The aerodynamics of hovering flight in Drosophila
1 Institute of Neuroinformatics, University/ETH Zürich,
Switzerland
2 California Institute of Technology, Mail Code 138-78, Pasadena, CA 91125,
USA
* Author for correspondence (e-mail: steven{at}ini.phys.ethz.ch)
Accepted 21 March 2005
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Summary |
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Key words: fruit fly, Drosophila melanogaster, flight, aerodynamics, power, biomechanics, behavior
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Introduction |
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Although many prior studies have succeeded in capturing wing kinematics in
free flight using high speed cine or video
(Wakeling and Ellington, 1997;
Willmott and Ellington, 1997
),
such methods do not provide a measure of instantaneous aerodynamic forces. One
means of circumventing this limitation is to employ computational fluid
dynamic (CFD) simulations to estimate forces from known kinematics (e.g.
Sun and Wu, 2003
;
Wu and Sun, 2004
;
Sun and Lan, 2004
). These
methods show great promise as computation power increases and algorithms
improve, although concerns still remain regarding the accuracy of simulating
unsteady 3D flows. Another, complimentary, approach is to measure the
instantaneous forces by `replaying' free-flight kinematics on a dynamically
scaled physical model equipped with suitable force sensors
(Fry et al., 2003
and
supporting on-line material at
http://www.sciencemag.org/cgi/data/300/5618/495/DC1/1).
Although there are limitations to the accuracy with which the morphology and
motion of an insect wing can be replicated by the robot, this method does
permit a time-resolved estimate of aerodynamics, and therefore an analysis of
energetics and control. Furthermore, experiments with physical models are
essential for testing the accuracy of CFD simulations.
Another common approach in the study of insect flight is the use of
tethered preparations for the measurement of wing kinematics, flight forces
(Cloupeau et al., 1979;
Buckholz, 1981
; Zanker,
1990a
,b
;
Dickinson and Götz, 1996
)
and energetic costs (Lehmann and
Dickinson, 1997
). The downside of this approach is that tethering
may alter the mechanical properties of the thorax or influence an insect's
behavior due to unnatural sensory stimulation. Although the recent generation
of tethered flight simulators can provide detailed visual stimuli, most cannot
provide mechanosensory feedback (but see
Sherman and Dickinson, 2003
).
Realistic sensory conditions are especially important for studies of flight
control, because sensory feedback from eyes, ocelli and (in flies) halteres
provides essential reafferent feedback from body motion
(Nalbach, 1993
;
Nalbach and Hengstenberg,
1994
; Dickinson,
1999
; Sherman and Dickinson,
2003
).
In this study we present detailed measurements of the continuous time course of 3D wing motion and flight forces measured in free flying and tethered fruit flies Drosophila melanogaster. We examine the aerodynamic mechanisms underlying hovering flight by comparing the measured time course of aerodynamic forces with those predicted from a quasi steady-state model. Next, we examine flight control by analyzing the time course of aerodynamic forces and torques in all six degrees of freedom (d.f.). We also assess the effects of tethering on flight performance by comparing wing kinematics, forces and moments generated under free and tethered conditions. Finally, we compare power requirements derived from measured instantaneous kinematics and forces with estimates based on time-averaged models. In sum, the results provide a detailed view of the aerodynamics, control and energetics of hovering flight in Drosophila.
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Materials and methods |
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We also measured the wing kinematics in tethered flying flies, in
combination with an optoelectronic wing beat analyzer
(Götz, 1987;
Lehmann and Dickinson, 1997
)
that measured the stroke amplitude and frequency of both wings in real time.
Sequences with roughly similar stroke amplitudes of both wings were used for
the analysis to ensure that the fly was not attempting to turn. The methods
applied to extract the kinematics were the same as those for free flying
flies, whereas the body position remained fixed. For tethered flight, we
analyzed a total of 59 stroke cycles (1550 frames) from five different
flies.
Measurement of aerodynamic forces
To measure the aerodynamic forces produced by a single wing, its motion in
body centered coordinates (Fig.
1C) was played through a newly developed dynamically scaled
flapping robot (Fig. 1D,E). The
device was similar to one described previously
(Dickinson et al., 1999),
except that wing motion was controlled by feedback-driven servo-motors and not
stepper motors, and the gearing mechanism was improved to lower the effects of
backlash (a more detailed description is given in
Dickson and Dickinson, 2004
;
note that our experiments were performed without translation). Inertial forces
due to the wing mass of the robot were below the noise limit of our sensor and
thus did not contaminate the measurements of aerodynamic force. It seemed
justified to perform the measurements in absence of a second wing, because in
none of the sequences we analyzed did the animals exhibit a clap-and-fling
behavior (Weis-Fogh, 1973
). A
control measurement using the robot in a two-wing configuration confirmed the
absence of detectable wingwing interactions. Another potential source
of error is that the forces produced by a slowly translating fly were measured
on a stationary robot. Although adequate for hovering, the motion of the fly
through the air must influence the flow and forces on the flapping wings to
some degree.
The stroke frequency of the robotic fly was precisely adjusted to match the
calculated Reynolds number of the flapping fly wings
(Dickinson et al., 1999). The
magnitude of aerodynamic forces acting on an actual fly,
FFly, is related to those measured in the robotic model,
FRobot, according to the relationship:
![]() | (1) |
Calculation of torques
The aerodynamic forces measured on the robot were in wing-centered
coordinates and required transformation into a fixed reference frame
(Haslwanter, 1995). The torque
acting on the body based on the aerodynamic forces could be calculated after a
number of assumptions were made. The fly's body was assumed rigid and the
center of mass was assumed to lie in the sagittal plane, half way along the
measured long axis of the fly. To calculate the magnitude of moment about the
center of mass of the body, we assumed that the center of pressure of the wing
was located 70% along the line connecting the base and tip. A similar value
has been confirmed by computational fluid dynamics
(Ramamurti and Sandberg,
2002
), and experimental measures of the distribution of chord-wise
circulation along the wing (Birch and
Dickinson, 2001
; Birch et al.,
2004
). The following calculations (Eq.
2,
3,
4,
5) apply to a single wing. The
effect of both wings was obtained by subsequently adding the contributions of
the right and left wing. The aerodynamic torque produced by a wing,
TAero, is a 3D vector originating at the center of mass of
the fly, and can be calculated from the vector product of the moment arm
r and the aerodynamic force FAero as:
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|
![]() | (3) |
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According to Newton's second law, FAcc can be calculated
from the product of wing mass, MWing, and wing
acceleration:
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Consistent with much of the literature, we present the specific power
P*, i.e. power normalized to muscle mass:
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Estimation of instantaneous specific muscle power
The total instantaneous power required to move the wing,
P*Mech, is the sum of instantaneous aerodynamic
and inertial power:
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![]() | (8) |
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Time-averaged models
One goal of the paper is to compare power estimates derived from
instantaneous forces and kinematics with those derived from prior
time-averaged models. For this comparison, we use Ellington's influential
models of induced, profile and inertial power
(Ellington, 1984e;
Lehmann and Dickinson, 1997
),
based on the time-averaged forces and kinematics. In this case, however, we
can use kinematic parameters derived directly from our 3D kinematic analysis
and compare the results with our more direct estimates of flight power based
on instantaneous free-flight kinematics and forces (Eq.
3,
4,
5,
6,
7).
Induced power
Specific induced power is estimated according to RankineFroude
theory (also see Ellington,
1984d,e
)
from:
![]() | (10) |
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Profile power
Specific profile power was evaluated according to Ellington's model:
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|
Inertial power
Inertial power can be estimated
(Ellington, 1984e) from:
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Results |
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A functional interpretation of the complex wing motion is possible by
associating the measured wing trajectory with the resulting aerodynamic forces
(Fig. 2B). The general pattern
of the wing stroke is a `U-shaped' wing trajectory. During the downstroke, the
aerodynamic force increases to a maximum around mid-stroke and then decays
toward the end of the downstroke (also see
Fig. 2C). Wing reversal and
supination are nearly synchronous, a phase relationship previously predicted
to generate near optimal lift for a reciprocating pattern of wing motion
(`symmetric case' in Sane and Dickinson,
2001). The wing then begins to move backward and downward at an
increasingly high velocity, causing a large force peak during the early phase
of the upstroke. The average force generated by each half stroke (see inset of
Fig. 2B) has a significant
horizontal component, due to the high angle of attack. The horizontal force
components cancel over the course of a stroke cycle, as expected for a
hovering force balance, leaving only a vertical component of the total force
vector.
Aerodynamic mechanisms
The simplest quasi-steady model, based on a single translation term,
roughly follows the time course of the measured forces
(Fig. 2C). Ignoring differences
in the time course, the mean of the translational term is 89% of the mean of
the measured force. However, this calculation is based on force coefficients
measured in the absence of a periodic wake. Experiments indicate that
translational forces drop roughly 10% due to induced flow within the wake
(Birch and Dickinson, 2003),
which would reduce the estimated contribution of translational forces to about
80%. Nevertheless, translational forces provide the main source of lift in
hovering flight. The estimates of the translational component here are higher
than prior estimates of 65%, based on phase-reconstructed tethered flight
kinematics (Dickinson et al.,
1999
). This difference is most likely due to the increased
aerodynamic angle of attack during the early part of each half stroke, a
consequence of the `U-shaped' trajectory of the wing, which is more prominent
in the kinematics of free flying animals. This motion, though less extreme, is
reminiscent of the upward force created by the vertical plunge during the
downstroke in dragonflies, which as recently discussed by Wang et al.
(2003
), is due primarily to
wing drag. Analogously, the forces diminish toward zero as the stroke
continues, because the aerodynamic angle of attack decays as the wing deviates
upward.
However, when time-dependent effects may contribute to force production,
comparisons of measured forces and theoretical models based on mean values may
be misleading. For example, the time course of the measured forces exhibits a
shoulder at the start of each half stroke that is not captured by the
translational component. Addition of quasi-steady terms resulting from
rotational forces (Sane and Dickinson,
2002; Walker,
2002
) yields a prediction for the mean total force of 103%, but
the measured forces still show a consistent advance relative to the
multi-component quasi-steady model throughout the entire stroke
(Fig. 2C). There are several
possible explanations for this temporal shift. First, the quasi-steady model
does not take into account any influence of the spatial and temporal dynamics
of the wake, such as wake capture, the elevation in force immediately
following stroke reversal (Dickinson et
al., 1999
), or the decrement in forces observed in long strokes
due to the periodicity of downwash (Birch and Dickinson,
2001
,
2003
). Such effects would be
consistent with the observed phase advance of measured forces relative to the
quasi-steady model. Second, the quasi-steady model does not take into account
the added mass force, which would be expected to increase total force at the
start of the stroke, and decrease it at the end of the stroke
(Birch and Dickinson, 2003
).
Although it is possible to derive a quasi-steady term for added mass force
(Sane and Dickinson, 2002
),
these models do not accurately predict its time course when accelerations are
large. This failure is due to a well-characterized hysteresis in added mass
force, in which the force lags behind wing acceleration
(Sarpkaya and Isaacson, 1981
).
Without more detailed information on the flow structure it is difficult to
exclude the importance of wake effects or added mass in contributing to the
unsteady component of the forces generated by the flapping wing.
Time history forces and moments
The most obvious requirement of hovering flight is the generation of a mean
vertical force that precisely offsets the fly's own body weight. As shown in
Fig. 3A, each wing provides a
vertical force peak during the middle of each half stroke, with a considerably
higher contribution from the upstroke. The relatively small variance among all
six flight sequences is confirmation that the differences in wing motion for
hovering and low advance ratio flight are quite small (note S.D.
envelopes in Fig. 3). Because
lift and drag are closely coupled, the wing creates large thrusts during each
half stroke (Fig. 3B), but they
sum nearly to zero over the course of a complete stroke cycle. Unlike forward
thrust and lift, sideways thrust (sideslip) cancels instantly due to the
bilateral symmetry of the wing motion (Fig.
3C). Similarly, because each wing contributes to yaw and roll
torque with an opposite sense of direction, these moments sum to zero at each
point in the stroke cycle, provided the motion of the two wings is bilaterally
symmetric (Fig. 3D,E).
Comparing the average peak magnitude of yaw torque generated by one wing
(48±14 nN m) indicates the precision with which flies must maintain
symmetric wing motion. A net average yaw torque of less than 2 nN m is
sufficient to initiate a body saccade, during which the angular velocity of
the body can exceed 2000° s1
(Fry et al., 2003). Thus, even
a small bilateral change in wing motion would cause the animal to rotate.
The sign and magnitude of pitch torque is the same for both wings, resulting in a strongly fluctuating time course (Fig. 3F). As with forward thrust, pitch torque averages to zero over a complete cycle. These results also confirm that our methods provide reliable measurements of forces and torques.
Effects of tethering
Although tethered flight has previously been used as a proxy for free
hovering flight, the effects of tethering have not been examined in careful
detail. Our data reveal four consistent differences between the free and
tethered conditions. (1) The stroke duty cycle is shifted such that downstroke
takes up 60.4% of the cycle in tethered flight, as opposed to 53.8% in free
flight (Fig. 4A). (2) The time
course of stroke deviation is distorted in tethered flight. The wing continues
to deviate downward to a minimum just prior to the ventral reversal, rather
than moving upwards to create the `U-shaped' trajectory characteristic of the
free-flight pattern. (3) Compared to free flight
(Fig. 4B), the total stroke
amplitude is lower in tethered flight due to a reduction in the ventral
(forward) extent of the stroke (Fig.
4C). (4) The stroke plane is tilted forward by about 12° with
respect to the body axis under tethered conditions
(Fig. 4C). These differences,
though subtle, are large compared to the changes in kinematics responsible for
free-flight maneuvers. For example, during body saccades the maximum
difference in stroke angle between the wings is only 10° (fig. 3F in
Fry et al., 2003). Although
the time course of the morphological angle of attack is quite similar under
free and tethered conditions, the difference in stroke deviation means that
that the aerodynamic angles of attack will vary throughout the stroke. The
consistency of the tethered flight kinematics is indicated by the remarkably
close match of our kinematics with those reported by Zanker
(1990a
, replotted as dotted
lines in Fig. 4A), who
reconstructed the wing motion by varying the phase of wingbeat-triggered
stroboscopic images. The clap-and-fling is absent in our tethered kinematics,
although it has been observed previous studies
(Götz, 1987
), an
observation for which we have no immediate explanation.
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Specific inertial power, P*Acc, represents the power expended to accelerate the mass of the wing (Eq. 4 and 6). It is slightly lower in magnitude compared to aerodynamic power and follows a very different time course. Within each half stroke, P*Aero first increases as the wing accelerates, and then reverses sign as the wing decelerates. The average inertial power required to accelerate the wing (assuming that wing deceleration accrues no cost and there is not elastic storage) amounts to 71 W kg1.
The total mechanical power, P*Mech, is the instantaneous mechanical power required to move the wings, calculated from the sum of P*Aero and P*Acc. Around the middle of the downstroke, P*Acc becomes negative, due to the sign reversal of the inertial forces acting on the wing. P*Mech consequently decreases toward the end of the downstroke and becomes negative for a short period as the decelerating wings yield more power than is required to overcome aerodynamic forces. A similar pattern repeats during the upstroke.
The actual power that the flight muscles must generate,
P*Musc, is not necessarily equivalent to
P*Mech, because negative mechanical power could
be stored in elastic elements within the flight motor and recovered at the
start of the next stroke (Dickinson and
Lighton, 1995). The possible range of muscle power may be
estimated by considering the consequences of 0% and 100% storage, with the
added assumption that dissipated excess negative work does not accrue a net
cost (also see Materials and methods). With no elastic storage,
P*Musc=P*Mech,high=115
W kg1; with 100% elastic storage,
P*Musc=P*Mech,low=97
W kg1 (Table
1). In the case of perfect elastic storage there is no cost to
wing acceleration and P*Musc consequently is
equal to
(differences in the values for P*Mech,low and
presented in
Table 1 are due to rounding
errors). Thus, the largest possible effect of elastic storage would amount to
a reduction of only 19%, slightly higher than prior calculations for D.
hydei (Dickinson and Lighton,
1995
). In summary, provided there is no cost to negative work, the
contribution of inertial cost is relatively small and may be further
attenuated due to elastic storage. This suggests that for
Drosophila at least estimates of flight costs based solely on
aerodynamic power provide a reasonable estimate (within about 20%) of true
energetic requirements.
Comparison with model-based estimates of muscle power
We calculated flight costs (Eq.
3,
4,
5) based on Ellington's
time-averaged models (Ellington,
1984e), for comparison with our new power estimates based directly
on instantaneous kinematics and forces. Ellington's equations use a series on
non-dimensional kinematic parameters that we were able to accurately calculate
based on our instantaneous measurements. New and prior values for these input
parameters are given in Table
1. The models predict two components of aerodynamic power: induced
power (P*ind), the cost associated with
accelerating a stream of air downwards to generate lift
(Eq. 10), and profile power
(P*pro), the cost associated with the drag
acting on the wings (Eq. 12).
The time-averaged estimate for inertial power
(P*acc) combines the cost of accelerating both
the wing mass and added mass.
Using our new values for mean stroke amplitude, , mean lift,
, and k, a wake periodicity
correction factor, we calculated a P*ind value
of 17.2 W kg1 for our free-flight experiments and 18.0 W
kg1 for our tethered flight experiments. Both these values
are just slightly lower than prior estimates based on tethered flight under
conditions (21.4 W kg1), in which moving visual patterns
were used to make the flies generate a lift force equal to body weight
(Lehmann and Dickinson,
1997
).
Profile power, P*pro, is linearly dependent
on the mean profile drag coefficient
, which is difficult to
predict without precise knowledge of wing kinematics and forces. For purposes
of comparison, we estimated
in two ways. First, a
simple estimate based on the Reynolds number (Re) was obtained by
applying Ellington's 7/sqrt(Re) approximation, which yields a value
of 0.57 for our free-flight data (tethered: 0.61), similar to values used in
previous studies (0.50.7, Dickinson
and Lighton, 1995
; Lehmann and
Dickinson, 1997
). This simple approximation for
yields a
P*pro,Re value of 36 W kg1
for our free-flight experiments (tethered: 25 W kg1),
similar to the previous estimate from tethered flies (38.4 W
kg1; Lehmann and
Dickinson, 1997
; Table
1). The second, more accurate, value for
was obtained from
measured profile drag and wing kinematics
(Eq. 14), and yields a much
higher value of 1.46 (tethered: 1.36), consistent with recent measurements of
wing drag coefficients at the angles of attack used throughout the stroke
(Dickinson et al., 1999
;
Sun and Tang, 2002
). Using
these higher, more accurate numbers for the mean drag coefficient results in a
substantially higher P*pro value of 94 W
kg1 (tethered: 56 W kg1).
A comparison of the muscle power predicted by the time-averaged models and
those based on instantaneous power must be made with close scrutiny of the
underlying assumptions. Muscle power must be at least as high as the predicted
aerodynamic power, if inertial components are assumed negligible. The total
aerodynamic cost predicted by Ellington's model using the improved estimates
for P*pro, is 111 W kg1 for
free flight and 74 W kg1 for tethered flight (calculated
from the sum of P*ind and
P*pro). This value is slightly higher than our
estimates of P*Aero based on instantaneous
forces and kinematics, although the power associated with the acceleration of
added mass, P*acc,virt
(Eq. 15), is not included, which
would tend to increase the magnitude of the discrepancy. As a component of
inertial power in Ellington's models, added mass has been previously assumed
to function similarly to the wing mass, as such accruing no extra cost given
even moderate elastic storage. Assuming that added mass does contribute to
flight cost, even when the cost of wing mass inertia is nullified by elastic
storage, we calculate flight power from the sum of
P*ind, P*pro and
,
yielding 134 W kg1. The term of
in this rough calculation
follows from Ellington's original definitions and assumes that negative work
accrues zero cost. This value is substantially higher than our estimate based
on instantaneous forces, in which added mass contributions are also included
in aerodynamic power (97 W kg1). In summary, the results
suggest that time-averaged models are likely to overestimate the power
requirements of hovering flight.
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Discussion |
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Because we independently measured the kinematics of both wings, we were able to determine the time history of forces and moments controlling all six degrees of freedom (Fig. 3). Our results indicate that the need to balance thrust, lift and pitch imposes severe constraints on the spatio-temporal pattern of wing motion. In particular, the speed and precision with which flies must control torque suggests that rapid sensory-motor feedback circuits are required to regulate wing motion. This control takes place within a complex set of constraints imposed by the requirements of balancing six degrees of motion within each stroke, the physiology of the flight apparatus, and the dynamics of the oscillatory wing motion. A detailed comparison with data from freely flying flies shows that the wing motion of tethered flies is distorted in a stereotyped manner (Fig. 4), resulting in a large tonic pitch-down moment (Fig. 5). Finally, we provide a direct estimation of muscle power based on instantaneous mechanical power, which yields a value almost twice that of prior estimates based on time-averaged models. Much of this discrepancy can be explained by a 2.5-fold underestimate of the actual mean drag coefficient in prior models. However, if implemented with more accurate input parameters, including more realistic values for the mean drag coefficient, the time-averaged models tend to overestimate flight cost, which brings into question their reliability.
Requirements for flight control
To hover stably and efficiently, a fly must generate sufficiently large and
precisely balanced flight forces, while minimizing energetic costs a
substantial challenge given that all forces and moments in hovering flight are
produced by the two rapidly reciprocating wings. Although our analysis does
not address flight control experimentally, it does reveal the requirement for
highly precise and fast control of wing kinematics on a stroke-by-stroke
basis. Because a flapping wing creates high force transients during the stroke
cycle, even the slightest variation in wing motion can rapidly alter the fly's
orientation, implying that the same rapid sensory-motor control systems are
being used during hovering flight as for saccadic turning maneuvers
(Fry et al., 2003). For
example, a net average torque of less than 2 nN m, or 4% of the peak yaw
torque produced by a single wing during hovering flight, is sufficient to
initiate a saccade, during which the angular velocity of the body can reach
2000 deg. s1 within a few wing beats
(Fry et al., 2003
). The
control of roll is even more sensitive because the moment of inertia around
the roll axis is considerably lower compared to that of pitch and yaw, due to
the elongated shape of the body.
In contrast to body motion outside the mid-sagittal plane (yaw, roll and sideways thrust), body motion within the mid-sagittal plane (pitch, upward and forward thrust) requires symmetric changes in stroke pattern that act over the course of a stroke cycle, imposing severe constraints on the spatio-temporal pattern of force production. The high sensitivity of net pitch torque to changes in the stroke pattern is illustrated in tethered flies, where a moderate distortion of the spatio-temporal pattern of wing motion leads to a strong nose-down pitch torque (Fig. 5F).
The neural and morphological mechanisms that allow flies to exert the
observed precise kinematic control remain poorly understood. The complex
morphology of the wing hinge, together with the resonant dynamics of the
oscillating wing motion, may provide mechanical constraints that permit a
limited number of steering muscles to generate gradual and precise
modifications of stroke patterns for flight control and maneuvering
(Tu and Dickinson, 1996;
Balint and Dickinson, 2001
).
There are several lines of indirect evidence for the importance of such
mechanical constraints in flies. As shown in this study, the pattern of wing
motion is highly stereotyped and frequency-invariant
(Fig. 4), at least during slow
flight. The comparison between free and tethered hovering data (Figs
3,
5) demonstrates how various
kinematic parameters change in concert, including stroke amplitude, mean
stroke position, stroke phase, stroke plane angle and, to a lesser degree,
angle of attack. Further, the wing kinematics observed in free and tethered
flight during turning maneuvers indicate that flight control is mediated by
highly correlated changes in stroke amplitude, deviation, angle of attack and
frequency (tethered flying Calliphora:
Balint and Dickinson, 2004
;
free flying Drosophila: Fry et
al., 2003
). From the perspective of flight control, the intrinsic
coupling of flight motor dynamics and mechanics represents the physical
backdrop upon which the animal's flight control system evolved the appropriate
control inputs. Determining how the nervous system has coevolved with the
musculo-skeletal system to produce such impressive performance despite these
constraints represents a formidable challenge for future research.
Effects of tethering
Tethered flight preparations have been used extensively to study the
neurobiology, physiology and behavior of flight in Drosophila and
many other insects and is likely to remain an important experimental paradigm.
To more accurately assess the potential behavioral artifacts introduced by
tethering, we performed experiments in free flying flies and compared the
results with those obtained from tethered flies under otherwise similar
experimental conditions. Tethered flies generate a pattern of wing motion that
is clearly different from those obtained during free flight
(Fig. 4). The fact that our
measurements of tethered flies are consistent both between flies and with
Zanker's earlier measurements (Zanker,
1990a) suggests that the distortions introduced by tethering are
stereotyped and result from a consistent mechanical or sensory effect. Such
distortions might result from a tonically activated equilibrium reflex
response, elicited by unnatural reafferent sensory feedback under tethered
conditions. The work of David
(1978
) indicates that fruit
flies regulate flight speed in part through changes in body pitch, such that
the animal pitches nose-down to increase flight speed, much like a helicopter.
Further, in free flight Drosophila exhibit a preferred ground speed
of about 10 cm s1
(David, 1982
). Thus, one
possibility is that stationary tethering might alter the animal's flight
velocity control system, resulting in a continuous attempt to accelerate
forward. An alternative, but not mutually exclusive, line of reasoning is that
tethering introduces a direct mechanical artifact that distorts the action of
the indirect flight muscles and the wing hinge. At this point we cannot
distinguish between these two hypotheses. However, recent measurements of wing
kinematics in free flying individuals at higher advance ratios are similar to
those of tethered flies (R. Sayaman, unpublished observation). This supports
the view that the artificial sensory conditions during tethered flight elicit
an inappropriate, but not necessarily unnatural, stroke pattern. In spite of
the significant changes in wing kinematics, tethered flies nevertheless show a
similar pattern of translational forces and a meaningful optomotor response
(e.g. Götz, 1968
;
Heisenberg and Wolf, 1993
).
The recent insights into free-flight dynamics
(Fry et al., 2003
) and those
from the present report may serve a better interpretation of tethered flight
responses with respect to their function in free flight under real-world
conditions.
Power
The utility of capturing the instantaneous kinematics and forces of freely
hovering animals is particularly evident in the estimation of mean muscle
power. Only under free-flight conditions is it guaranteed that a fly generates
the power required to hover and that its aerodynamic force output truly
reflects hovering conditions. Unlike time-averaged estimates of flight power,
the measured instantaneous mechanical power allows a more direct calculation
of mean muscle power and thus an improved estimate of muscle efficiency and
the effects of elastic storage. Our value for Drosophila melanogaster
(97 W kg1) is almost exactly the same as a recent estimate
(95.7 W kg1) based on a computational fluid dynamics (CFD)
model of flight forces and power costs in a closely related species, D.
virilis (Sun and Tang,
2002). These experiments confirm previous studies
(Dickinson and Lighton, 1995
)
that even a moderate amount of elastic storage would be sufficient to
eliminate the cost of inertial power, provided that negative work accrues no
energetic cost.
These direct power estimates also allow us to test the accuracy of
time-averaged models used to predict flight power. The results of this
comparison indicate that prior estimates are likely to have underestimated the
cost of hovering flight by a factor of nearly two
(Lehmann and Dickinson, 1997).
The primary reason for this underestimate is that prior work employed
Ellington's 7/sqrt(Re) estimate for the mean drag coefficient, which
our direct force measurements indicate is approximately 2.5 times too low,
confirming recent findings (Dickinson and
Götz, 1996
; Dickinson et
al., 1999
; Ellington,
1999
; Usherwood and Ellington,
2002
). Given that Drosophila fly at a relatively low
Reynolds number, the error for larger hovering insects is likely to be even
greater. However, in the case of Drosophila at least, the assessment
of time-averaged estimates is more complicated. In addition to providing
better estimates of the mean drag coefficient, our 3D free-flight data provide
more accurate values for the kinematic input parameters to the time-averaged
models. When these values are used, the models slightly overestimate
aerodynamic power compared to the values based on instantaneous forces (111 W
kg1 vs 97 W kg1;
Table 1). This discrepancy is
greater, however, if the added mass term is combined with the time-averaged
estimate, which results in a value of 134 W kg1. The
comparison also reveals the difficulty related to the reliable measurement or
estimation of the input parameters used in Ellington's models. Estimating
muscle power based on high-speed videography is likewise based on a number of
basic assumptions including, for example, the positions of the center of
pressure and mass on the wing. Furthermore, our calculation of specific muscle
power critically depends on knowledge of the body mass, which we inferred
indirectly from a regression on wing length. In spite of these limitations,
the parameters used in our calculations are well founded by independent
measurements and subject to a limited margin of error. In particular, the
calculations of instantaneous power do not require higher order terms that
tend to amplify errors. Although the time-averaged models have provided a
remarkably useful tool for insect flight research prior to recent advances in
high speed imaging and data processing, we suggest that the more direct
measurement of flight power or numerical simulation may present a more
practical solution for future research.
List of symbols
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