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First published online October 17, 2008
Journal of Experimental Biology 211, 3478-3489 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.017590
Kinematics of slow turn maneuvering in the fruit bat Cynopterus brachyotis
1 Department of Ecology and Evolutionary Biology, Brown University, Providence,
RI 02912, USA
2 Division of Engineering, Brown University, Providence, RI 02912, USA
* Author for correspondence at current address: Department of Organismal Biology and Anatomy, University of Chicago, Chicago, IL 60637, USA (e-mail: jiriarte{at}uchicago.edu)
Accepted 6 September 2008
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Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Key words: aerodynamics, bat, flight, kinematics, turning
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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).
However, most studies of the morphological basis of bird and bat maneuvering
have been restricted to analyses based on fixed-wing aerodynamics
(Rayner and Aldridge, 1985;
Aldridge, 1987;
Warrick, 1998), even though
flying animals turn using unsteady dynamics, violating the assumptions of
steady-state aerodynamic theory. Although the degree to which assuming
fixed-wing models introduces error in analysis is unknown, predictions derived
from steady-state models have been applied extensively in the bird and bat
research communities, particularly when looking for morphological correlates
of flight performance and its ecological implications (e.g.
Aldridge, 1986a;
Norberg and Rayner, 1987;
Kalcounis and Brigham, 1995).
Recent information on maneuvering flight of birds
(Warrick and Dial, 1998;
Warrick et al., 1998;
Hedrick and Biewener, 2007;
Hedrick et al., 2007) and
insects (Fry et al., 2003;
Card and Dickinson, 2008) has
expanded the discussion beyond the assumption of fixed wings. These studies
emphasize the importance of temporal sequences of wing movements to understand
the mechanical basis of turning behavior. Although bats are believed by some
to be the most maneuverable flying animals for their size, no analogous
studies have been performed for bats. In the present study, we evaluate the
morphological and aerodynamic mechanisms used by bats to carry out 90 deg.
turns at slow speed by analyzing wing and body kinematics in detail.
To successfully complete a turn, an animal must translate its center of
mass (CoM) along the flight path (i.e. change its flight direction) and rotate
its body around its CoM to align its body orientation with the new direction.
The magnitude of change in direction of flight is a function of the impulse
(force · time) perpendicular to the original direction of
movement. Impulse is the result of the centripetal force produced by the
change of the orientation of the net aerodynamic force generated by the body
and wings. Two basic strategies to produce a turning force include banked and
crabbed turns (Fig. 1). In a
banked turn, the animal rolls around its cranio-caudal axis, tilting the
vector of the vertical component of the net aerodynamic force (i.e. lift in
level flight) laterally and towards the center of the turn
(Fig. 1). These turns are used
by most fixed-wing aircrafts (Filippone,
2006). By contrast, in a crabbed turn, the animal yaws into the
turn, orienting the forward component of the net aerodynamic force (i.e.
thrust in level flight) towards the center of the turn, without the need of
adjusting the vertical component vector
(Fig. 1). In both cases, the
reorientation of aerodynamic forces produces a laterally oriented force that
drives the organism into a turn. Both banked and crabbed turning mechanisms
require the rotation of the body about its CoM.
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), locusts
(Berger and Kutsch, 2003),
dragonflies (Alexander, 1986),
gliding frogs (McCay, 2001)
gliding mammals (Bishop and Brim-DeForest,
2008) and birds (Warrick and
Dial, 1998; Hedrick and
Biewener, 2007). However, crabbed turns are also phylogenetically
widespread having been described in some dipterans (see
Dudley, 2002), dragonflies
(Alexander, 1986), gliding
frogs (McCay, 2001) and
gliding mammals (Bishop and Brim-DeForest,
2008).
For both banked or crabbed turns, body rotation results from an asymmetry
in aerodynamic forces between left and right wings, an asymmetry in the
inertial forces produced by the two wings or a combination of both.
Aerodynamically generated force asymmetries can be expected as the result of
differential changes in wing shape, such as changes in wing surface area,
angle of attack, or camber or maybe due to differences between left and right
wings in kinematic parameters, such as relative velocity (see
Dudley, 2002). By contrast,
inertially generated force asymmetries can be produced by differences in
motion between left and right wings. Inertial forces can produce net changes
in body orientation over a wingbeat cycle even when no external torques are
applied due to conservation of angular momentum
(Hedrick et al., 2007).
Moreover, little is known about the kinematics and aerodynamics of turning
in bats, including whether they use primarily one, the other or both turning
mechanisms. Whereas aerial maneuvers have been qualitatively described for two
bat species (Norberg, 1976),
and kinematics of the CoM have been analyzed for six other species performing
180 deg. turns (Rayner and Aldridge,
1985; Aldridge,
1987), no detailed analysis of body orientation and/or wing
kinematics has yet been carried out for bats.
Photographs of bats performing flying maneuvers sometimes show the body
rolled toward the direction of turning
(Norberg, 1976). Based on this
evidence and the widespread use of banked turns in organisms as
morphologically and phylogenetically diverse as insects, amphibians, birds and
mammals, we predicted that bats would also use banked turning and, therefore,
maneuver by rolling their body to reorient the lift force vector.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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).
Lightweight spherical beads covered with reflective tape were attached to the pelvis on the skin overlying the pubic symphysis (pvs marker) and just lateral to the sternum (Rch and Lch, right and left chest markers, respectively). Chest markers were placed medial to the glenohumeral joint to ensure they remained in the field of view of the cameras as much as possible throughout the wingbeat cycle. Three additional anatomical landmarks on each wing: the wrist and the distal part of the distal phalanges of the 3rd and 5th digit (wst, d3 and d5, respectively) (Fig. 3) were marked with small circular pieces of reflective tape.
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). The
three-dimensional position of each marker was reconstructed from the
two-dimensional video files using the DLT coefficients derived from the
calibration frame. Because of the great range of motion of the wing during
flight, in some cases, markers were not visible in at least two cameras and
the spatial position of the marker could not be resolved, resulting in gaps in
the data. This was the case for the wst and d5 markers at
the beginning and at the end of the downstroke, in particular. Gaps, however,
were relatively short and the curves were interpolated and filtered with the
`Generalized Cross Validatory Spline' (GCVSPL) software
(Woltring, 1986). The spline
smoothing coefficients were adjusted to produce a filter cut-off frequency of
approximately 45 Hz, nearly five times greater than the wingbeat frequency.
The quintic spline method also allows the direct calculation of higher-order
derivatives and, therefore, provides greater accuracy in calculating
velocities and accelerations (Walker,
1998). First and second derivatives of positional data were
calculated from the spline coefficients, assuming no error and, hence, without
further filtering. To test the accuracy of our experimental setup, a spherical marker bead was thrown in a parabolic path through the calibrated space in front of the camera. Our calculation of its downward acceleration based on kinematic reconstruction was within 0.5% of 9.81 ms–1. We also moved a rigid card with attached reflective markers at known separation distances similar to the intermarker distances on the wings of the bats as described above. Measurement error based on kinematic reconstruction was no more than 3% from the actual distances, with mean absolute errors ranging from 0.3 mm to 1.2 mm.
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) and elevation (
) of
the body with respect to the global coordinate system
(Fig. 4). However, because of
these rotations, the last rotation angle does not accurately represent the
bank orientation of the body with respect to the global coordinate system.
Therefore, bank angle (
) was calculated as the angle between the line
connecting the two chest markers and the horizontal
(Xg–Yg) plane
(Fig. 4, inset).
Body angles: yaw, pitch and roll
Rotations about the body-centered Xb,
Yb and Zb axes were designated roll,
pitch and yaw, respectively (Fig.
4), following aerodynamic conventions
(Phillips, 2004). Body angular
velocities were calculated by applying a classical transformation from the
angular velocities of the Euler angles, commonly used in rigid body dynamics
(Phillips, 2004). Because bats
were recorded mid-turn, they already had an initial `pitch' and `roll' angles
relative to the global coordinate system. These angles were added to the
angular velocity cumulative sum and represent the angular body position with
respect to the beginning of the recorded portion of the turn. Yaw initial
orientation was arbitrary but because it has no systematic effect on flight
control, all trials started with 0 deg. yaw angle [following Card and
Dickinson (Card and Dickinson,
2008)]. Body angular accelerations were calculated as the first
derivative of the body angular velocities over time.
Determination of CoM
Although the wings of the bat comprise a relatively small fraction of the
overall mass of the bat (Thollesson and
Norberg, 1991), the motions and accelerations associated with wing
flapping may produce substantial inertial effects. As a result of these
morphing motions, the CoM of the bat will not correspond to a fixed anatomical
location on the bat during flight. To account for the wing displacements in
the determination of the location of the CoM, we constructed a mass model
representation of the bat.
The mass model is a time varying, discrete mass approximation of the bat mass distribution based on the location of the markers. To develop the discrete mass system representing the bat, we partitioned total body mass into individual components or regions. The wing membrane, wing bones and trunk were treated as separate masses, which were combined to form the total mass model.
To model the mass distribution of the membrane, we constructed a
triangulation of the wing geometry at each time step. The large-scale, base
triangulation was developed using the location of the marker positions at any
given time, and a subsequent subdivision of these triangles was performed to
give a mesh of fine-scale triangular elements
(Fig. 5). Each triangle element
(Ti) on the membrane was assigned a constant thickness
(1x10–4m) and density
(1x103kgm–3), based on measured
characteristics of bat wing membrane skin
(Swartz et al., 1996). A
resulting discrete point mass (mi) for each triangular
membrane element was computed based on the volume of that triangular membrane
and assigned a position at the centroid of the triangle element. To model the
distribution of mass among and within each of the wing bones, we constructed a
curve between the markers at the endpoints of the bones. The curve for each
bone in the wing was defined from the location of the markers, and the mass of
the fourth digit, that we did not track, was divided equally between the third
and fifth digit. Given the tapered shape of bat bones
(Swartz, 1997), the
cross-sectional radius of each bone element of the model was defined by a
quadratic function with respect to the length of the bone. We assigned a
constant density to the bones (2x103kgm–3).
Using the distribution of bone radii distribution and the location of the bone
elements in space, the line was subdivided into smaller line-elements from
which discrete mass points were defined. The mass of the wings was scaled such
that the constructed distribution represents 16% of the total body mass,
according to measurements of bats of similar size
(Thollesson and Norberg,
1991). The mass and moment of inertia of the wing with respect to
the shoulder were compared with measured values
(Thollesson and Norberg, 1991)
to ensure that the model represents the physical reality. Finally, the bat's
body was defined as a three-dimesional ellipsoid divided into discrete mass
points.
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The discrete mass representation of the membranes, bones and body were
combined with detailed kinematic records of motion of each landmark to
determine the CoM of each of the mass elements, mi, using
the equation:
 | (1) |
represents the position vector of the CoM,
represents the
position vector of the i-th discrete point mass and
mT represents the total mass of the bat.
Calculation of kinematic parameters
Velocity, acceleration, changes in heading and curvature
Net body velocity (Vb) and acceleration
(Ab) vectors were calculated as the first and second
derivatives of the position vector of the CoM in the global coordinate system.
The global trajectory of the bat (i.e. the flight direction) in the horizontal
plane was defined as the bearing angle (
) and was calculated as
the angle between the horizontal component of the net body velocity vector
(Vb,xy) and the Xg
axis (Fig. 6). Changes in
heading can be described as a rate of turning known as curvature (
).
Curvature is defined as the inverse of the radius of the curved path and is
calculated by the equation:
 | (2) |
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); however, we ignored induced velocity because our analyses
focus on comparisons between left and right wings, and induced velocities are
similar for the two wings.
The difference in the surface area between left and right wing was estimated in two ways: (1) by calculating the wrist angle, a measure of the flexion/extension of the wing as a proxy and (2) by the lateral distance of the wingtip marker (d3) with respect to the midline of the body in a body coordinate system. Wrist angle was defined as the interior angle of the triangle formed by the chest (ch), wrist (wst) and wingtip (d3) markers for each wing. Thus, when wrist angle and the lateral distance of the wingtip marker are large, wing surface area is also expected to be large.
Downstroke, upstroke and stroke plane angle
Downstroke and upstroke phases of the wingbeat were defined by positive and
negative velocities of the wrist in the Zb direction,
respectively. The vertical (
v) and horizontal
(h) stroke plane angles were defined as the major axis of
the projection of the wingtip with respect to the body on the
Xb–Zb and
Xb–Yb planes, respectively
(Fig. 7). These major axes were
estimated by fitting a least-square line for each wingbeat.
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Wingbeat consolidation and statistical analyses
To avoid the problem of autocorrelation and pseudoreplication among
wingbeats, kinematic parameters were calculated from one representative
wingbeat per trial. We defined the representative wingbeat as the one with a
heading angle the closest to 45 deg. from the initial orientation of the
flight. This wingbeat represented a mid-turn wingbeat and usually represented
the maximum angular velocities of both heading and body angles. In some cases,
angular velocities peaked ±1 wingbeat from the wingbeat defined by the
heading angle criterion. In such a case, the angular velocity criterion was
used. For most of the analyses, a sample size of 32 trials was used and values
are reported as means ± s.e.m., unless specifically indicated.
Statistical analyses were conducted using JMP 6 (SAS Institute, Cary, NC, USA)
and MATLAB R2006a (The Mathworks, Natick, MA, USA). Regression analyses were
performed with general linear models (GLM) to control for differences among
individuals.
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Changes in bearing occurred almost entirely during the downstroke (Fig. 8), with a mean change of 16.0±0.8 deg. wingbeat–1. Depending on the flight speed, we captured between two and four wingbeats within the calibrated space. Extrapolating the mean change in heading during a wingbeat cycle to the whole turn, C. brachyotis would complete a 90 deg. turn in approximately 6–7 wingbeats. This is likely to be an overestimation as the change in heading tends to peak towards the middle of the turn. From a preliminary study of C. brachyotis performing the same task, a 90 deg. turn was completed in approximately 6–9 wingbeats (J.I.-D., unpublished). Bats reached maximum changes in bearing of 416.9±26.4 deg. s–1 near mid-downstroke, producing turns with a minimum turning radius of 0.290±0.031 m (curvature of 5.53±0.62 m–1), approximately 0.8 wingspans. Mean curvature during downstroke was 3.36±0.33 m–1.
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Changes in body angles
Pitch angle showed high within-wingbeat variation, reaching a minimum at
mid-upstroke and a maximum at mid-downstroke, with a mean change of
12.1±0.9 deg. half-stroke–1
(Fig. 9D). Yaw angle increased
constantly throughout the wingbeat (Fig.
9D) and showed a difference of 13.7±1.0 deg. between the
end and the beginning of the wingbeat (paired t-test,
t31=12.4, P<0.0001) that resulted from
positive yaw angular velocities throughout the wingbeat
(Fig. 9E). By contrast, roll
angle decreased over the wingbeat, decreasing during the upstroke and
remaining constant during downstroke (Fig.
9D). Over a wingbeat cycle, roll angle decreased
–4.3±1.1 deg. (paired t-test,
t31=–4.0, P<0.0001). Yaw angular
velocity was positive throughout the wingbeat, in contrast to roll angular
velocity, which was mostly negative (Fig.
9E).
Pattern of change in heading and flight direction
Heading and bearing angle varied in a similar fashion throughout the
wingbeat cycle, with changes of similar magnitude but with a clear offset
between them (Fig. 8A;
Fig. 10A). Heading angular
velocity peaked at the upstroke–downstroke transition, although bats
changed bearing the most at the middle of the downstroke
(Fig. 10B), indicating that
changes in heading preceded changes in flight path during the turn. The
difference between heading and bearing angle peaked at the
upstroke–downstroke transition and reached a minimum at the end of the
downstroke (Fig. 10A).
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Wingbeat kinematic parameters
Bats flew using wingbeat frequencies of 9.2±0.1 Hz, with upstrokes
comprising 56±2% of the stroke cycle. Wingtip speed with respect to the
body showed a sinusoidal variation with a frequency of nearly half of wingbeat
frequency (Fig. 11A). Wingtip
speed reached a minimum of 4 m s–1 at mid-upstroke and a
maximum of about 8 m s–1 near the end of upstroke and at
mid-downstroke (Fig. 11A).
Wrist velocity showed less variation during the stroke cycle with a mean speed
near 3 m s–1 (Fig.
11B). Mean downstroke speed was 6.31±0.11 and
2.97±0.10 m s–1, for the wingtip and wrist,
respectively. During a half-stroke, angle of attack changed from approximately
50 deg. at the beginning of downstroke to approximately 20 deg. at the end of
downstroke with a mean of 26.7±0.7 deg. Vertical stroke plane angle,
v, was 52.7±4.8 deg.
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Wing kinematics in the body coordinate system were very similar for the inside and outside wings; however, small but statistically significant asymmetries were observed. Mean wingtip speed of the inside wing was 7% faster (a difference of 0.27±0.15 ms–1; paired t-test, t31=1.82, P=0.08), particularly during the upstroke (Fig. 11A). No significant differences in speed between the two wings were observed at the wrist (Fig. 11B). These differences are mostly due to higher wingtip lateral velocities of the inside wing during the beginning and the end of the upstroke (see Fig. S1F in supplementary material).
The angle of attack of the inside wing during downstroke was 9% larger (a
difference of 2.7±0.9 deg., paired t-test,
t31=3.15, P<0.01) than the outside wing
(Fig. 11C). Also, the wrist
angle, a measure of the extension of the hand, and likely of the surface area
of the wing, was larger in the inside wing by 3.3±0.7 deg. (paired
t-test, t31=4.18, P<0.001)
(Fig. 11D). Even though elbow
angle was not measured, we believe that this angle reflects overall wing
extension as we also found no major differences in the distance of the wingtip
to the midline of the body throughout the wingbeat (see Fig. S1A,E,I in
supplementary material). The largest kinematic difference was found in the
horizontal stroke plane angle h. The asymmetry in
h during turning was 10.8±2.8 deg. (paired
t-test, t31=3.86, P<0.001),
indicating that the outside wing moved more parallel to the long axis of the
body than the inside wing, which had an overall direction more oriented
towards the midline.
Kinematic correlations with changes of direction
In a banked maneuver, the centripetal force that produces the turn depends
on the bank angle: the greater the bank angle, the greater the centripetal
force and the tighter the turn. In such a case, the rate of change in
direction angle is expected to be proportional to the bank angle
(McCay, 2001). However, in a
crabbed maneuver, the change in direction should be related to the rate of
change in heading rather than heading orientation
(Warrick et al., 1998;
Hedrick and Biewener, 2007).
Both heading angular velocity and mean bank angle during the downstroke are
significantly correlated with the peak rate of change in direction (GLM,
r2adj=0.88, F4,44=92.7,
P<0.0001 and GLM, r2adj=0.72,
F4,44=32.48, P<0.0001, respectively)
(Fig. 12). In a multiple
regression model, controlling for individual effects, only heading angular
velocity was significant (GLM, r2adj=0.89 for
the whole model; heading angular velocity effect, β=0.82,
F1,43=63.5, P<0.0001; bank angle effect,
β=0.13, F1,43=1.6, P>0.2). The partial
correlation between heading rate and bearing rate when controlling for bank
angle was rheading|bank=0.80 (two-tailed
t-test, P<0.0001) whereas the partial correlation between
bank angle and bearing rate when controlling for heading angular velocity was
rbank|heading=0.14 (two tailed t-test,
P>0.05).
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Based on the instantaneous acceleration of the CoM estimated from the mass
model (see Materials and methods), it is possible to calculate the total
instantaneous centripetal acceleration (Ac,total) necessary
to produce a turn with a radius 1/ using:
 | (3) |
is the
curvature of the turn. Given the symmetry in the wing kinematics in the body
coordinate system, we can estimate the centripetal component produced by the
banked orientation of the body by assuming that the net aerodynamic force is
oriented perpendicular to the bank angle
(Fig. 13A). Thus, the bank
component of the centripetal acceleration was estimated as:
 | (4) |
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DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Kinematic mechanisms affecting change in heading
Discussions of turning in flying vertebrates have focused almost
exclusively on the reorientation of the lift vector by rolling the body into a
bank as the mechanism for the generation of the centripetal force
(Norberg, 1990;
Dudley, 2002). However, the
results of the present study demonstrate that for C. brachyotis,
change in heading was the best predictor of the change in flight direction. By
rotating their bodies horizontally during upstroke, bats reoriented their
major body axis in the direction of the turn. As a consequence, when
aerodynamic force was produced during downstroke, the thrust vector was
already oriented in the direction of the turn
(Fig. 1). Although the bank
angle did not significantly explain changes in flight direction when changes
in heading are considered, the banked orientation of bats observed during
downstroke will probably produce centripetal forces nonetheless. Our
estimations of the centripetal accelerations produced by the banked
orientation suggested that bats produced approximately 70% of the necessary
force required to generate the observed turn. This figure, however, is likely
to be an overestimate. We assumed that the net aerodynamic force was
perpendicular to the horizontal plane of the bat but the larger angle of
attack of the inside wing compared with the outside wing during the downstroke
would probably displace the net aerodynamic force vertically, reducing the
role of banking with respect to changes in heading. Thus, we hypothesize that
there will be a synergistic effect of the changes in heading during upstroke
and the banked attitude of the body that will increase the amount of
centripetal force produced by either a banked or crabbed turning alone.
Offset between changes in heading and changes in flight trajectory
It has been suggested that when a flying organisms does not bank, changes
in bearing in crabbed turns are functionally linked to changes in heading
angles such that flight trajectory would change only when yaw angle changes
(Warrick et al., 1998). This
assumes, however, that the yaw moment is produced by differential generation
of thrust between the left and right wings during downstroke. This is clearly
not the case for C. brachyotis where changes in heading preceded
changes in bearing (Fig. 10A).
The offset between body rotations and changes in flight trajectory implies
that, at the beginning of the downstroke, a bat's cranio-caudal axis is
already partially oriented toward the turn. As a consequence of this body
orientation, the forward component of the net aerodynamic force (NAF) will
also have a centripetal component that adds to the centripetal component
produced by the bank per se. Therefore, we hypothesize that the
temporal offset between body rotations and changes in flight direction
increases the centripetal force generated in comparison with that generated by
a banked turn alone. Such an offset could be particularly important for turns
during slow flight, in which air flow over the wings is relatively slower and,
thus, contributes less to lift generation than in high-speed flight.
Changes in body orientation prior to changes in heading may also improve
the ability of a flying animal to orient the head to the direction of travel,
therefore, improving spatial orientation. Insectivorous bats clearly orient
the head toward the insect when pursuing maneuverable or erratic prey, and
change their flight direction accordingly to keep the body aligned with the
head direction (Ghose and Moss,
2006). In this case, rotating the body before changing heading
would facilitate alignment of the head and body to increase prey location and
obstacle avoidance success (Ghose and
Moss, 2003; Ghose et al.,
2006). The fact that we observe alignment of the body with flight
direction on a non-echolocating, fruit-eating bat suggests that this
phenomenon would be important not only when emitting echolocation calls and
listening for returning echoes and when capturing prey but also simply to
maneuver successfully in three-dimensionally complex environments.
Effect of body-based rotation angles (yaw, pitch and roll) on turning
Because bats adopt a banked attitude during the turn, changes in heading
angle require changes in both pitch and yaw. Our results indicated that pitch
is particularly relevant, showing significant variation throughout the
wingbeat cycle and higher angular accelerations than yaw. This suggests that a
significant portion of the change in heading derives from changes in pitch.
This, in turn, informs our understanding of the forces necessary to rotate the
body. A body's rotation about its CoM depends on its mass moments of inertia
and on the moments about each axis. In organisms with elongated bodies, it is
assumed that the moment of inertia around the roll axis is smaller than the
moment of inertia around the yaw and pitch axes (e.g.
Dudley, 2002), suggesting a
faster rotational response to roll than to yaw or pitch. However, rolling
moments in these bats seem to be mostly compensatory with changes in the
opposite direction to the turn, resulting in an approximately constant bank
angle (Fig. 9A,D).
There are advantages of employing modulation of pitch to perform turns.
Assuming that there is a trade-off between bilateral wing motion asymmetry and
the efficiency of lift and thrust that are produced, pitch can be adjusted by
bilaterally symmetrical changes in wingbeat kinematics that shift the net
aerodynamic force vector either anterior or posterior to the CoM whereas
changes in yaw and roll require bilateral asymmetries
(Dudley, 2002). Furthermore,
pitch modulation may also require less force than is required to produce
rotational changes in yaw. The contribution of the wings to the total mass and
to the moment of inertia can be considerable in bats
(Kirkpatrick, 1990;
Watts et al., 2001). For
example, in a study of eight bat species, the mass of one wing accounted for
approximately 8% of the total body mass and contributed to approximately 93%
of the roll inertia (Thollesson and
Norberg, 1991). In this example, yaw inertia is expected to be
larger than pitch moment of inertia assuming that the pitch rotational axis
passes through the wings. It is interesting to note that flapping fliers, even
in straight, level flight at constant velocity show up and down pitching
moments during upstroke and downstroke, respectively. However, comparison of
the effect of these `natural' pitching moments with those observed on turning
flight is not straightforward due to the banked orientation of the bat.
Pitching moments are the result of both inertial and aerodynamic effects.
Although changes in pitch due to inertial forces are not expected to change
when bats are in a bank turn compared with when they are in straight flight,
we do expect changes in how aerodynamic forces would affect pitch due to the
differences in orientation of the gravitational force with respect to the net
aerodynamic force.
Mechanisms of heading rotation
Changes in heading are essential to the completion of the turn. However, we
observed that heading rotation in the direction of the turn occurs mostly
during the upstroke. This portion of the wingbeat cycle, at least for slow
flight, has been believed to be inactive aerodynamically
(Norberg, 1990;
Spedding et al., 2003). How,
then, are bats able to change their heading orientation during the upstroke?
One mechanism for producing changes in heading is to generate more thrust with
the outside wing than with the inside wing, producing a torque in the
direction of the turn. This could potentially be accomplished by a backward
flick with the tip of the wing, which has been reported in some bats at the
beginning of the upstroke when flying at low speeds
(Aldridge, 1986b;
Norberg and Winter, 2006).
Such a backward flick is observed in C. brachyotis at the wingtip
(Fig. 15A) but not at the
wrist (Fig. 15B) and it is
unlikely to produce a global yaw moment because the backward velocities of the
outside and the inside wing do not differ significantly during the upstroke
(Fig. 15A).
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An alternative means by which to produce changes in heading is the use of
asymmetric movements of the wings during upstroke. Left–right asymmetry
could generate inertial torques that differ between the inside and outside
wing. Such a mechanism is an effective way to produce what is called a
zero-angular-momentum rotation, a maneuver used by self-righting cats (e.g.
Arabyan and Tsai, 1998) and
gymnasts (Yeadon, 1997). In a
zero-angular-momentum rotation, body segments are rotated with respect to each
other during flight, hence, the whole body will rotate as a consequence to
conserve angular momentum. By this mechanism, changes in heading would arise
from differences in the movement of the left and right wings in the horizontal
Xg–Yg plane. It is difficult to
predict the effect of wing and body inertia on the body rotation without
modeling the time-varying contribution of a morphing wing and rotating body
during the turn but considering the wing masses of bats are approximately 16%
of total body mass (Thollesson and
Norberg, 1991), it could be expected that inertial reorientation
of the body may be important. Inertial contributions to body rotations during
turning have been estimated for birds where asymmetries in the amplitude of
the wingbeat were capable of transient changes in roll of only 6 deg., with a
net change of 1.6 deg. per wingbeat
(Hedrick and Biewener, 2007).
Bats are expected to have a smaller moment of inertia for roll than for yaw
and pitch, hence, the magnitude of changes due to inertial reorientation in
yaw are not likely to be great enough to account for the mean change of 20
deg. in yaw observed during upstroke. However, this issue cannot be resolved
without modeling the inertial effect of the observed wing kinematics.
Bat turning compared with other flying organisms
Experiments on maneuvering in birds show that pigeons and cockatiels use
banked turns (Warrick and Dial,
1998; Hedrick et al.,
2002). In pigeons, roll acceleration increases and decreases
during a single wingbeat, and changes in acceleration are correlated with
left–right asymmetry in downstroke wing velocity
(Warrick and Dial, 1998).
Cockatiels show a similar roll acceleration profile, with changes in roll
orientation within each wingbeat correlated with wing motion asymmetries
(Hedrick and Biewener, 2007).
This within-wingbeat variation, however, is probably the result of inertial
forces produced by wing kinematic asymmetries and, therefore, tends to cancel
out over a complete stroke cycle (Hedrick
and Biewener, 2007). In fact, changes in cockatiel flight
direction were best explained by changes in roll orientation between
wingbeats, which are not correlated with changes in roll within each wingbeat,
and that are likely to be the combined result of both inertial and aerodynamic
effects (Hedrick and Biewener,
2007). When compared with birds performing similar turns, bats
produced tighter maneuvers, allowing them to complete the turn in a smaller
number of wingbeats than pigeons and cockatiels
(Warrick and Dial, 1998;
Hedrick and Biewener, 2007).
These differences could be the result of differences in size. C.
brachyotis are approximately 10 times smaller in mass than cockatiels,
and it has been suggested that maneuverability is inversely related to body
size (Aldridge, 1987;
Stockwell, 2001). Whether
these differences in turning performance are a consequence of differences in
size or due to differences in turning mechanisms is not known.
Studies of 180 deg. turns in microchiropteran bats have shown that bats
initiate turns by flying upwards and decelerating
(Rayner and Aldridge, 1985;
Aldridge, 1987). We observed
similar patterns in the present study, although our bats maintained their net
forward speed throughout the recorded portion of the turn. The curvatures of
the turns observed in our experiment were 3–23 times smaller (i.e.
greater turning radius) than those observed for other bat species performing
180 deg. turns (Aldridge,
1987). Such variation is to be expected, considering the
differences in task and body sizes.
The use of a combination of crabbed and banked mechanisms to produce a turn
is likely to increase the maneuverability of bats compared with a mechanism
that employs lift alone. The net influence of the crabbed component on turning
is not readily quantified but it is clearly important for insects
(Dudley, 2002). For example, in
dragonflies capable of both banked and crabbed turns, the latter strategy
produces turns at much higher rates, with changes in direction of 180 deg. in
less than three wingbeats (Alexander,
1986). This degree of maneuverability is similar to that observed
in the bats during the present study, where 180 deg. turns can be achieved in
3–4 wingbeats, although no information is available on whether the turn
was banked, crabbed or a combination of both
(Tian et al., 2006).
Our findings show that bats use a combination of crabbed and banked mechanisms to produce centripetal accelerations required to perform a turn. C. brachyotis changed its heading during upstroke and, thus, reoriented the body in such a way that the forward component of the net aerodynamic force produced during downstroke was aligned with the direction of travel. Therefore, the reorientation of the body and the bank angle of the body acts synergistically to produce a centripetal force. Bats seemed to actively change their yaw and pitch whereas changes in roll were compensatory to maintain a constant bank attitude. Reorientation during a wingbeat cycle is probably the result of the combination of aerodynamic and inertial forces, and future research should include estimation of how asymmetries in wingbeat kinematics to estimate the magnitude of inertial reorientation.
LIST OF ABBREVIATIONS
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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  K. Phillips BATS TAKE TURNS BY BANKING AND CRABBING J. Exp. Biol., November 1, 2008; 211(21): ii - ii. [Full Text] [PDF]
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