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First published online July 20, 2007
Journal of Experimental Biology 210, 2593-2606 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.002071
Aerodynamic force generation, performance and control of body orientation during gliding in sugar gliders (Petaurus breviceps)
Department of Ecology and Evolutionary Biology, Brown University, Providence, RI 02912, USA
Present address: Section of Ecology and Evolution, University of California, Davis, Davis, CA 95616, USA (e-mail: kvwbishop{at}ucdavis.edu)
Accepted 24 March 2007
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Key words: aerodynamics, biomechanics, gliding, mammal, stability
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;
McCay, 2001a;
Socha, 2002), and in those
that do, the wings differ from human engineered wings in shape, material
composition and typical flight speeds, so standard aerodynamic theory may not
accurately predict animal gliding performance
(Bishop, 2006;
McGuire and Dudley, 2005;
Socha et al., 2005). Form and
function are related to evolutionary fitness through the performance of tasks
that are critical to survival and reproduction, such as locomotion. An
understanding of the mechanistic basis for differences in performance provides
valuable groundwork upon which to base studies of the evolutionary history and
diversity of organisms.
In addition to being interesting as a form of locomotion in its own right,
mammalian gliding is of particular interest due to its probable role in the
evolution of flapping flight in bats. It is widely believed that bats evolved
from an arboreal gliding ancestor similar to extant gliding mammals
(Clark, 1978;
Smith, 1976;
Norberg, 1985;
Norberg, 1990;
Simmons, 1995;
Bishop, 2006). The transition
from gliding to flapping flight presents an interesting evolutionary problem.
Gliding and flapping flight are very different behaviors and may be subject to
different optimization criteria (Padian,
1982). Mammalian glider wings differ consistently from bat wings
in being very short spanwise relative to their breadth, i.e. having a very low
aspect ratio (AR), and in being nearly rectangular in shape compared to the
tapering wings of bats. Experiments with physical models of wings have shown
that wings with AR<2, as possessed by all mammalian gliders, have very
different aerodynamic properties than wings with AR>2, as are typical of
bats, such that aerodynamic performance declines with increasing AR in the low
AR range and improves with increasing AR in the high AR range
(Torres and Mueller, 2001;
Shyy et al., 2005;
Galvao et al., 2006). How is
it possible to move between two such highly specialized states without going
through a phase in which the animal is not particularly well adapted for
either function?
Previous studies have modeled an evolutionary transition from gliding to
flapping flight by assuming that flapping evolved to extend the flight
distance (Bock, 1965;
Parkes, 1966;
Norberg, 1985). Flight
distance is extended by altering the production of aerodynamic forces. The
measure of performance most commonly used is the glide ratio
(Vernes, 2001;
Stafford et al., 2002;
Jackson, 2000), the horizontal
distance traveled divided by the vertical drop, because it is a measure of the
distance an animal can travel from a given height. In a steady,
non-accelerating glide, the glide ratio is determined by the ratio of lift to
drag, the aerodynamic forces perpendicular to the direction of travel and
parallel and opposite to the direction of travel, respectively. The
lift-to-drag ratio can be increased by increasing lift, decreasing drag,
and/or by producing thrust, defined as a force that opposes drag. Flapping has
often been assumed to have evolved as a means to increase lift and thrust
(Bock, 1965;
Parkes, 1966;
Norberg, 1985), thereby
increasing the distance traveled, but other potentially important features of
flight performance have typically been overlooked.
Although producing lift and thrust is certainly important to powered
fliers, controlling the flight trajectory is equally important, and the roles
of stability and maneuverability have often been overlooked in discussions of
gliding performance. It has been suggested that flapping movements of wings
such as those possessed by mammalian gliders would cause rotational
instabilities that could lead to problems with control
(Caple et al., 1983). It is
possible, however, that gliding animals actively employ movements of the limbs
to counter rotational motions during a glide
(Bishop, 2006), and further,
that the earliest flapping behavior may have been used for stability and
maneuverability and not simply for lift or thrust. Specialized gliding frogs
have been found to be passively unstable gliders
(McCay, 2001b), suggesting
that they must actively control their glide using body movements. An
understanding of how gliding flight is actively controlled by limb movements
is critical to developing viable models of a gliding to flapping
transition.
In this study, I examined gliding in a specialized mammalian glider, the marsupial sugar glider Petaurus breviceps (Waterhouse) using video images captured using two high-speed video cameras to reconstruct the 3D coordinates of anatomical landmarks. I investigated the relationships among details of gliding kinematics, aerodynamic force generation, and gliding performance, with special attention to assessing control of the glide trajectory by documenting the relationship between limb movements and body rotations. Finally, I compared gliding in P. breviceps to that in a placental gliding specialist, Glaucomys volans, the southern flying squirrel. Relationships between kinematics and performance that are similar in diverse groups of mammals may have also governed gliding in bat ancestors and provide a basis for hypotheses about the origin of flapping in bats. This study represents the first interspecific comparison of the detailed 3D kinematics of gliding in mammals and is the first to document the role of limb movements in controlling body rotations in any glider.
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Data collection
Four sugar gliders were purchased at weaning through the pet trade and
maintained in the animal care facility at Brown University in accordance with
institutional animal care procedures (IACUC #16-06). The animals were housed
in a large cage with ample room to run and jump, and food and water were
provided ad libitum. After a 1-week period of hand taming, the
animals were trained on a daily basis to jump, then glide to a PVC pole with a
diameter similar to that of a large tree trunk. Training continued until adult
size had been reached (approximately 4 months old), at which time filming
commenced. At the time of filming, the mean body length of the animals was
13.5 cm from the tip of the nose to the base of the tail and their mean body
mass was 73.4 g. During the training period, the gliders were given food
rewards to promote gliding behavior, but no food rewards were given during
filming to ensure the accuracy of weight measurements taken just before data
collection.
Filming took place in a large garage with a high ceiling at Harvard University's Concord Field Station using a pair of high-speed digital video cameras (Redlake, PCI-1000, San Diego, CA, USA) at a framing rate of 250 Hz and an image size of 480x420 pixels. The cameras were placed beneath the glide path at an angle of approximately 90° to one another (Fig. 1). The volume of space visible in both cameras was calibrated in three dimensions using a 0.57 mx0.49 mx0.41 m pre-measured calibration frame (Peak Performance, Inc., Englewood, CO, USA). The cameras were positioned such that the calibration frame occupied as much of the field of view as possible in both cameras so that when the animal was visible in both cameras, it was also in the calibrated volume, with some exception at the edges of the field of view. Consequently, the volume of the overlapping fields of view of the cameras was approximately equal to the volume of the calibration object.
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Using medical adhesive [Silastic(R) type A, Dow Corning, Midland, MI, USA], 6 mm diameter spherical reflective markers were attached to the skin at the sternum, pubic symphysis (hereafter called the pelvis marker), right and left hip joint, left wrist, left ankle and the center of the free edge of the left patagium (Fig. 2). The markers were placed ventrally so that they could be seen from the cameras beneath the glide path. Because an additional point on the body axis was needed for some of the analyses, the nose was also treated as a landmark.
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Video sequences were digitized using DLT DataViewer 2 (http://faculty.washington.edu/thedrick/digitizing/) in Matlab (MathWorks, Natick, MA, USA). I estimated digitizing error by digitizing the same trial five times and computing the standard deviation for the five trials for each point in each frame, then taking the mean of the standard deviations of all the frames for each point. This mean of the standard deviations is henceforth referred to as the digitizing error. The digitizing error was approximately 1 mm in each dimension for points with a visible marker and 3 mm for points that were visually estimated.
Random digitizing error can have a disproportionate effect on computed quantities. To estimate this effect, I simulated random noise by adding or subtracting a random number up to the estimated digitizing error to each coordinate (x,y,z) for a representative trial. I then calculated resulting distances, velocities, accelerations and angles for the data with simulated error. This was repeated 10 000 times; digitizing error for the computed quantities was estimated as the standard deviation of the replicate computations. For linear measurements the digitizing errors were all less than 0.5 mm, a maximum of a 2% error. For velocity measurements, the standard deviations were less than 0.002 m s–1, representing a 0.04% error. Accelerations had standard deviations of less than 0.01 m s–2 and errors of less than 5%. Angular measurements were accurate to 0.5° with errors of less than 3%, with the exception of roll angle, which had errors of nearly 6%. The residuals of direct linear transformations, which incorporate both the spatial error of the calibration and digitizing error, were approximately 1 mm for all points in all dimensions.
Performance measures
An animal may glide for many reasons, such as to reach distant food
resources, escape predators, or return to nest sites. Because the relevant
aspect of performance depends on the specific task being performed, no single
measure of performance is sufficient to characterize overall gliding
performance. Therefore, in this study I consider three aspects of performance
that appear to be particularly important: glide angle, glide velocity and
stability. All of these performance parameters depend on the animal's
manipulation of the aerodynamic forces it generates.
The position data were smoothed using a curve-fit method to compute whole body velocity and acceleration. A second-degree polynomial was fit to the position data for the sternum and pelvis markers and the polynomials were differentiated twice to compute the velocity and acceleration, respectively. The second-degree polynomials provided a good fit to the data, with maximum residual errors of 0.6%, indicating that acceleration was essentially constant in all directions over these short glide segments.
The position of the center of mass was estimated using a cadaver specimen that was frozen in a position similar to that adopted in flight, and was found to lie near the midpoint between where the sternum and hip marker were placed. Because the center of mass is approximately mid-way between the sternum and the hip, and assuming minimal spinal flexion, whole-body velocities and accelerations were estimated as the averages of those of these two landmarks. To the extent that the center of mass deviates from this position, a small error will be introduced into the estimates by any body rotations about the center of mass.
Glide angle is defined as the angle between the animal's glide trajectory
and the horizontal (Fig. 3).
More shallow glides have lower glide angles, while steeper glides have higher
glide angles. In a steady, non-accelerating glide, the glide angle is
inversely proportional to the lift-to-drag ratio
(Fig. 3A), so horizontal
distance traveled from a given height is maximized by generating a large
amount of lift relative to drag. Glide angle was computed for each frame as:
 | (1) |
is the glide angle, Vy is the vertical
component of the whole body velocity, and Vx is the
forward component of the whole body velocity.
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A positive horizontal acceleration indicates that the resultant aerodynamic force is inclined forward with respect to the vertical. The forward component of the resultant force was computed as the animal's body mass x the forward component of its acceleration. The two forces acting vertically on a gliding animal are its body weight and the vertical component of the resultant aerodynamic force (Fig. 3B). In a non-accelerating glide the vertical resultant aerodynamic force is equal to body mass x acceleration due to gravity. However, the net downward acceleration in these glides indicates that the entire body weight was not balanced by the vertical component of the resultant aerodynamic force. Therefore, I computed the vertical component of the resultant aerodynamic force by subtracting the measured vertical acceleration from the acceleration due to gravity before multiplying by the body mass.
Because drag is defined as a force acting parallel and opposite to the
direction of the whole-body velocity, I used the angle between the opposite of
the velocity vector and the resultant aerodynamic force vector to decompose
the resultant aerodynamic force into lift and drag components
(Fig. 3B). The reference angle
between drag and the resultant aerodynamic force is:
 | (2) |
 | (3) |
is the reference angle between
drag and the resultant aerodynamic force. To compare airfoils of different
sizes operating at different speeds, these forces are typically converted into
dimensionless force coefficients:
 | (4) |
Body rotations
The three rotational axes are pitch, roll and yaw. The rotational angles of
the body were computed sequentially in the order: yaw, pitch, roll. Rotating
the coordinate system in this order results in an animal-centered coordinate
system with the frontal plane of the body as the x–z
(horizontal) plane. The coordinates for all of the points were recalculated
after each rotation, and the next rotation was computed based on the new
coordinates, resulting in an animal-centered coordinate system with the
horizontal plane defined by the sternum, right hip and left hip of the animal
and with the sternum at the origin. This plane will henceforth be referred to
as the body plane. For the purposes of this discussion I use capital letters
to refer to global coordinates and lower case letters to refer to
animal-centered coordinates.
Yaw is defined as a rotation about the Y (vertical) axis. I
defined the body axis by taking the mean of the left and right hip coordinates
(hereafter called `mean hip') and computing the coordinates of the sternum
with respect to that point. I define the yaw angle as the angle between a line
joining the sternum and mean hip and the X (forward) axis, computed
as:
 | (5) |
Pitch is defined as a rotation about the mediolateral axis and was computed
as the angle between a line connecting the sternum and mean of the hip
coordinates and the yaw-corrected x axis:
 | (6) |
Roll is defined as rotation about the anteroposterior axis. It is computed
as the angle between a line connecting the right and left hip and the yaw and
pitch-corrected z (lateral) axis:
 | (7) |
Limb positions
The angle of attack of a wing is defined as the angle between the chord
line, a line joining the leading and trailing edges of the wing, and the
direction of the oncoming airflow. In the case of a gliding animal, the
direction of the oncoming airflow is determined by the glide angle. The angle
of attack is important aerodynamically because within a range of low angles of
attack, lift increases with increasing angle of attack. Beyond this range, as
the angle of attack gets larger, the airflow begins to separate from the
surface of the wing and the wing begins to stall. At angles of attack beyond
the stall angle, lift decreases with increasing angle of attack. Because more
of the wing's surface is exposed to the air flow as angle of attack increases,
drag increases with increasing angles of attack up to 90°.
Angle of attack was computed for each frame as the angle between the chord
line, a line connecting the wrist and ankle, and the velocity vector
(Fig. 3B) using the following
equation:
 | (8) |
The camber of a wing is defined as its curvature from leading edge to
trailing edge. A gliding mammal can theoretically control the camber of its
wings in one or both of two ways. It can move its fore- and hindlimbs closer
to one another, increasing the slackness of the wing membrane and allowing
more billowing. Gliding mammals also have musculature within the wing membrane
(Johnson-Murray, 1977;
Johnson-Murray, 1987;
Endo et al., 1998). Although
the function of these muscles has not been tested, they are positioned such
that if contracted or relaxed they may allow lesser or greater billowing of
the wing.
The amount of lift a wing can generate increases with increasing camber up to the point when the airflow becomes detached from the wing surface. Increasing camber also exposes more of the wing area to the airflow, so drag is expected to increase with increasing camber. I estimated camber height as the perpendicular distance of the patagium point from the chord line (Fig. 3B). I define relative camber here as the ratio of the camber height to the chord length. Relative camber corrects for overall size when comparing wings of different individuals and/or species.
All limb positions of the animals other than angle of attack and camber were defined with respect to the animal-centered coordinate system, to separate movements of the limbs from the overall movements of the animal.
The angle of attack of the wing depends on both the angle of the body with respect to the glide trajectory and the position of the limbs with respect to the body. Of these components of angle of attack, the one over which a gliding animal has the most direct control is the angle of the chord line with respect to the body, which I am calling the chord angle. I computed chord angle as the projection on the x–y (sagittal) plane of the angle between a line connecting the wrist and ankle and the body plane. This angle is significant aerodynamically because changes in this angle represent a movement of the limbs that tends to change the angle of attack of the wing.
The elevation of the wingtips above the body, or dihedral, affects the
passive stability of a flying body (Bertin,
2002). The elevation angle is here defined as the projection on
the y–z (transverse) plane of the angle between a line
connecting the sternum and wrist and the z (mediolateral) axis. I
consider the elevation angle to be zero when the wrist is vertically even with
the body plane. A positive angle indicates that the forelimbs are held above
the plane of the body, i.e. at a dihedral, whereas a negative angle indicates
that the forelimbs are held below the plane of the body, which is called an
anhedral.
Protraction is defined as movement of a limb toward the head. The
protraction angle is here defined as the projection on the
x–z (frontal) plane of the angle between a line
connecting the sternum and wrist and the z (mediolateral) axis. I
consider the protraction angle to be zero when the forelimb is perpendicular
to the body axis as seen from a dorsal view, and positive when the forelimb
moves toward the head. This angle is analogous to the sweep angle of an
airplane wing, which affects its lift-generating performance
(Bertin, 2002). A positive
protraction angle indicates a forward-swept wing, whereas a negative
protraction angle means the wing is swept backward.
For each time step, I estimated the area of a single wing by taking the mean of the 3D distance between the sternum and wrist and the 3D distance between the pelvis and the ankle and multiplying it by the chord length computed for that time step, and doubled this quantity to estimate the total wing area. I computed the wing loading by dividing the weight of the animal measured just before the trial (mass x acceleration due to gravity) by the estimated wing area. Errors inherent in using this estimate will affect the exact values of wing loading and force coefficients. However, there is no bias in this estimator that affects comparisons among individuals, and use of this approach allows direct comparisons with earlier studies.
Statistics
Means are reported ± 1 standard deviation (s.d.). Correlations
between the average value of variables over the glide sequence were estimated
using Pearson correlation coefficients with a significance level
P=0.05. When more than one factor is predicted to have an effect, I
used a stepwise multiple regression analysis, also with a significance level
P=0.05, to estimate the effect of each of the factors while holding
the others constant. Multiple regressions were done using average values for
each glide sequence. To account for intra-individual variation, regressions
were performed separately on each individual. This approach gave results
consistent with those for the individuals combined (data not shown).
To estimate correlations between variables within a glide sequence I used
cross correlation analysis (Chatfield,
1992). Cross correlation is a technique that estimates the
correlation between two sets of time series data, taking into account that the
correlation may not be instantaneous. The correlation is computed at a number
of positive and negative time lags and the lag with the maximum absolute value
for the correlation coefficient is taken to be the true time lag for the
relationship. This technique does not assume a causal relationship between the
two variables. For the cross correlation, data were smoothed using a lowpass
Butterworth filter with a cut-off frequency of 25 Hz and the mean value for
the trial was subtracted from each data point to remove the effect of overall
trends (detrended) before computing the cross correlation functions
(Fry, 1993). The cut-off
frequency was selected based on visual inspection of the raw data along with
the filtered data and chosen based on the elimination of point-to-point
variation while maintaining overall trends. A sensitivity analysis indicated
that varying the cut-off frequency by ± 5 Hz does not affect the
conclusions based on the cross correlation. 95% confidence limits (CL) were
estimated at each time lag as:
 | (9) |
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Overall, the wing tended to be held in a position that increased its angle of attack compared to that determined by the pitch of the body (Table 2). The chord angle with respect to the body plane averaged over the glide sequence was positive in all but one trial, with a mean for all trials of 15.2±6.7°. In 35 of the 49 trials, the chord angle remained positive throughout the glide sequence (Fig. 6A). In the remaining 14 trials, chord angles started out small and positive and became negative in the last half of the glide sequence.
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When averaged over a glide sequence, the forelimb tended to be held in a slightly elevated (wrist higher than the body plane) posture in most cases. The mean elevation angle for all trials was 6.7±7.8° (Table 2). There was one individual who used much higher positive elevation angles than the rest, but the more typical pattern was to hold the forelimb fairly close to horizontal with respect to the body plane. In the majority of video sequences, the elevation angle changed from positive to negative, or the reverse, at least once (Fig. 6B).
The forelimbs were kept in a strongly protracted position for all glides, at no time in any glide was the wrist observed to be posterior to the sternum (Fig. 6C). The mean protraction angle over all trials was 30.6±5.4° (Table 2).
Gliding performance
The resultant velocities used by the sugar gliders had a mean of
5.08±0.30 m s–1. In 54 of 55 cases the gliders
accelerated in the forward direction with an average horizontal acceleration
of 2.1±0.6 m s–2. The downward accelerations were
small, with a mean of 1.0±0.5 m s–2. This corresponds
to an average upward acceleration due to aerodynamic forces (i.e. with gravity
subtracted) of 8.8 m s–2
(Table 1).
The observed glides were fairly steep, with a mean glide angle of 49.6±2.5° (Table 1). The glide angles decreased over the captured glide sequence in nearly all of the glides, indicating that the animals were flattening their glide trajectory at this point in the glide. The mean rate of change in the glide angle was –11.1±5.6° s–1. Only two glides had glide angles that increased over the captured glide sequence. There was a small, but significant correlation between downward acceleration and the rate of decrease in the glide angle (r=–0.276, P=0.042).
Relationship between limb position and aerodynamic forces
In general, the sugar gliders produced more lift than drag. The lift
coefficients averaged 1.48±0.18
(Table 1). The mean drag
coefficient for all trials was 1.07±0.13
(Table 1). The mean
lift-to-drag ratio was 1.39±0.16
(Table 1). All of the gliders
performed similarly in terms of lift and drag production, although they did
not all use the same range of angles of attack
(Fig. 4) or camber
(Fig. 5).
It should be noted that lift and drag are affected by multiple factors; angle of attack and camber are both likely to contribute to the production of aerodynamic forces. Therefore, a simple relationship between forces and any one factor is difficult to interpret from a graphical representation and multiple regression techniques must be used to control for the effects of the other factors. However, graphs are provided to illustrate the range and magnitude of these variables. Examination of Fig. 6 and Fig. 7 suggests that there was little to no relationship between either angle of attack or camber with lift coefficient, drag coefficient, or lift-to-drag ratio. This was supported in the case of lift coefficient by a stepwise multiple regression analysis (Table 3), which removed both angle of attack and relative camber from the model as factors affecting lift.
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Multiple regression analysis for drag, however, retained both angle of attack and relative camber as significant factors (Table 3). A model that included only angle of attack accounted for only 6.2% of the variation in drag coefficient, but together with relative camber accounted for 24% of the variation in drag. In the model including both angle of attack and relative camber, angle of attack was strongly positively correlated with drag, whereas relative camber was strongly negatively correlated with drag.
Similarly, for lift-to-drag ratio, angle of attack on its own accounted for only 12.8% of the variation in lift-to-drag ratio, but a model including both angle of attack and camber accounted for 41.1% (Table 3). This model indicates a strong negative correlation between angle of attack and lift-to-drag ratio and a strong positive correlation between relative camber and lift-to-drag ratio.
Relationship between aerodynamic forces and performance
Glide angle was strongly related to both wing loading and to the production
of lift (Fig. 7). A stepwise
multiple regression removed drag coefficient as a factor contributing to glide
angle and retained lift coefficient and wing loading
(Table 3). Wing loading alone
accounted for 68.1% of the variation in glide angle and wing loading and lift
coefficient together accounted for 80.9%. The model including both wing
loading and lift coefficient shows that wing loading was very strongly
positively correlated with glide angle, indicating that heavier wing loading
produced steeper instantaneous glide angles, and lift coefficient was
negatively correlated with glide angle, in keeping with the expectation that
more lift produces a shallower glide.
Velocity was affected by both wing loading and aerodynamic force production
(Fig. 8). A stepwise multiple
regression analysis retained wing loading, lift coefficient and drag
coefficient as factors contributing to velocity
(Table 3). Wing loading alone
accounted for 57.2% of the variation in velocity, wing loading and lift
coefficient together accounted for 75.0% and wing loading, lift coefficient
and drag coefficient accounted for 79.3%. Wing loading was strongly positively
correlated with velocity, whereas lift and drag coefficients were both
negatively correlated with velocity. This is consistent with aerodynamic
theory, which predicts that more heavily wing-loaded animals will glide faster
(Norberg, 1990) and that at
steep glide trajectories, both lift and drag contribute to the force opposing
gravity and will tend to slow the animal down.
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Pitch
There were no limb positions that had significant correlations with pitch
in more than 70% of the trials for all of the gliders pooled, but there were
strong associations for the individuals taken separately.
For Ind 2, there was a significant positive correlation between pitch and chord angle in 79% of trials, indicating that nose-up rotations in pitch were associated with movements of the limbs that tend to increase the angle of attack. For the trials with significant positive correlations, the lags at which the maximum correlation coefficients occurred were mostly zero and positive, with small negative lags in two out of 14 total trials for this individual. In this and all subsequent comparisons, a positive lag indicates that changes in limb position led changes in rotations and negative lags mean that changes in rotation led changes in limb position.
For Ind 3, pitch had significant correlations with chord angle and protraction. Chord angle had a significant positive correlation with pitch in 79% of 14 trials, indicating that nose-up rotations in pitch were associated with movements that tend to increase angle of attack. In trials with significant positive correlations, the maximum correlation coefficients occurred mostly at a lag of zero, with the exception of two trials, one with a lag of 2 and one of –3. There was a significant negative correlation between pitch and forelimb protraction in 71% of trials for this individual, indicating that movements of the forelimb toward the head were associated with nose-down changes in pitch. Of the trials for this individual with significant negative correlations, the maximum correlation occurred at positive lags in most trials, but one trial had a lag of zero and three had negative lags.
In Ind 4, pitch was most often correlated with relative camber and protraction. There was a significant negative correlation between pitch and relative camber in 76% of the 17 trials for this individual, such that nose-up pitching rotations were associated with decreases in camber. For trials with significant negative correlations, the lags for the maximum correlation were mostly large and positive, ranging from 3–10, with the exception of three with large negative lags. Protraction was significantly negatively correlated with pitch in 71% of the trials for this individual, indicating that nose-up rotations were associated with movements of the forelimbs away from the head. For trials with a significant correlation, the lags were mostly positive, with one zero lag and two trials with negative lags.
Roll
For all trials pooled there were strong associations of both chord angle
and elevation angle with roll. In 71% of trials there was a significant
positive correlation between chord angle and roll, indicating that movements
of the limbs that tend to increase the angle of attack on the left side were
associated with rolling rotations to the left (right hip higher than the
left). When correlations were positive, lags were mostly zero and positive,
but six of the 35 trials had small negative lags. Limb elevation had a
significant positive correlation with roll in 76% of all trials, indicating
that movements of the limbs above the plane of the body on the left were
associated with rolling rotations to the left. For trials with a significant
positive correlation, most of the lags were zero or small and positive (up to
three), but three trials had small negative lags. Ind 2 showed the same
correlations as the pooled data. Roll was positively correlated with limb
elevation and forelimb protraction in Ind 3, but not with chord angle.
In Ind 4 there was a very strong association between relative camber and roll. Roll and relative camber had a significant negative correlation in 94% of the trials for this individual, meaning that increases in camber on the left were associated with rolling rotations to the right. Lags were mostly small and negative, up to –3; five trials had a lag of zero, and one had a positive lag of one.
Yaw
There was a tendency in the pooled data set for yaw to be associated with
protraction of the forelimb. In 76% of trials protraction was significantly
negatively correlated with yaw angle, indicating that yawing rotations to the
right were associated with movement of the left forelimb away from the head.
In trials with significant negative correlations, lags were either zero or
small and negative, with the exception of four trials with small positive lags
(
3) and one with a lag of nine. This trend was driven primarily by Ind 2
and Ind 4, who had a significant negative correlation between protraction and
yaw in 79% and 100% of their trials, respectively.
In Ind 2 there was a significant negative correlation between yaw and relative camber in 86% of trials, all with zero or small positive lags. This indicates that increases in camber on the left were associated with yawing rotations to the left. This individual had no strong associations between yaw and any other limb position.
In addition to protraction, Ind 4 also showed strong associations of yaw with both chord angle and limb elevation. This individual had a significant negative correlation between yaw and chord angle in 71% of trials, indicating that movements of the limbs that tend to increase angle of attack on the left were associated with yawing rotations to the left; all trials with significant negative correlations had negative lags ranging from –10 to –2. Wing elevation was significantly negatively correlated with yaw in 82% of trials for this individual, indicating that movements of the forelimbs above the plane of the body on the left were associated with yawing rotations to the left, all with negative lags (ranging from –9 to –4).
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Body rotations result from asymmetrical generation of forces across the respective body axes. In addition, rotations can be induced by inertial effects when the position of the center of mass changes due to movement of the limbs. I estimated the moment of inertia of the gliders by dividing frozen cadaver specimens of sugar gliders into 1 cm sections in both the longitudinal and transverse directions and weighing each segment. The estimated moments of inertia were very small, 1.4x10–5 kg m2 for the roll axis and 9.9x10–5 kg m2 for the pitch and yaw axes, so these body rotations are expected to be very sensitive to changes in force at the wings. Therefore, the very short lag times reported here are credible.
Although this analysis of the relationships between limb movements and body rotations provides valuable insights, there are some limitations that should be noted. First, the captured glide sequences are shorter than the oscillation period of the limb movements and body rotations. This means that the lag at which the actual maximum correlation occurs may be greater than the number of lags tested, in which case it would not be detected. Because these tend to be oscillating phenomena, there are alternating positive and negative peaks in the cross correlation function that decrease with increasing distance from the actual time lag of the function. If the actual time lag is greater than the number of lags tested, it is likely that the next highest correlation coefficient will have the opposite sign. These errors are most likely to have occurred where the absolute value of the actual lag is high, because nearby lower correlation peaks are more likely to be detected when correlation maxima have small or zero lags, verifying that it is a true maximum. It should be noted that errors resulting from the short length of the video sequences are conservative because there is a tendency to miss significant correlations rather than to detect false correlations, and that analysis of longer glide sequences would be expected to yield even stronger correlations.
Another limitation to this approach is that cross correlation analysis does not take into account the interaction between the effects of multiple limb movements on each body rotation. Different limb movements can produce force asymmetries that will result in a single kind of rotation, therefore it is possible that some correlations were missed due to these interactive effects. A related limitation is that only one wing was marked and the effect of the position of one wing on body rotations depends entirely on what the opposite wing is doing. I make the conservative assumption here that the opposite wing is stationary; it is likely that even stronger correlations would be detected if measurements had been taken from both wings.
Gliding performance
Glide angle
The glides performed by the sugar gliders were quite steep. Glide angles of
sugar gliders averaged over whole glides in the wild have been estimated to be
29.69±1.10° (mean ± s.e.m., N=13)
(Jackson, 2000), while the
gliders in this study used glide angles around 50°. This is probably due
to the fact that gliders launched from a relatively low height and did not
have time to reach their minimum glide angle, or that they chose a glide angle
according to their intended target. It is interesting to note that the
vertical accelerations were very small, indicating that the gliders were
generating nearly enough aerodynamic force to balance their weight at this
point in the glide, and their glide angles were rapidly decreasing.
Glide angle is geometrically defined by the ratio of vertical velocity to horizontal velocity. Assuming similar launch velocities, the horizontal and vertical velocities at a given time after the launch are determined by the horizontal and vertical accelerations since launching, which are in turn determined by the horizontal and vertical forces. For a given lift and drag coefficient, a more heavily wing-loaded animal will have a greater vertical acceleration (and therefore velocity) at a similar time in the glide than one with lower wing loading because its weight is greater relative to the aerodynamic force it produces, and it will therefore glide more steeply. The findings of this study are consistent with this prediction. Glide angle was strongly associated with wing loading (Table 3), such that animals that were heavier relative to their wing size had steeper instantaneous glide angles.
Velocity
Glide velocity depends on both wing loading and aerodynamic forces. The
body weight of the animal is supported by the vertical component of a
resultant aerodynamic force, which is the vector sum of lift and drag. The
resultant aerodynamic force is given by:
 | (10) |
is the density of
air, V is the glide velocity, CR is a
dimensionless force coefficient and S is the planform area of the
animal. In a steady, non-accelerating glide the resultant aerodynamic force is
equal to the animal's body weight. Substituting body weight for the resultant
aerodynamic force and solving for velocity gives:
 | (11) |
The results of this study conform well to these predictions. Wing loading had a strong positive correlation with velocity (Table 3), indicating that more heavily wing-loaded sugar gliders do, in fact, glide faster than those with lower wing loading. In addition, lift and drag coefficients were negatively correlated with glide velocity (Table 3). A higher lift coefficient means that more lift is generated for a given velocity and wing area, and the same is true for drag coefficient. This means that with higher force coefficients, the amount of force needed to support the body weight is achieved at a lower speed.
Comparison with flying squirrels
By making comparisons of the gliding behavior and performance of diverse
groups of gliding mammals, we can begin to discover general rules for
mammalian gliding. Studies that link the details of gliding kinematics with
ecologically important performance variables are instrumental in providing a
mechanistic basis for such rules. McGuire and Dudley
(McGuire and Dudley, 2005)
made an interspecies comparison of gliding performance in closely related
gliding lizards, examining the effect of body size on gliding performance in
nearly isometrically scaled lizards. In addition, Socha and LaBarbera
(Socha and LaBarbera, 2005)
examined the effect of body size on gliding performance in tree snakes of
different ages in the same species. To date, there have been no interspecies
comparisons investigating convergence in gliding behavior.
A comparison with a study of southern flying squirrels (Glaucomys
volans) under conditions similar to those in this study
(Bishop, 2006) shows that both
species exhibit similar gliding performance in terms of glide velocity, but
sugar gliders had significantly higher glide angles than flying squirrels
(Table 1). Sugar gliders also
used significantly higher angles of attack than flying squirrels, whereas
flying squirrels used significantly greater amounts of relative camber than
sugar gliders (Table 1).
Flying squirrels generated more lift and less drag than sugar gliders. Despite large differences in lift and drag production between flying squirrels and sugar gliders, their glide angles were remarkably similar, indicating that the squirrels did not use the additional lift to flatten their glide trajectory during this phase of the glide, but rather to accelerate horizontally (Table 1). Flying squirrels had both greater horizontal and vertical accelerations than sugar gliders, but had velocities during the captured portion of the glide that were statistically indistinguishable from those of sugar gliders (Table 1). This suggests that flying squirrels would ultimately reach a higher glide velocity than sugar gliders, as expected based on their greater wing loading (Table 1).
There are several possible reasons why flying squirrels tend to produce
greater lift coefficients than sugar gliders. One is that their wings are more
cambered in flight (Table 1).
In addition, flying squirrels possess a well-developed forewing structure
called a propatagium that is present, but much smaller, in sugar gliders. This
flap of skin anterior to the forelimb attaches distally at the pollex and
proximally at the zygomatic arch on the cheek in flying squirrels
(Johnson-Murray, 1977), but
only extends from the neck to the center of the antebrachium in sugar gliders
(Johnson-Murray, 1987). It is
possible that this structure behaves as a leading edge flap which, when
deflected downward, enhances the camber of the wing and can also help to delay
stall at high angles of attack (Wilkinson
et al., 2006).
Glide velocity was related to both force production and wing loading in the sugar gliders in a manner consistent with conventional aerodynamic theory. More heavily wing-loaded animals tended to have faster glides, and greater production of aerodynamic forces (both lift and drag components) tended to result in slower glides. In flying squirrels, however, velocity was positively correlated with lift coefficient, but not correlated with either drag coefficient or wing loading. If the glides were steady, one would expect a negative correlation between velocity and lift coefficient (Eqn 4). This result is explained by the fact that in the flying squirrels high lift coefficients were associated with horizontal acceleration, rather than simply balancing the body weight. In the case of the squirrels, increasing the lift coefficient increased the forward acceleration, which in turn contributed to greater overall velocity.
Conclusions
Although it may be very important in certain circumstances to travel as far
as possible, if an animal cannot control its body orientation and trajectory
well enough to arrive safely at a desired location, optimizing glide distance
does not do much good. Although more information is needed on the effect of
limb movements on flight performance, particularly oscillations intermediate
in amplitude between the small ones seen in this study and those seen in
powered flight, models of the origin of flapping flight should not focus
solely on the generation of lift and thrust forces without taking into account
the role of stability and maneuverability. This study suggests that limb
movements are used differently by individual gliders to control body
rotations, so movements in all orientations should be considered when
investigating the effects of these intermediate amplitude limb movements, and
not focus solely on dorsoventral `flapping'.
It is possible that small amplitude movements of the wings, like those
found in this study, which came about primarily for the purpose of maintaining
stability, may have secondarily increased lift generation
(Norberg, 1985) or thrust
(Vandenberghe et al., 2004),
and could have improved other components of glide performance such as glide
angle or velocity. Studies of low aspect ratio wings at Reynolds numbers
relevant to vertebrate flight have shown that up to aspect ratios of 
2,
aerodynamic performance declines with increasing aspect ratio, particularly at
the high angles of attack used by the gliders in this study
(Torres and Mueller, 2001;
Shyy et al., 2005). But, at
higher aspect ratios, aerodynamic performance increases with increasing aspect
ratio. This may have presented a kind of adaptive barrier during the
transition from a low aspect ratio glider wing to a high aspect ratio bat
wing. It is possible that limb movements leading to flapping behavior provided
the means for overcoming this transition between aerodynamic regimes.
Understanding the relationships between kinematics, force production and
gliding performance across species in the context of disparate performance
parameters, not only improves our understanding of and appreciation for
gliding as a form of locomotion, but will also lead to more fruitful
hypotheses regarding the origin of flight in bats.
List of abbreviations and symbols
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Acknowledgments |
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Footnotes |
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References |
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