbeat frequency of
155.2±5.3 Hz (S.D.), and two individuals flew at a mean of
129.4±4.5 Hz (S.D.)
(Fig. 21A). Both modes were
represented within each frequency group. In order to make comparisons across
the population of the effects specific to changes in wingbeat trajectory, in
the preceding sections we normalized the instantaneous wingtip velocities
within each stroke with respect to the cycle period. With this normalization,
the relationship between downstroke deviation and the calculated mean force
magnitude overlapped for the two frequency populations
(Fig. 21B).
|
However, when we reproduced each fly's kinematics using the dynamically scaled mechanical model, we scaled the wingtip velocities so as to maintain the observed differences in wingbeat frequency. Therefore, we evaluated the difference between our calculated force magnitudes and the forces measured using the mechanical model by making the comparison separately within the two frequency groups. Fig. 21C illustrates how the relationship between downstroke deviation and the calculated mean force magnitude differs for the two frequency groups when tip velocity is not normalized. The mean difference between the downstroke regressions was 8.8x105±6.8x106 N (S.D.), and the mean difference between the upstroke regressions was 1.3x104±2.1x105 N. The relationship between downstroke deviation and the measured mean force magnitude was very similar, although the scatter among points was larger (Fig. 21D). The mean difference between the downstroke regressions was 4.9x105±1.4x106 N, and the mean difference between the upstroke regressions was 1.4x104±1.7x105 N. Therefore, the effect of the difference in wingbeat frequency on differences in measured force magnitude was very similar to its estimated effect on differences in calculated force magnitude, although the limited range of frequencies did not permit a correlational analysis. The regressions from Fig. 21C and Fig. 21D are shown together in Fig. 21E. For each wingbeat cycle, the measured force was always larger than the calculated force, primarily due to the unexplained force transient at the beginning of each half-stroke as shown in Fig. 3C. However, the relationship between downstroke deviation and force magnitude was roughly the same for calculated and for measured forces.
The difference between calculated and measured forces was roughly the same for both frequency groups (Table 1). The difference was only slightly smaller for Mode 1 strokes than for Mode 2 strokes, given the standard deviations (Table 1). Overall, we found no systematic variation in the magnitude of error between measured and calculated forces, and the random error was small relative to the total mean forces. Therefore, the trends described using the theoretical quasi-steady model were preserved in our measurements using the dynamically scaled mechanical model.
|
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Discussion |
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|
Each of the three mechanisms of neuromuscular control downstroke
deviation, mode and dorsal amplitude involved multiple aerodynamic
mechanisms for their effect on the total aerodynamic force vector. Although we
organized our analysis according to the three features of wing motion we found
to be independently controlled with respect to patterns of muscle activity,
these categories also encompassed many of the types of kinematic variation
that have been noted previously. Kinematic parameters that have been measured
in prior studies of insect flight include stroke amplitude, stroke position or
deviation, stroke inclination, degree of pronation/supination, differences in
reversal timing, speed and timing of rotation, and wing deformation (e.g.
camber and torsion) (for review, see
Kammer, 1985;
Taylor, 2001
). Due to the
difficulty of obtaining simultaneous, multi-dimensional, time-resolved images
of wing motion, these kinematic parameters have generally been considered as
separate categories of modulation. However, our results suggest that these
features of the wing stroke are not varied independently and that their
coupling has important functional consequences. For instance, changes in
downstroke deviation involve concerted modulation of almost all the components
listed above. The combined result is a positively correlated change in both
the geometrical angle of attack and the tip velocity during the downstroke,
which alters the force magnitude without large changes in force inclination.
Conversely, the concerted changes associated with dorsal amplitude involve a
negative correlation between the angle of attack and the tip velocity during
the upstroke, and this allows for changes in force inclination without changes
in force magnitude.
Rather than the independent control of each kinematic mechanism of force
generation (i.e. angle of attack, tip velocity and rotation), it is the
ability to control specific complexes of these parameters during downstrokes
and upstrokes independently that allows Calliphora to control
specific forces and moments. It has long been suggested that flies tend to
generate the majority of lift during the downstroke and thrust during the
upstroke (Buckholz, 1981;
Nachtigall, 1966
). However,
kinematic mechanisms by which lift and thrust are de-coupled through
differences in downstroke and upstroke kinematics have only been hypothesized
(Kammer, 1985
;
Nachtigall and Roth, 1983
). We
have found that Calliphora can indeed control the angle of attack and
tip velocity of downstrokes and upstrokes independently. In addition,
modulation of ventral rotation complements the modulation of the downstroke
translatory force, and modulation of dorsal rotation complements modulation of
the upstroke translatory force.
The manipulation of wing kinematics over each stroke involves a system of
mechanical linkages that converts the configuration changes imposed by the
steering muscles at the wing hinge to the concerted changes in stroke
kinematics. The coupling among kinematic parameters may result from these
mechanical linkages or from stereotyped patterns of motor neuron activation.
Within the group of steering muscles recorded in this study, we noticed a
strong tendency for low-frequency basalare muscle activity to be paired with
elevated I1 activity and for high-frequency basalare muscle activity to be
paired with elevated activity in the muscles of pterale III during steering
reactions. The aerodynamic analysis indicates that this gross coupling results
in a tendency to actively pair the smallest roll with the smallest yaw torques
and the largest roll with the largest yaw torques. This is consistent with the
strongest turns measured in free flight being a characteristic banked turn
(Schilstra and van Hateren,
1999; Wagner,
1986
). Similarly, some of the correlations between kinematic
parameters such as angle of attack and tip velocity quantified in this study
may be due to consistent patterns of activity in the other unrecorded muscles,
rather than to a coupling within the mechanical linkage. Conversely, variation
in the temporal firing patterns of the recorded muscles III1 (within Mode 2)
and I1 (within Mode 1) may have contributed to some of the unexplained
variation within our dataset.
Tests of the context dependence of the concerted changes in wing motion
observed in this study will depend on improvements in several inter-related
areas of analysis. Additional neuromuscular and kinematic mechanisms of
control will most likely be identified through an increase in the number of
muscles recorded and a larger range of quantified kinematic variation. We have
probably not captured the full range of wing motion within
Calliphora's repertoire. For example, although our results suggest
that changes in wingbeat frequency are independent of other changes in wing
kinematics, we cannot discount the possibility that frequency may be
controlled together with other aspects of the wingbeat trajectory but simply
were not observed in our study. In addition, another level of complexity
involves variations in wing deformation, which we were unable to measure.
Although there is no evidence that flies can actively alter wing deformation,
especially during rotation when wing torsion is most pronounced
(Ennos, 1988), we observed
considerable wing deformation through the duration of the upstroke. Such
effects might be due to either wing inertia or aeroelastic effects
(Combes and Daniel, 2003
).
Whether or not they are controlled by the fly, unsteady mechanisms such as
Nachtigall's swing mechanism (Nachtigall,
1979
,
1981
) or Ellington's flex
mechanism (Ellington, 1984b
)
may cause additional modulations of force unaccounted for in this study. A
better understanding of the control of forces will also require empirical
tests of the factors affecting the wake capture force. Although we found no
evidence for controlled variation in the wake capture force, it may vary with
respect to some as yet unknown kinematic variable or with varying free-flight
conditions.
Most importantly, although we have identified examples of kinematic
mechanisms by which different aspects of the aerodynamic force vector can be
controlled independently in one wing, the total aerodynamic output will depend
on the coordinated control of both wings. The contribution of the total
aerodynamic forces and moments to motion of the body will depend on a number
of other factors. These include the effects of gravity, inertia, body drag,
advance ratio, the effective angle of attack during body translation, and
changes in the center of mass due to motion of the legs and abdomen. A more
complete description of the influence of wing kinematics on flight behavior
will require analysis of the aerodynamic forces within the context of the
combined effect of all these factors on the resultant body orientation, flight
direction and flight speed. For instance, forward velocity can also contribute
to differences in the relative contribution of downstrokes and upstrokes to
the flight path (Dudley and Ellington,
1990b; Ellington,
1995
; Willmott and Ellington,
1997b
). Future experiments incorporating the effect of body
translation on the aerodynamics of force generation will be an important step
toward a better understanding of flight control.
As our understanding of the variability of wing kinematics and its contributions to insect flight behavior improves, it will be interesting to begin comparisons across species. Although Diptera share a similar flight-related musculo-skeletal architecture, species differ in their ability to hover, fly backwards or sideways. Continued studies of the independent kinematic control parameters of flies may lead to a better understanding of the diversity of flight behaviors and pursuit strategies among flying insects.
List of symbols
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Acknowledgments |
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References |
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