A 3-D kinematic analysis of gliding in a flying snake, Chrysopelea paradisi
1 Department of Organismal Biology and Anatomy, University of Chicago,
Chicago, IL, 60637, USA, USA
2 Leica Geosystems (Singapore) Pte Ltd, 25 International Business Park,
#02-55/56 German Centre, Singapore 609916
* Author for correspondence (e-mail: jjsocha{at}midway.uchicago.edu)
Accepted 9 March 2005
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Summary |
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Key words: snake, gliding, flight, locomotion, performance, kinematics, Chrysopelea paradisi
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Introduction |
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Most previous studies of vertebrate gliders (e.g. flying squirrels) have
determined aerial performance using distance between takeoff and landing
locations to estimate angle of descent and speed (e.g.
Jackson, 1999;
Vernes, 2001
;
Young et al., 2002
; but see
McGuire, 1998
as a counter
example). However, because lift production is a non-linear function of
airspeed (Vogel, 1994
), the
descent of any gliding animal starting from rest must follow a non-linear
path. Therefore studies that focus on a `steady phase' of gliding may not
adequately characterize gliding performance. Furthermore, gliders modify body
posture and wing shape to effect changes in trajectory (e.g.
Jackson, 1999
). How such
behavioral regulation relates to changes in performance is unknown. Compared
with the relatively minor bodily adjustments used by other gliders,
correlations between posture and performance may be most clearly and easily
identified in the large-scale body oscillations used by
Chrysopelea.
The genus Chrysopelea is composed of five species
(Mertens, 1968) and is likely
to be a monophyletic assemblage. Chrysopelea are active, diurnal
snakes that inhabit lowland rainforests and whose diet primarily consists of
lizards and, occasionally, birds and bats (Tweedie,
1960
,
1983
). Beyond these broad
observations of lifestyle, there have been no in-depth studies regarding how
they use aerial locomotion in the wild. In general the advantages of being
able to aerially locomote, even if the animal can only move downward in still
air, are obvious - the ability to fall safely while gaining horizontal
distance potentially broadens an animal's behavioral repertoire and ecospace
(Norberg, 1985
,
1990
;
Rayner, 1981
). Both
parachuters (sensu Oliver,
1951
), which fall a greater vertical distance than travel
horizontally, and gliders, which travel farther horizontally than they fall,
can use aerial locomotion to cross space between trees or to move to the
ground. In the descent, kinetic energy of movement comes from the conversion
of potential energy due to relative height. Not only does this entail less
energy usage than in crawling down the tree, across the ground and up the
target tree (Caple et al.,
1983
; Dial, 2003
;
Norberg, 1981
,
1983
), but in flying the
glider may encounter fewer predators along the way; however, both assertions
are generally lacking in empirical data
(Scheibe and Robins, 1998
). If
they do encounter a non-flying predator while perched on the tree, they can
become airborne to escape. Furthermore, they can potentially chase prey that
become airborne themselves, although this seems to be the least likely to
actually occur in the wild.
In this study, we examine the aerial locomotor ability of C.
paradisi, which has been hypothesized by Mertens
(1970) to be the `true glider'
(i.e. glide angle less than 45° during steady gliding;
Oliver, 1951
) of the flying
snakes. In particular, we address the following questions: is C.
paradisi's aerial descent simply drag-based parachuting in which the
snake falls as a projectile, or does the snake generate enough lift to
introduce a significant horizontal component to the trajectory? If C.
paradisi is indeed a true glider, at what point in the trajectory does
the snake reach equilibrium (Vogel,
1994
)? To answer these questions, it is necessary to examine how
the speed of the snake and the shape of the trajectory change through time and
space, and if changes in aerial locomotor posture relate to changes in
performance within the trajectory. We used video photogrammetry to obtain the
3-D coordinates of the head, midpoint and vent of individual snakes throughout
their aerial trajectory in a semi-natural setting. These coordinates are used
to describe the snake's aerial kinematics in detail and estimate the basic
characteristics of snake flight, and represent the first full 3-D analysis of
any glider's trajectory. Results from this study will serve as a framework for
future studies of flying snake aerodynamics, morphology and evolution.
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Materials and methods |
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Snakes were housed in a non-public animal room in the Reptiles Division of the Singapore Zoological Gardens, following standard approved zoo protocol. Snakes were kept in 10-gallon aquaria with copious branches and water and were fed wild-caught geckos once per week. No trials were conducted on the two days following feeding. Because the animal room was open-air, temperature (25-32°C) and relative humidity (50-70%) were similar to ambient conditions of the snakes' natural habitat. Animal care and experimental procedures were approved by the University of Chicago Animal Care and Use Protocol Committee (IACUC #70963).
Experimental setup
Snakes were launched from a specially constructed scaffolding tower in an
open, grassy field in the Singapore Zoological Gardens
(Fig. 1A). Snakes jumped under
their own power (see Movies 1 and 2 in supplementary material) from a branch
secured to the tower at a height of 9.62 m. The branch was approximately
straight, tapering in diameter from 4 cm at the base to 2 cm at the tip, and
protruded 1 m from the edge of the tower's platform. The branch was chosen for
its sufficient roughness, as Chrysopelea have trouble gripping smooth
or debarked branches. A blue fabric sheet was hung adjacent to the branch to
shield a stand of trees in the rightmost corner of the field from the snake's
field of view. The coordinate system was defined relative to the front of the
tower, with the x-axis oriented to the side, the y-axis
oriented to the front and the z-axis oriented vertically.
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Two Sony DCR-TRV900 digital videocameras (NTSC standard; Tokyo, Japan) were
positioned at the top of the tower, approximately 12 m above the ground. In
accordance with standard photogrammetric techniques, they were placed as far
apart as possible (2 m) to maximize base:height ratio, important to the
3-D coordinate reconstruction process. Each camera was attached to a mount and
secured to the side of the tower to minimize camera vibration. Vibration was
not discernible in the video records; digitization of a fixed point throughout
the snake's trajectory showed only random scatter and was consistent among
trials. For all recording, videocameras used effective shutter speeds between
1/1000-1/2000 s and recorded at 30 interlaced frames per second. Video
sequences were synchronized post hoc by matching short-duration, high
amplitude peaks in the audio signals. The smallest effective focal length
(nominally 4.3 mm, equivalent to 41 mm in 35 mm photography) was used to
obtain the largest field of view. For some trials, the cameras were placed
lower on the tower (at a height of approximately 5.8 m) to record a closer
view of the gliding phase of the trajectory; such trials were used to obtain
greater details of the snake's postural changes in mid-glide.
A grid of evenly distributed reference points (known as control points;
Fig. 1A), covering an area
approximately 8x8 m, was placed within the field of view of the
videocameras as a position and orientation reference. These points provided
the photogrammetric basis for the 3-D determination of the snake's landmark
positions. The relative positioning of the control points was determined to
the nearest 0.1 mm using standard terrestrial surveying techniques. The
digitizable region of the glide arena (in which the videocameras overlapped in
view) was roughly pyramidal in shape, with the apex located at the cameras and
the base ranging from 2 to 14 m from the bottom of the tower.
Two Nikon SLR cameras (models F5 and F100; Tokyo, Japan) were used to
capture specific aspects of posture throughout the trajectory. Two primary
setups were used. The first was at the base of the tower, with the camera
angled upward to record a ventral view of the snakes after takeoff. The camera
was aligned such that the lens axis was roughly perpendicular to the ventral
surface of the snake during that point in the trajectory. In addition to
posture, these images were used to calculate wing loading. The second general
position was on a scaffolding tower located 10 m to the side of the main
launch tower. These views were used to obtain lateral posture information.
Glide trial protocol
During glide trial days, snakes were kept in individual cotton reptile
sacks placed in a styrofoam container in a shaded location. Ambient air
temperature was recorded every half hour with a Kestrel 2000 digital
anemometer/thermometer (Nielsen-Kellerman, PA) placed in the open air at eye
level on the launch platform. On days with wind, we coordinated the release of
the snake onto the branch with lulls in the wind and noted the wind speed
immediately after the glide. A better indicator for the presence and relative
magnitude of wind during a glide was the blue sheet, which waved in winds as
low as 0.1 m s-1 and could be evaluated a posteriori from
the video records. For most trials there was no noticeable wind; the maximum
recorded windspeed was less than 0.5 m s-1.
To create visible landmarks, the snakes were marked on the dorsal surface at the head body junction (hereafter, `head'), body midpoint and vent with a one-centimeter band of non-toxic white paint (Wite-Out, Waterman-BIC, Milford, CT, USA). For the young juvenile snake, care was taken to use a minimum amount of paint to prevent a significant increase in mass of the snake.
In preparation to launch, snakes were removed from the reptile sack and placed by hand onto the proximal end of the branch, with the snake's head facing away from the tower. The snake usually moved toward the end of the branch and either stopped or began takeoff immediately. In trials in which the snake hesitated, it was gently prodded on the posterior body and/or tail to try to elicit an escape response. Snakes that didn't respond within 10 min were removed from the branch.
Upon landing of the snake, two volunteers recaptured the snake and marked the location where the tail contacted the ground. The X, Y position of landing was measured with a tape measure, using a spot on the ground directly below the tip of the branch as the origin (Fig. 1A). Because the volunteer marking the landing location was running and the snake usually kept moving after hitting the ground, there was a fair degree of placement error, sometimes as high as 50 cm. This error was minimized through verification or adjustment from the video records. Landing location data were used to corroborate the reconstructed 3-D trajectory coordinates.
Individual snakes were sampled multiple times per day, with at least 15 min of recovery between trials. Observational days were usually followed by a day of rest. A total of 237 trials were recorded, averaging 11 per snake. Here we analyze a subset of these trials in detail to illustrate specific aspects of kinematics and behavior and to characterize gliding in C. paradisi. Fourteen trajectories, each representing a different snake, were chosen using the following criteria. (1) The size of the snakes spanned the range of the sample. (2) The quality of video was high (in regard to lighting, ease of identifying landmarks and percentage time that the snake's landmarks were in view). (3) Because this study is concerned with the limits of aerial ability, the `best' glide for each snake was used, where `best' was evaluated as greatest horizontal distance traveled. When the farthest trajectory for a particular snake was not suitable for analysis (according to criterion 2), the next farthest trajectory was used.
3-D coordinate reconstruction
The video records of the glide trials were transferred at highest quality
to a Macintosh G4 computer via Firewire (IEEE 1394) using Adobe
Premiere (version 6.0) software, with a raw image size of 720x480
pixels. Sequences were deinterlaced to yield 60 video fields per second and
exported as a series of `pict' image files, which were converted to
maximum-quality `jpg' files using Adobe Photoshop. Synchronization was
verified (and/or adjusted) by comparing movements of the snake in both sets of
images. The jpgs were imported into ERDAS Imagine software (version 8.4; Leica
Geosystems GIS and Mapping, LLC, Atlanta, USA) and converted to ERDAS's
proprietary `img' format. This image conversion process had no effect on image
quality. The head, midpoint and vent landmark points of the snake were
digitized in each image pair at a sampling frequency of 30 Hz. (Although 60 Hz
data were available, an initial test determined that the additional temporal
resolution was unnecessary for interpreting spatial, glide angle and velocity
patterns.)
ERDAS Imagine w/Orthobase (version 8.4) software was used to reconstruct
spatial coordinates in three dimensions. Orthobase uses the direct linear
transformation (DLT) method (Abdel-Aziz and
Karara, 1971) to transform coordinates between a 2-D image plane
through the camera and a 3-D object space. DLT is a commonly used method to
reconstruct 3-D coordinates in studies using non-metric cameras (e.g.
Ambrosio et al., 2001
; de
Groot, 2004; Douglas et al., 2004; Gruen,
1997
; Meintjes et al.,
2002
). Assuming collinearity, the image point, focal point of the
camera and object point are calculated to lie on a straight line (one per
point per image) based on the following DLT equations:
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![]() | (1) |
where X, Y and Z are the spatial coordinates of a point
in 3-D object space, x and y are the two coordinates of that
point mapped onto the 2-D image space, x and
y
are nonlinear systematic errors,
x and
i are random
errors, and the coefficients Li are the 11 DLT parameters.
In this study, each stereo video image pair contained complementary views of
the control points on the ground and the snake in the air. Thus each control
point, for which both the image space and object space coordinates were known,
generated four equations. Using data from a minimum of six control points and
the exterior orientation (the spatial and angular position of the cameras),
the DLT parameters were calculated; these descriptors constitute the DLT
model. After the model was determined, the digitized x, y coordinates
of the snake's head, midpoint and vent landmarks in each image pair were used
used to calculate their respective X, Y and Z coordinates.
See (Chen et al., 1994
;
Gruen, 1997
;
Maas, 1997
;
Yuan and Ryd, 2000
) for
further details of DLT methodology.
Quality of data
The combination of the large size of the glide arena, the low base/height
ratio of the cameras and the small amount of variation in the vertical axis of
the reference points constrained the quality of data produced in this study.
Given the small variance in vertical axis of the reference points,
photogrammetric reconstruction was made possible by measuring the position of
the cameras (within a few centimeters) and restricting their adjustment in
Orthobase such that they were relatively fixed in space (using a standard
deviation of 5 cm). RMS coordinate error (which includes digitization error,
equipment error and 3-D reconstruction error) was calculated for each landmark
using Orthobase. In general, error increased throughout the trajectory
(Fig. 2), a standard feature of
stereophotogrammetric systems (e.g. Yuan
and Ryd, 2000); as the snake moved away from the cameras, it
became smaller in the field of view and the relative parallax between images
decreased, increasing the error. For the midpoint, the mean RMS error ranged
from 1-4 cm, 2-14 cm and 3-13 cm in the X, Y and Z axes,
respectively (Fig. 2). At any
point, the error in the X-axis coordinates was 2-4 times smaller than
that in other two axes. The midpoint error was slightly lower than that of the
head and vent, likely because these points were more difficult to identify.
Landmarks were also generally more difficult to digitize on small snakes
because their dorsal surface areas were smaller. Color contrast was a problem
in the latter portion of the trajectory, with the green and black snake
obscured against the green grass background. In future studies, increased
spatial resolution can be accomplished by increasing the contrast between the
snake and the background, by using more cameras, by placing the cameras closer
to the area of interest and farther apart from each other, and by introducing
more vertical variation amongst the control points.
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Trajectory variables
The following variables were used to characterize the gliding performance
of snakes throughout their aerial trajectory. In general, a glider's
trajectory begins when its entire body becomes airborne. In this study we
ignored the snake's initial upward motion in the air (due to takeoff) and
defined trajectory as starting from the downward motion of the snake. In
practice, the trajectory was taken as the path of the snake from the video
field preceding the first downward movement of the snake's midpoint coordinate
(at the apex of the arc resulting from takeoff) to the video field in which
the midpoint contacted the ground. Although the takeoff was ignored, in all
the trials analyzed here the snake launched itself from the branch using a
vertically looped takeoff, usually of the `J-loop' variety
(Socha, 2002b).
Glide angle ()
Glide angle is the angle between a tangent to the local trajectory of the
snake and the horizontal plane (Fig.
1B). In equilibrium gliding, it is a simple function of the
lift-to-drag ratio (see below) and therefore a primary descriptor of
performance for a glider. The smaller the glide angle, the more horizontal the
trajectory and the farther the snake can potentially travel. Glide angles
reported here are instantaneous glide angles, (n), defined
as the angle between a best-fit line through the coordinates and the
horizontal plane, calculated using sets of three temporally consecutive
coordinates. Because the coordinates are 3-D, the principal axis of variation
(analogous to a least-squares line in 2-D; see
Socha, 2002a
) was used as the
best-fit line. Glide angles were calculated for all three landmarks, but only
those calculated from the midpoint coordinates are reported here; midpoint
coordinates were the most abundant (in terms of visibility) and gave glide
angles with the least variation, which provides evidence that the midpoint was
the landmark closest to the snake's true center of gravity in any postural
configuration. For comparison, glide angles for a theoretical projectile
launched at the same height and initial velocity as the snakes were also
calculated using standard equations of motion. Because a theoretical
projectile encounters no drag or lift, it offers the most conservative
estimate of change in glide angle due to aerodynamic forces on the snake.
Glide ratio
The ratio of horizontal distance traveled to height lost is also commonly
used to describe a glider's performance. The larger the glide ratio, the
farther the glider can travel in a given vertical drop. In equilibrium, glide
ratio is equivalent to the ratio of lift-to-drag and is equal to the cotangent
of the glide angle (Vogel,
1994):
![]() | (2) |
We used the minimum glide angle to calculate glide ratio for a trajectory.
Ballistic dive angle (BD)
At the start of a trajectory, a glider has low airspeed. Because lift
production is approximately proportional to the square of the speed, the
magnitude of lift is initially small. The phase in which the glider falls like
a projectile is the ballistic dive. A glider dropped from rest (with a
positive angle of attack) would start with a glide angle near 90°, and
this angle would decrease as speed increased. Because the snake's jumping
takeoff imparted a horizontal velocity component to the initial trajectory,
glide angle was initially near zero, increased to a maximum and decreased
thereafter. We defined the ballistic dive angle as the maximum glide angle
during this phase. Because the shape of the trajectory is a smooth spatial
function, there was no spatially discrete end to the ballistic dive.
Therefore, the end of the ballistic dive phase was defined as the point where
the glide angle began to decline.
Trajectory shallowing rate
As a glider's speed increases, increasing lift production causes the
trajectory to shallow. The trajectory shallowing rate is defined as the rate
at which the glide angle changes during the shallowing phase. It was
calculated using a least-squares regression of the glide angle as a function
of time. Only the linear portion of the curve (chosen by eye) were used in the
regression.
Speed and acceleration
Airspeed is the speed of the snake along its trajectory, calculated using
the distance between the 3-D coordinates. Prior to calculating airspeed, the
digitized coordinates were smoothed using QuickSAND software
(Walker, 1997) with a Lanczos
five-point moving regression, which takes a weighted average of the two
smoothed values immediately prior to and following the point of interest
(Walker, 1998
). This algorithm
was chosen because it most accurately reproduced the original data. Sinking
speed and horizontal speed were calculated in the same manner as airspeed
using the vertical and horizontal components of the coordinate data,
respectively. Acceleration in the initial trajectory was estimated using
least-squares regression of speed vs time in the region of linear
increase (identified by eye).
Heading angle
Heading angle is the direction of travel of the snake along its trajectory
as projected onto the horizontal plane, using three temporally successive
(X, Y) positions of a given body landmark to define a local flight
path. The angle between a least-squares fit line calculated using these three
points and the Y-axis was the instantaneous heading angle. Separate heading
series were calculated for the head, midpoint and vent, all of which
oscillated about the heading of the snake's center of gravity. Because the
center of gravity was not digitized, an average heading of the snake was
estimated based on the three landmarks. The average heading data were used to
determine the relative positioning of the three landmarks and can be used to
identify turns, which are not examined here.
Postural variables
The following variables were used to quantify the snake's aerial locomotor
behavior, defined as postural changes in the trajectory
(Emerson and Koehl, 1990;
McCay, 2001
). It should be
noted that performance and behavior variables may be conflated, as it is
difficult to assess a priori which features are actively under the
control of the snake, which are a consequence of aerodynamic forces, or
both.
Excursion (vertical, fore-aft, lateral)
The most straightforward metric of posture is the relative spatial
positioning of the landmarks. We defined vertical, fore-aft and lateral
excursion as the perpendicular distance of the head or vent to the midpoint
(relative to the snake's direction of travel, calculated using instantaneous
heading angle).
Horizontal body angle (HBA)
The horizontal body angle is the angle between the body axis and the
horizontal plane (Fig. 1B). It
was calculated as an indicator of the degree of incline of the snake's body
relative to the ground. Horizontal body angles were calculated for both the
anterior (head to midpoint, HBAA) and posterior (midpoint
to vent, HBAP) segments.
Body angle of attack (B)
In aerodynamics, angle of attack is the angle between the chord line of the
airfoil (the line that connects the leading and trailing edges) and the
direction of oncoming airflow. Angle of attack is an important aerodynamic
parameter - in general, as angle of attack increases, lift and drag increase
until stall occurs (Bertin,
2001). Because the orientation of local camber was not recoverable
based on the coordinate data alone, a body angle of attack was used as a proxy
for true angle of attack. We defined body angle of attack as the angle between
a segment's body axis and the local trajectory, calculated by summing the
instantaneous glide angle and the horizontal body angle:
![]() | (3) |
Body angle of attack was calculated separately for both the anterior and posterior body segments.
Wing loading
Wing loading is the weight of the flyer divided by the projected area of
the wings. For gliders, the higher the wing loading, the higher the speed
needed to maintain an equilibrium glide
(Vogel, 1994). Wing loading
also affects maneuverability - all else being equal, gliders with higher wing
loading are generally less maneuverable
(Emerson and Koehl, 1990
;
Norberg et al., 2000
). Because
it is not known which parts of the snake act as a functional `wing', the
projected area of the entire snake (including the tail) was used to calculate
wing loading (WL):
![]() | (4) |
where S is area and W is the weight of the snake. Projected area was measured in NIH Image (version 1.62, National Institutes of Health, Bethesda, Maryland, USA) using photographs of the ventral surface of the snake (e.g. Fig. 3). Images were internally calibrated using the known snout-vent length of the snake. The coordinates of the midline of the snake were digitized in NIH Image, smoothed in QuickSAND and interpolated to 1000 points; the total distance between these points was used as the reference length. Any non-orthogonality of the snake in relation to the camera was not systematically taken into account. Error due to non-orthogonality is a function of the cosine of the angle of rotation away from planar, resulting in an underestimate of wing area. Therefore wing loading values presented here err in the direction of overestimation. For example, if a snake was rotated 25° from orthogonal when photographed, its calculated wing loading would be overestimated by 9%.
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Undulation frequency
The snake's swimming-like motion is a form of aerial undulation, in which
traveling waves pass posteriorly down the body. Undulation frequency, the
frequency of these waves, is defined as the frequency of side-to-side movement
of the head or vent about the axis determined by the direction of travel. It
was calculated as the inverse of the undulation period, the time interval
between successive peaks of head lateral excursion.
Undulation wave height
Undulation wave height is the maximum height of the traveling waves, a
measure of the total width of the snake in the aerial `S' posture
(Fig. 3). Because it was not
possible to measure this width directly from the video records, the maximum
lateral separation between the midpoint and the vent was used as a proxy for
wave height. It is a proxy because the maximum lateral separation between the
midpoint and vent is 50% SVL (by definition) and it is theoretically possible
for the wave height to be greater than 50%. Undulation wave height was
calculated as one-half the vertical distance between peaks of the head and
vent lateral excursion. The maxiumum side-to-side speeds of the head and vent
were estimated by multiplying the undulation wave height by the undulation
frequency.
Undulation wavelength and wave speed
The wavelength of the snake's traveling waves was measured in NIH Image
using the same ventral-view photographs used to calculate wing loading. Wave
speed was calculated by multiplying the wavelength by the undulation
frequency.
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Results |
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Airspeed, sinking speed and horizontal speed changed in characteristic fashion. All initially increased in the first portion of the trajectory as the snake accelerated downward (Fig. 5). After this initial acceleration, the speeds transitioned to stable or to a different rate of increase or decrease. On average (see Table 1), the airspeed at the beginning of the trajectory was 1.7±0.6 m s-1 and increased at a rate of 7±1 m s-2. At 1.11±0.31 s and 4.13±0.51 m height lost, the airspeed transitioned to level at a speed of 8.9±1.4 m s-1 or tapered off. This shift in airspeed is defined as the airspeed transition point. The sinking speed started from zero and increased at a rate of 6±2 m s-2 (which is lower than 9.8 m s-2, the acceleration due to gravity). At a slightly earlier time than the airspeed (0.93±0.22 s and 3.79±0.47 m height lost), the sinking speed transitioned to level at a speed of 6.1±0.7 m s-1 or began decreasing. The horizontal speed initially increased at a rate of 4±1 m s-2 and transitioned at a later point in the trajectory (1.58±0.15 s and 5.6±0.60 m height lost). Just after the transition points for each speed type, the variance in the pooled speed data temporarily decreased (Fig. 5).
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In most trials, the three speeds were never constant at the same time, which indicates that force equilibrium was not achieved. One trial was a possible exception, with a 0.4 s span near the end of the sequence where both glide angle and speed may have been constant (Fig. 6). This possible equilibrium glide occurred after a shallowing glide in which the snake dropped 4.3 m in vertical height in about 1 s. In two other trials, shallowing rate decreased near the end of the sequence, suggesting that these snakes were approaching a steady glide angle. In these three trials that came closest to equilibrium, the snakes were relatively small, with masses of 3-27 g.
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Postural changes
Snakes used two behaviors: a body-organizing behavior early in the
trajectory in which the snake formed a characteristic `S' posture
(`S-formation'; Fig. 7A) and a
high-amplitude form of undulation (`aerial undulation') thereafter. The snake
began S-formation in an approximately straight posture with the three
landmarks generally aligned along the fore-aft axis
(Fig. 8). The snake was
initially oriented in a `nose-up' position, with the head higher than the
midpoint and the midpoint higher than the vent. As the snake fell through the
ballistic dive, an initial traveling wave formed anteriorly approximately 1/8
SVL from the head (Fig. 7A). As
the wave began moving caudally, the head and vent were pulled inward toward
the midpoint in the fore-aft direction, forming a wide `S' shape in overhead
view, and the body pitched forward into a `nose-down' position. This phase
lasted about 0.7 s.
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The end of the S-formation graded into aerial undulation with the anterior segment of the snake beginning to move from side-to-side. As the initial traveling wave passed posteriorly down the body, the posterior segment began undulating about 0.2 s later. In full-body aerial undulation, the vent moved in phase with the head (Fig. 9C). Snakes undulated at an average frequency of 1.4±0.3 Hz, with posteriorly directed waves traveling at an average speed of 0.24±0.03 m s-1 and with an average wavelength of 0.20±0.07 m. The head and vent moved side-to-side with average maximum speeds of 0.18±0.05 and 0.30±0.07 m s-1 and average wave heights of 20±3 and 34±5% SVL, respectively.
|
Aerial undulation was not simple sinusoidal motion in a single plane. The undulation was complex, with the head and vent moving both fore and aft in the horizontal plane, and up and down in the vertical plane (Figs 8, 9, 10). The fore-aft excursion of the head was relatively small and the head never crossed behind the midpoint. The fore-aft and vertical excursions of the vent were larger than those of the head - in most undulatory cycles, the vent dropped below the level of the midpoint and moved forward, sometimes crossing in front of the midpoint. These postural changes were similar from cycle to cycle.
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The horizontal body angle, which was determined by the relative vertical positioning of the landmarks, varied throughout the undulatory phase. The horizontal body angle of the anterior segment was much less variable than that of the posterior segment, which swung downward in every cycle. In one trial in which the cameras were positioned closer to the end of the trajectory, the anterior horizontal body angle remained at approximately 0° (Fig. 11A), indicating that the anterior segment of the snake was undulating in the horizontal plane. Correspondingly, the body angle of attack for the anterior segment was about 35° (Fig. 11B), but declined near the end of the sequence as the glide angle decreased. The posterior body angle of attack was similar to that of the anterior for part of the cycle, but when the posterior segment dropped below the anterior, the angle increased to as much as 110°. Thus, the coordinate data show that different parts of the body experienced different orientations to the oncoming airflow and that these orientations changed through time. There is no `characteristic' body orientation for the snake as a whole.
|
The lateral view photographs provide further evidence of body orientation throughout the trajectory (Fig. 7B-D). Although qualitative in nature, these data help fill in the gaps in the coordinate data, which only provide positional information on three points on the body and do not directly indicate how each local body segment was oriented to the oncoming airflow. The images show that the snake's head and first few centimeters of the body were consistently angled downward with respect to the horizon, regardless of how the rest of the body was configured. The images also confirm that the horizontal body angle is a reasonable proxy for body orientation for the anterior half of the snake. However, it is a misleading metric for the posterior half of the body, which displays much more dramatic displacements in the vertical axis. For example, Fig. 7D shows a case in which the vent is below the midpoint. Here the horizontal body angle for the posterior segment (approximately 90°) would seem to indicate a perpendicular orientation relative to the horizon. However, the photograph shows that the body in between the midpoint and vent consists of two mostly-straight body segments angled laterally downward and connected by a curved segment. In each segment, the ventral surface faces downward, not at a 90° angle to the horizon.
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Discussion |
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Glide speed
The speed patterns provide indirect evidence of the snake's aerodynamic
characteristics. Not surprisingly, the airspeed increased as the snake
initially fell. In most trials, there was a transition point where this
initial acceleration decreased abruptly, indicating a marked decrease in net
force on the snake. Although this shift occurred just after aerial undulation
became fully developed on average, there is no clear pattern within individual
trials that relates kinematics to the timing of this transition. Young et al.
(2002) noticed a similar
transition in speed in parachuting geckos (Ptychozoon kuhli). After
the transition point, airspeed became constant in some trials and increased at
a lower rate in other trials, a difference that may be size dependent. Sinking
speed also shows a transition point almost concurrent with the airspeed
transition. Unlike airspeed, sinking speed decreased in some trials,
indicating that the upward force on the snake was increasing. The temporary
decrease in variance just after the speed transition points may indicate a
physical constraint in which snakes are confined to pass through a particular
speed.
These interpretations are clouded by the variance in the speed data, which
exhibit oscillations of varying period and amplitude (e.g. Figs
6 and
8). These oscillations may be
real, perhaps the result of the midpoint of the snake not coinciding with the
effective center of gravity; as the snake undulates and changes orientation,
the lift-to-drag ratio may be constantly changing, which would constantly
change both speed and glide angle. Unfortunately, errors in the coordinate
data were too high to identify and correlate such fine-scale changes in
posture, speed and glide angle. Observed oscillations in speed may also be due
to systematic error - as the snake moved away from the cameras, the coordinate
position error increased. Furthermore, identifying the center of gravity as
posture changes requires data on the tail's position, which were not collected
here; although the tail is thin, it may have a significant effect on center of
gravity due to its length (30% SVL). Trends in speed should be easier to
identify in the future using longer glide sequences and improved
photogrammetric resolution.
In a comprehensive study of gliding lizards (Draco spp.), McGuire
(1998) observed a
stabilization of airspeed in some trials, but did not observe glides in which
the rate of increase of airspeed simply declined after a transition. In some
trials, the airspeed of the lizard increased and then decreased with no
equilibrium period, which he termed a `velocity peak'. In most of these
velocity peak glides, the lizard landed on a pole; he therefore interpreted
the decrease in airspeed as a preparation for landing. He also observed a
concomitant upswing in the trajectory before landing, so it is likely that the
lizard was stalling in preparation for landing. Flying squirrels are known to
perform a similar stalling behavior by radically increasing the angle of
attack, which slows the squirrel down and pitches the body upward, enabling an
upright landing on a vertical substrate
(Nachtigall, 1979
;
Scholey, 1986
;
Stafford et al., 2002
). None
of the snakes in this study landed on a vertical substrate, but they were not
observed to slow down before landing on the ground.
Glide equilibrium
When discussing a glider's aerodynamic characteristics, it is important to
describe its performance during equilibrium. A glider can use a range of
equilibrium speed and glide angle combinations (usually plotted in a `glide
polar'; see Fig. 12). This
range defines the limits of its gliding ability. If in equilibrium at its
minimum speed, the glider maximizes its time aloft; at its minimum glide
angle, it maximizes its distance traveled. By definition, force equilibrium
assumes that the forces are balanced on the glider and that there is no net
acceleration. There was no unequivocal evidence for force equilibrium on the
snakes given the combination of changing airspeed, sinking speed and
horizontal speed in most analyzed trials. It is therefore unclear what factors
contribute to how quickly the snakes may attain equilibrium. It is also
unclear whether or not the snakes even attempted to reach equilibrium - if
they purposefully chose landing locations closer than their ability limits,
equilibrium gliding may not have been necessary. Regardless, it is
theoretically possible for C. paradisi to attain equilibrium if
launched from a sufficient vertical height. Indeed, unobstructed space for
aerial locomotion is ample in the mixed dipterocarp forests it inhabits;
canopy stratification is pronounced and emergent trees of prominent heights
are frequent (Dudley and DeVries,
1990; Richards,
1996
; Whitmore,
1992
). Considering that some snakes may have reached equilibrium
in this study when jumping from a height of 9.6 m, it is possible that snakes
attain equilibrium gliding when taking off from greater heights in the wild,
given sufficient horizontal space. Conversely, equilibrium gliding may be
precluded due to insufficient space between trees, or because fully developed
snake flight may be inherently unsteady, independent of starting height.
|
One trial showed evidence of equilibrium, with a short region where all
three speeds were approximately constant
(Fig. 6) before the snake
traveled out of view of the cameras. This snake, which consistently gave long
glide sequences, was the second smallest of the sample. It also gave the
farthest glide on record - in an exceptional glide, the snake traveled a
horizontal distance of 21 meters, 30% farther than any of the other 237
trials. Because only the first 1.2 s (22%) of this trajectory was
recorded by videocameras (and therefore was not included as one of the 14
`full' trajectories described herein), the unknown portion of the trajectory
could be not be determined. However, the lowest measured glide angle for
recorded trajectories was 13°, which suggests that the glide angle
in the exceptional trajectory reached magnitudes equal to or lower than
13°. Furthermore, because this glide was much longer than all
others, it seems the likeliest to have contained an equilibrium component.
By comparison, McGuire
(1998) found a large
percentage (48% of 150) of his flying lizard glide trials to have an
equilibrium component. Because the lizards were launched from a lower starting
point (6 vs 9.6 m), they must have used shorter ballistic dives
and/or higher shallowing rates than did the snakes. Generally lizards with
lower wing loadings reached equilibrium more often, but the largest lizards
(with wing loadings of about 25 N m-2) reached it too.
Given that equilibrium did not conclusively occur in the trials analyzed here, the average minimum glide angle reported for C. paradisi, 28°, is a conservative estimate. The range of minimum glide angles was 13-46°, with smaller gliders attaining lower glide angles than larger snakes. However, it is possible that these snakes are capable of reaching the same minimum glide angle given a great enough starting height. If true, then ballistic dive distance, trajectory shallowing rate, or horizontal distance traveled may be better indices of performance than minimum glide angle for C. paradisi.
Aerodynamic characteristics
To estimate the range of aerodynamic characteristics in C.
paradisi, we calculated basic aerodynamic parameters using performance
values from two representative snakes, one small and one large (M=11 and 63 g,
SVL=47 and 83 cm). The small snake was the best glider of any snake observed.
The calculations below use data from a phase late in the trajectories where
the airspeed had stabilized or was approaching stability.
To estimate the snake's average lift coefficient late in the trajectory, we
assume that all forces acting on the snake are in equilibrium and that the
airfoil is parallel with the ground. L and D are the lift
and drag forces, which act perpendicular and parallel to the glide path,
respectively; R is the resultant aerodynamic force, which acts
upward; W is the weight of the snake; and is the angle of
attack. An equation for lift coefficient at high Reynolds numbers
(Vogel, 1994
) is:
![]() | (5) |
where CL is the lift coefficient, is the density
of air at 30°C (the field temperature), S is the projected
surface area in plan view and U is the airspeed. The lift force is
equal to the resultant aerodynamic force, R, multiplied by the cosine
of the glide angle,
![]() | (6) |
Because the forces are in equilibrium, the total aerodynamic force is equal
to the weight of the snake,
![]() | (7) |
Substituting this expression for lift and wing loading (WL) for
the weight divided by the projected surface area in equation 5 yields the
following:
![]() | (8) |
Using an air density value of 1.165 kg m-3 (at 30°C), airspeeds of 7 and 10 m s-1 for the small and large snakes, respectively, and wing loadings of 18 and 31 N m-2, the average calculated lift coefficients are 0.63 and 0.53. Actual instantaneous lift coefficients during the glide may vary substantially.
The Reynolds number (Re) can be calculated following Vogel
(1994). Here we assume that
the flattened width of the snake (approximately twice the resting width;
Socha, 2002b
) is the
characteristic length. Using a kinematic viscosity of
15.94x10-6 m2 s-1 at 30°C and
flattened widths of 1.2 and 2.4 cm for the small and large snakes,
respectively, the approximate Reynolds numbers are 5000 and 15,000.
Aspect ratio is calculated using the span and area of a flyer's wings. However, it is not immediately obvious which parts of the snake act as a functional `wing' from which to measure span. If the span is considered to be the maximum length of snake between curves, the aspect ratios of the small and large snake are about 13 and 11, respectively. If it is assumed that only the relatively straight sections of body between the curves as the `wing', the aspect ratios are about 10 and 8, respectively. In either case, the `aspect ratio' of C. paradisi is high relative to most other vertebrate gliders.
The maximum lift-to-drag ratios can be estimated using the minimum glide angle. With minimum observed glide angles of 20° and 40° for the small and large snake, respectively, the lift-to-drag ratios are 2.7 and 1.2. In the trial with the lowest recorded glide angle (13°), the calculated lift-to-drag ratio is 4.3.
Postural influences on snake flight
Although no precise association of changes in posture with changes in glide
angle were observed here, these data suggest four hypotheses for behavioral
influences on lift generation. (1) In the S-formation phase, more of the
snake's body becomes perpendicular to the airflow as the snake pulls itself
into the `S' posture. In effect, the body changes from a spear to a biplane.
This orientation drastically increases the snake's aspect ratio and therefore
should generate more lift than a mostly straight snake. (2) As the snake
pitches downward, the chords of the snake's `airfoils' rotate into a position
more favorably aligned with the oncoming airflow, suggesting that the snake
becomes less like a bluff body (in which drag dominates) and more like an
airfoil (in which lift dominates). (3) The shallowing glide, in which the
glide angle decreases, begins only after the snake is fully undulating,
suggesting that aerial undulation itself may increase the lift-to-drag ratio.
(4) Alternatively, changes in glide angle are solely a result of increasing
speed due to gravity, suggesting that behavior has no effect on lift
generation. Considering the abrupt transitions in speed, this scenario seems
unlikely. A combination of physical modeling, live-animal manipulation (for
example, attaching weights to the body to alter mass distribution in flight)
and statistical analyses of performance (e.g.
Socha and LaBarbera, 2005) are
required to fully explore these behavioral hypotheses.
Aerial undulation is one of the most striking features of flying snake
locomotion. No other glider makes such pronounced undulatory motions with its
body while gliding, and its function during flight is uncertain. Given that
lateral undulation is the dominant mode of locomotion in snakes
(Pough et al., 2001), aerial
undulation may simply be a behavioral vestige with no function. Some
non-flying snakes, in fact, undulate when dropped from a height (J.J.S.,
unpublished). However, given its prominent role in gliding flight, it seems
unlikely that aerial undulation has no aerodynamic function. The side-to-side
movement adds an additional component of speed over the body in addition to
the airspeed; however, this component is so small that the additional lift it
would produce is marginal. Other possible functions include dynamic
stabilization, in which the changing posture would serve to move the centers
of gravity and pressure in a way that allows controllable flight. In a study
that explicitly addressed the effects of behavior on performance
(Socha and LaBarbera, 2005
),
undulation amplitude was found to have a far greater influence than undulation
frequency. Future modeling studies will detail how specific aspects of
undulation amplitude and frequency relate to aerodynamic force production or
stabilization.
Aerial undulation is kinematically different from lateral undulation on the
ground, in which constant points of contact are maintained, or in water, in
which traveling waves increase in magnitude along the body. In a pilot study,
Socha (1998) recorded the
aerial, terrestrial and aquatic locomotion of four C. paradisi
specimens. For both water and land, the forward speed of the snake was much
lower (1.0-1.5 m s-1), the undulation frequency was almost three
times higher (3-4 Hz) and the wave height was smaller (
7-14% SVL) than in
air. The shape of the waves was also different - during terrestrial
locomotion, the waves were similar in size along the snake, and in swimming,
the size of the waves increased as they traveled down the body. Although both
are consistent with normal snake locomotion, neither undulation pattern
resembles the high-amplitude `S' shape used in aerial undulation.
Comparative performance
Considering the snake's unorthodox locomotor style, it is relevant to ask
how snake aerial locomotion compares to that of other vertebrate gliders.
However, it is first important to note the differences in methods employed in
other gliding studies. Generally, animal gliders have been observed moving
through the air in their natural habitats. In these field studies (e.g.
Ando and Shiraishi, 1993;
Jackson, 1999
;
Scholey, 1986
;
Stafford et al., 2002
;
Vernes, 2001
), glide
performance was estimated using a few basic measurements, such as takeoff
height, landing height and horizontal distance (all usually estimated using a
rangefinder or maps); and aerial time (measured with a stopwatch). Because
straight-line distances were used to calculate performance in these studies,
true glide ratio (valid only at equilibrium) and speed are underestimated -
the animals actually achieved higher glide ratios and higher speeds during the
gliding portion of the trajectory than reported. For some gliders, performance
metrics have only been reported from anecdotal observations, with distances
estimated by eye (e.g. Hyla milaria, as described in
Duellman, 1970
). Whereas some
observers make credible judgments of distance, others seem implausible (e.g.
glide ratios of 11-13 for colugos, Lekagul
and McNeeley, 1977
; and a gliding distance of 450 m for a flying
squirrel, Nowak, 1999
). In
other studies, accuracy cannot be evaluated because glide performance is cited
without mentioning methods (e.g. Kawachi
et al., 1993
). These studies are important in establishing an
estimate of performance and placing gliding in an ecological context, but do
not address kinematic details. By contrast, McGuire's
(1998
) study of 11 species of
flying lizards (Draco), in which he digitized the lateral view of 150
trajectories at 12 Hz to calculate instantaneous glide angles, had a much
higher temporal and spatial resolution than previous studies. However, because
his reported glide ratio values were calculated based on the takeoff and
landing points only, peak performance was also underestimated.
The shape of C. paradisi's trajectory appears to be similar to
that of other gliders, with a steep initial dive followed by a shallowing
glide. Snakes lost 1-5 m of vertical height at a glide angle of 52-62°
during the ballistic dive. In comparison, Scholey
(1986) estimated that the
giant red flying squirrel, Petaurista petaurista (mass
1.3 kg),
traveled through the ballistic dive at a glide angle of 50-70° while
losing 6 m of vertical height, and Vernes
(2001
) used a regression of
horizontal distance vs vertical distance traveled to estimate a
ballistic dive of 1.9 m in the northern flying squirrel, Glaucomys
sabrinus (mass
93 g). However, these comparisons should be viewed
with caution because it is unclear to what extent the inferentially estimated
ballistic dive estimates correspond to the quantitatively measured ballistic
dives in this study.
When considering glide angle, C. paradisi are better gliders than
`parachuting' lizards (Ptychozoon sp.;
Marcellini and Keefer, 1976;
Young et al., 2002
), on a par
with some gliding mammals (e.g. Petaurus breviceps and Petaurista
leucogenys; Nachtigall,
1979
; Ando and Shiraishi,
1993
) and poorer gliders than some gliding lizards (Draco
spp.; McGuire, 1998
) and
flying fish (e.g. Cypselurus spp.;
Davenport, 1992
). As in these
other flyers, C. paradisi is potentially capable of using aerial
locomotion to move effectively between trees, chase aerial prey, or avoid
predators. Airspeed and wing loading also fall within the range of other
gliders (Rayner, 1981
).
There have been fewer accurate reports of body orientation during gliding.
Flying squirrels and flying lizards orient their bodies approximately parallel
with the horizontal plane while gliding
(McGuire, 1998;
Nachtigall, 1979
;
Scholey, 1986
), such that the
angle of attack is approximately the same as the glide angle. This also seems
to be true of C. paradisi when considering the anterior segment
alone. For these gliders, this parallel body orientation may be a compromise
between choosing an angle of attack that maximizes the lift-to-drag ratio and
keeping the landing location and nearby obstacles within the field of
view.
Chrysopelea and Draco, to our knowledge, are the only
predator/prey system in which both taxa glide. It has been suggested (R.
Dudley, personal communication) that their respective gliding abilities may
have been subject to an evolutionary arms race. This hypothesis is plausible
considering their respective gliding performance ability. Draco is
the better glider in some respects, with a higher lift-to-drag ratio, shorter
ballistic dive and likely higher maneuverability, but its range of performance
within the genus overlaps that of C. paradisi, and C.
paradisi may glide faster. Faster gliding is better for linear pursuit,
but once the Draco turns, its lower speed and lower wing loading
would make it difficult to intercept. Furthermore, Chrysopelea
actively hunt their prey, likely have good vision for a snake (see
Socha and Sidor, in press) and
as lizard eaters (Flower,
1899
) are reported to take Draco in their diet
(Wall, 1908
). Historically, it
cannot be determined if their gliding abilities co-evolved, but it is possible
in principal to observe whether Chrysopelea chases Draco in
the wild.
Conclusions
C. paradisi are true gliders. Their style of gliding flight, in
which a cylindrical animal becomes flattened, coils into an `S' shape and
aerially undulates, has no analogue in the natural or engineering world. Its
aerial undulation involves complex kinematics in which the body moves in and
out of the horizontal plane, for which the aerodynamic consequences are
unknown. Although the results presented here unequivocally show that C.
paradisi does produce significant horizontal motion in flight, it is
still unclear which aspects of behavior and morphology are responsible for the
generation of aerodynamic forces. Such forces allow the snake to gain
horizontal distance and to turn, all accomplished without tumbling over and
using no obvious morphological control surfaces. To address these aerodynamic
issues, we have begun to investigate snake flight using physical models to
isolate specific effects of shape, and plan to conduct parallel computational
modeling studies that examine the effects of the snake's complex
kinematics.
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Acknowledgments |
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Footnotes |
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