First published online May 5, 2005
Journal of Experimental Biology 208, 1817-1833 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01579
A 3-D kinematic analysis of gliding in a flying snake, Chrysopelea paradisi
John J. Socha1,*,
Tony O'Dempsey2 and
Michael LaBarbera1
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
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Fig. 1. Setup and trajectory variables. (A) Observational setup. Snakes were
launched from a horizontal branch (height=9.62 m) at the top of a scaffolding
tower in the Singapore Zoological Gardens. Two digital videocameras recorded a
stereo view of the trajectories from a position approximately 2.5 m above the
branch (Position 1). In a few trials, the cameras were placed lower on the
tower (at a height of approximately 5.8 m; Position 2) to record a closer view
of the gliding phase of the trajectory (grey arrows). Markers were placed in
the field in a rough 2 m grid to serve as reference points for the 3-D
coordinate reconstruction; an average of eight were used per trial. An
arbitrary reference system was set up with the Y axis projecting
forward (perpendicular to the tower face), the X axis to the side
(parallel to the tower face) and the Z axis in the vertical, with the
origin (+) located at the ground directly beneath the distal tip of the
branch. (B) Definition of glide angle and horizontal body angle. Sequence is a
side view of a trajectory at an early stage of the shallowing glide. Points
represent the head/body junction (triangles), body midpoint (circles) and vent
(squares) from one snake during one trajectory, sampled at 30 Hz. Temporal
sequence is from upper left to lower right. Instantaneous glide angle (inset,
lower left) was calculated as the angle between the principal axis of
variation of three consecutive midpoint coordinates and the horizontal plane.
Anterior and posterior horizontal body angles (inset, upper right corner) were
calculated as the angle between a line connecting the head to the midpoint and
the horizontal plane (HBAA) and as the angle between a
line connecting the midpoint to the vent and the horizontal plane
(HBAP). HBAP values were given a
negative sign convention relative to HBAA so that equal
angles indicate equivalent body postures. Scale bar, 20 cm.
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Fig. 2. Pooled RMS coordinate errors for midpoint coordinates. Each box represents
the standard quartiles of the pooled distribution of all 14 trajectories at
each spatial interval. Distance traveled represents the straight-line distance
from the respective trajectory coordinate to the takeoff location. Error bars
represent 10th and 90th percentiles, respectively.
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Fig. 3. Overhead view of snake's landmarks during aerial undulation. Points
represent the head/body junction (triangles), body midpoint (circles) and vent
(squares) from one snake during one trajectory, sampled at 60 Hz. Temporal
sequence is from bottom to top. As the snake moves forward along the
trajectory, traveling waves move posteriorly down the snake, producing a
side-to-side undulatory pattern in which the head and vent move out of phase
with the midpoint. At the beginning of the sequence shown, the head is moving
to the left relative to the midpoint. The frequency of undulation and maximum
wave height (approximately the distance between the paired black arrows) were
1.3 Hz and 24% SVL, respectively, with an average airspeed of 6 m
s-1. Scale bar, 20 cm.
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Fig. 4. Trajectory and glide angle summary plots. Fourteen trajectories, each from
a different snake, are shown. (A) Side view of trajectories. Data are
unsmoothed 3-D coordinates sampled at 30 Hz, rotated about the average heading
angle prior to plotting. The gray shading represents the range of trajectory
space of the trials. Trajectories are similar in the first 5 m of vertical
drop and then diverge, with the snakes shallowing at different rates. (B)
Pooled glide angle through time. Each box represents the standard quartiles of
the pooled distribution of all 14 trajectories at each time interval. Error
bars represent 10th and 90th percentiles, respectively. The dotted line
represents the glide angle of a theoretical projectile launched with an
initial horizontal velocity of 1.7 m s-1. For the snakes, glide
angle began near zero, increased rapidly and deviated from the theoretical
projectile early, approximately where aerial undulation began.
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Fig. 5. Summary plots of airspeed (A), sinking speed (B) and horizontal speed (C)
vs time. Each box represents the standard quartiles of the pooled
distribution of all 14 trajectories at each time interval. Error bars
represent 10th and 90th percentiles, respectively. All speeds increased early
in the trajectory. Airspeed and sinking speed leveled off after about 1 s. The
horizontal speed increased throughout. Note the temporary decrease in variance
near the transition points (indicating the end of the initial acceleration) in
each speed plot (arrows).
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Fig. 6. Speed and glide angle vs time for one trajectory with possible
equilibrium component. At a time late in the trajectory (gray shading),
airspeed (thick black line), sinking speed (thick gray line), horizontal speed
(thin black line) and glide angle (red line) all exhibit generally constant
values. The cause of the oscillations in glide angle is unknown - they may be
the result of the midpoint of the snake not coinciding with the effective
center of mass, or they may be due to an increase in coordinate error as the
snake moves away from the cameras. Mass=11 g, SVL=47 cm.
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Fig. 7. Photographs of postural changes during aerial trajectory. (A) Ballistic
dive, ventral view. Silhouettes composited from four consecutive photographs
(200 ms intervals) of a snake (M=36 g, SVL=69 cm) at the beginning of the
trajectory. Temporal sequence is from bottom to top. The white arrow shows the
location of the first traveling waving formed by the snake; in successive
silhouettes this wave can be seen moving posteriorly down the snake. The
snake's side-to-side width increases and head-to-tail length decreases as the
`S' is formed. Scale bar, 10 cm; because the snake moved closer to the camera
during the sequence, scale bar only applies to first silhouette. (B,C) Early
shallowing glide phase, lateral view. In the vertical axis, the vent moves
upward relative to the head and midpoint; in the lateral axis, the head moves
to the right and the vent moves to the left relative to the snake's fore-aft
axis. Interval between frames is 250 ms. (M=28 g, SVL=63 cm.) (D) Late
shallowing glide phase, posterolateral view. The anterior body is
approximately parallel with the ground and the posterior body is angled
downward. The white paint marks are the midpoint and vent landmarks. (M=68 g,
SVL=82 cm.)
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Fig. 8. Overhead and side views of postural changes through time in one trajectory.
Lines represent the connections between the head, midpoint and vent, as in
Figs 1 and
3. Postures are aligned by the
midpoint, with the gray arrows representing the snake's forward direction of
travel along the trajectory; temporal sequence is from left to right. The two
gaps represent missing data. One full undulatory cycle is represented between
the two dotted lines. In every cycle, the posterior segment swung lower than
the anterior segment and sometimes passed in front of the midpoint. This part
of the cycle (black bars) accounted for approximately half the cycle. (M=27 g,
SVL=63 cm.)
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Fig. 9. Vertical, fore-aft and lateral excursion vs time through one
trajectory. Excursion is the perpendicular distance, standardized by SVL, of
the head (triangles) or vent (crosses) relative to the midpoint. (A) Vertical
excursion. The head is initially higher than the other landmarks. The snake
then pitches forward (indicated by bar) such that the tail is higher than the
midpoint, which is higher than the head. At the arrow, the vent is brought
down to the level of the head and midpoint and rises up and down thereafter.
The head stays at a relatively constant level. (B) Fore-aft excursion. The
head and vent are brought towards the midpoint in the S-formation phase (bar).
The head shows little movement thereafter. The vent moves in slightly cyclic
fashion fore and aft. For both fore-aft and vertical excursion, the error
increases substantially in the last 0.4 s of the trajectory. C. Lateral
excursion. Head and vent undulatory movements are clearly shown. The head
undulation (first arrow) begins before the vent undulation (second arrow). In
fully developed undulation, the head and vent are in phase, with the head
excursion smaller than the vent excursion. The gray bar represents one full
undulatory cycle. (M=36 g, SVL=69 cm.)
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Fig. 10. Side view (A), overhead view (B) and rear view (C) of excursion during the
shallowing phase in one trajectory. The midpoint is at the center of each
figure. The beginning of the sequence is indicated with gray circles. The
first undulatory cycle is represented by thick lines. Missing data are
represented by dashes. For most of the sequence, the head is roughly
vertically aligned and about 20% SVL forward of the midpoint. The vent begins
slightly lower than midpoint and is moved downward and forward in the second
undulatory cycle. The head and vent move side-to-side in phase, with the head
using a smaller amplitude than the vent. (M=26 g, SVL=62 cm.)
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Fig. 11. Horizontal body angle (HBA) and body angle of attack vs
time in the shallowing glide phase of one trajectory. The videocameras were
stationed lower on the tower to record the end of the trajectory only; as such
an arbitrary time was assigned to the beginning of the sequence. A. Anterior
HBA (filled circles) was relatively constant around 0°, meaning that the
anterior body was oriented approximately parallel with the ground. The
posterior HBA (unfilled circles) cycled to greater than 60°, indicating
that the posterior body swung below the horizontal. B. The pattern of body
angle of attack is similar to that of HBA. However, because the glide angle
was decreasing throughout this sequence, both anterior and posterior body
angles of attack declined relative to HBA. Only the anterior body angle of
attack, which changed from about 40° to 20° during the sequence, is a
good proxy for true angle of attack. (M=16 g, SVL=54 cm.)
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Fig. 12. Performance and velocity polar diagrams (following
Tucker, 1998 ) for two
trajectories, representing a small snake (circles; M=11 g, SVL=47 cm) and a
large snake (triangles; M=83 g, SVL=85 cm). The straight lines indicate speed
combinations resulting in constant glide angle. (A) Performance diagram. The
larger snake had a higher initial airspeed and greater rate of sinking speed
increase. Sinking speed increased in linear fashion until reaching transition,
which occurred at a higher airspeed for the larger snake (arrow). (B) Velocity
polar diagram.
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© The Company of Biologists Ltd 2005
