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First published online September 19, 2008
Journal of Experimental Biology 211, 3181-3194 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.019844
The kinematic determinants of anuran swimming performance: an inverse and forward dynamics approach
Concord Field Station, Department of Organismic and Evolutionary Biology, Harvard University, Bedford, MA 01730, USA
e-mail: richards{at}fas.harvard.edu
Accepted 12 August 2008
 |
Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Key words: blade element model, forward dynamic model, hydrodynamics, frog, Xenopus laevis
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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;
Blake, 1979;
Fish et al., 1997). These
animals, often called `drag-based' swimmers, rely on resistive hydrodynamic
forces to swim (Daniel, 1984;
Vogel, 1994). Alternatively,
other paired-limb swimmers generate `lift-based' propulsion by the use of
modified morphology and kinematics allowing propulsive limbs to move
dorso-ventrally (e.g. Walker and Westneat,
2002; Johansson and Norberg,
2003). This has stimulated studies investigating how
lift-vs drag-based swimming relates to propulsive efficiency and
swimming speed (Vogel, 1994;
Fish, 1996;
Walker and Westneat, 2000).
More recent work has proposed that frog feet generate lift; however, no
evidence for `lift-based' propulsion has been found in swimming anurans
(Johansson and Lauder, 2004;
Nauwelaerts et al., 2005).
More broadly, these studies relate the kinematics of limb motion to
hydrodynamics as well as swimming performance. Using swimming frogs as a
model, the current study expands on previous work by exploring how swimmers
control different components of motion (i.e. foot rotation and translation) in
order to modulate hydrodynamic forces and swimming performance.
In addition to simple cranio-caudal rotation in rowing, aquatic tetrapod
limbs use joints to further control the propulsor's position with respect to
the body. For example, swimming turtles use proximal joints to control the
medio-lateral position of the forefeet to maximize drag-based thrust during
caudal limb rotation, but minimize drag during the recovery stroke
(Pace et al., 2001). Diving
grebes (Podiceps cristatus) also benefit from the additional range of
motion, generating lift-based thrust by using proximal joints (causing
backward and upward foot motion) while rotating the feet at distal joints
(Johansson and Lindhe Norberg,
2001). Given that jointed limbs confer diverse swimming modes
among species, is kinematic variability a means for controlling swimming
performance within a species? In addition, do the relative roles of limb
joints shift across different swimming behaviors to enable a broad range of
performance within individuals?
Studies of terrestrial locomotion have addressed how the functions of
different limb joints change to enable increases in speed (e.g.
Dutto et al., 2006), incline
(e.g. Roberts and Belliveau,
2005), acceleration (Roberts
and Scales, 2004; McGowan et
al., 2005) and stabilizing responses to substrate height
perturbations (Daley et al.,
2007). Such studies have shown that partitioning of limb function
(e.g. mechanical work production, absorption, stabilization) occurs across
individual limb joints. For example, in wallabies, the ankle serves to store
and return elastic energy during steady speed locomotion
(Biewener and Baudinette,
1995). However, during acceleration the roles of hind limb joints
in turkeys and wallabies change, with the ankle providing most of the
increased mechanical work required to increase speed
(Roberts and Scales, 2004;
McGowan et al., 2005).
Similarly, the ankle shifts from elastic energy recovery (producing little net
joint work) during steady level running in guinea fowl, to energy absorption
following an unexpected drop in substrate height
(Daley et al., 2007). By
analogy, limb joints during swimming may also have distinct functions (e.g.
work production, energy transmission between joints, or joint stabilization).
Presumably, these roles can change according to varying mechanical demands
across different swimming tasks (e.g. predator escape, prey capture and steady
swimming). Understanding how musculoskeletal dynamics enable diverse swimming
behaviors is therefore important for understanding the evolutionary and
ecological diversity of aquatic vertebrates.
Aquatic frogs are ideal models for exploring the differential use of limb
joints to modulate swimming performance. For example, work by Nauwelaerts and
Aerts addressed functions of anuran hind limb joints in swimming vs
jumping to explore how hind limb mechanics enable function across ecological
performance space (Nauwelaerts and Aerts,
2003). They used a novel and elegant approach of analyzing joint
kinematics patterns as functions of both propulsive impulse (`locomotor
effort') and locomotor mode. Their findings demonstrate that kinematic
variation within a locomotor mode (explained by variation in propulsive
impulse) can confound comparisons between jumping and swimming kinematics.
Consequently, their work gives compelling evidence that anurans modulate limb
kinematics to enable a range of performance within as well as between
locomotor modes. However, the mechanistic link between time-varying patterns
of joint motion and performance has not yet been explicitly examined in
swimming frogs.
Given the potential range of kinematics patterns available to frog hind
limbs (Kargo and Rome, 2002),
resolving the functional roles of individual joints may be a daunting task.
However, frog hind limbs move mostly in the frontal plane during swimming
(i.e. within the plane defined by the cranio-caudal and medio-lateral axes)
(Peters et al., 1996).
Therefore, the joint motions can be summed into three components:
cranio-caudal foot translation, medio-lateral foot translation (each caused by
hip and knee rotation) and cranio-caudal foot rotation (from ankle and
tarsometatarsal joint rotation). Several recent studies have speculated on the
importance of translational foot motion, observing that the foot is swept
through the water at nearly 90 deg. to flow for most of the power stroke, with
rotation delayed towards the end of limb extension
(Peters et al., 1996;
Nauwelaerts et al., 2005).
This suggests that foot rotation (via ankle extension) need not
directly aid in propulsion. Instead, the role of foot rotation may be to
straighten the foot parallel to flow to minimize drag just prior to the glide
phase (Peters et al., 1996;
Johansson and Lauder, 2004;
Nauwelaerts et al., 2005).
Johansson and Lauder further suggest that foot rotation serves to shed the
attached vortex from the foot, minimizing a retarding hydrodynamic force
incurred from fluid added mass as the foot decelerates late in the propulsive
phase (Johansson and Lauder,
2004).
Building on this earlier work, my study tests the hypothesis that
propulsion in Xenopus laevis is powered primarily by hip and knee
extension (causing foot translation) rather than foot rotation produced at the
ankle. Increases in speed from stroke to stroke, therefore, are expected to be
powered mainly by increases in translational thrust from the foot. For the
present study, a blade element model modified from an earlier study
(Gal and Blake, 1988b) was
used to dissect the components of thrust due to foot translation and rotation.
Additionally, the blade element kinematic analysis was coupled with a forward
dynamic approach to create a generalized anuran swimming model. This
simulation allowed the modulation of swimming performance through manipulation
of hind limb kinematic patterns. Along with a prior study of plantaris longus
muscle function during X. laevis swimming
(Richards and Biewener, 2007),
the current study provides a framework for interpreting the role of muscle
function in the context of the complex kinematics of jointed appendages.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Inverse model: estimating propulsive forces from joint kinematics input
Thrust was estimated as the sum of two independent hydrodynamic forces
acting at the feet: drag and added mass
(Daniel, 1984;
Gal and Blake, 1988b). In this
model, the feet were the only propulsive surfaces (i.e. propulsive
hydrodynamic effects of the cylindrical leg segments were not considered).
Propulsion was driven by extension of the hip and knee, causing both lateral
and aft-directed foot translation, as well as at the ankle, causing foot
rotation. All equations, therefore, could be expressed in terms of
translational velocity aft to the center of mass (vt),
lateral translational velocity (vl), rotational velocity
about the ankle joint (vr) and velocity of the center of
mass (vCOM) (Fig.
1). In the current study, all velocity components of the hind limb
were defined with respect to the coordinate system illustrated in
Fig. 1. Therefore, aft-directed
foot translational velocity was positive. The drag-based thrust force on each
foot was estimated from a blade element model modified from
(Gal and Blake, 1988b):
 | (1) |
is the water density, 
f is the foot angle
measured from the body midline, r is the distance along the foot,
vr and vt are the velocity components
defined above and a, b and c are dimensions of the foot
(assumed to be symmetric about its mid-axis; see
Fig. 1). Due to a lack of
published literature addressing the coefficient of drag
(CD) of a translating and rotating plate in the range of
Reynolds number (Re) of Xenopus laevis feet (Re
1000 to 20,000), CD was set constant at 2.0, the
maximum value for a flat plate at 90 deg. angle of attack at
Re=103 (Andersen et
al., 2005). A sensitivity analysis to CD was
performed by running 500 model iterations while randomly varying the
CD between 1.1 (Gal
and Blake, 1988b) and 2.0 at each iteration. All other input
parameters (e.g. foot kinematics, added mass coefficients) were left
unchanged. Simulated variation in CD resulted in a
negligible (<2%) variation in the net drag produced during the stroke,
suggesting that the findings predicted by the model are insensitive to
variation in CD within the range of 1.1 to 2.0.
From the added mass coefficients (m), added mass thrust was
calculated (see Appendix A) as:
 | (2) |
 | (3) |
Forward model: simulating swimming velocity from joint kinematics input
A forward dynamics approach was used to computationally solve the
time-varying acceleration and velocity of the frog body due to the
time-varying thrust estimated at the foot (Eqns
1 and
2). The following equations were
used (Nauwelaerts et al.,
2001):
 | (4) |
 | (5) |
; Nauwelaerts and Aerts,
2003). Since swimming acceleration and thrust are functions of one
another, the coupled ordinary differential equations (Eqns
1,
4 and
5) were solved simultaneously
using a numerical equation solver in Mathematica 6.0 (Wolfram Research,
Champaign, IL, USA).
Model verification: predicting swimming velocity from foot kinematics
To verify the numerical model, Xenopus laevis swimming was
recorded for four individuals across the entire range of their performance
(from slow swimming to rapid escape swimming). Joint kinematics data were
measured from video sequences filmed from a dorsal view at 125 frames
s–1 with a 1/250s shutter speed using a high speed camera
(Photron, San Diego, CA, USA), as detailed in a previous study
(Richards and Biewener, 2007).
Small plastic markers (0.5 cm diameter) were placed on the snout, vent, knee,
ankle and tarsometatarsal joint using a cyanoacrylate adhesive. Foot
kinematics were digitized in Matlab (The MathWorks, Natick, MA, USA) using a
customized routine (DLTdataviewer 2.0 written by Tyson Hedrick). Only strokes
with a straight swimming trajectory were analyzed.
Due to the large mass of the legs (11% of body mass), the position of
the center of mass (COM) was assumed to vary depending on the position of the
legs behind the body. Using a dead frog, the position of the COM (relative to
the snout) was measured with the ankle joint moved to various distances caudal
to the vent of the body (see Walter and
Carrier, 2002). Using the measured relationship between the
position of the ankle (relative to the vent) and the COM on the dead frog, the
instantaneous COM position on swimming frogs was estimated from the known aft
translational displacement of the foot.
|
f, vt, vr,
vl,
,
,
) for six trials
were then input into the swimming model (see above) to predict the frog's
swimming velocity and acceleration output
(Fig. 2A–C). Only the
power stroke, defined as the period of positive COM acceleration (i.e. the
period between the onset of swimming and peak COM velocity), was analyzed.
Simulated and observed COM velocity profiles were then compared to verify the
model.
Estimating net joint work and hydrodynamic efficiency
As an index of overall muscular effort required to extend the hip, knee and
ankle to power swimming, estimates of net joint work were obtained by inverse
dynamics (not to be confused with the inverse approach used to estimate
hydrodynamic forces from foot kinematics; see above). The net work required at
a given joint is the sum of internal work (from the inertia of the segments),
external work (from the thrust reaction force at the foot), and hydrodynamic
work (from the hydrodynamic forces acting directly on the segments; see
Appendix B). The thigh and shank were modeled as cylinders
(Fig. 1A) with uniformly
distributed mass, such that the COM lies at the center of each segment length.
Segment masses were measured from the Xenopus laevis frog used above.
Internal and external moments were calculated according to Biewener and Full
(Biewener and Full, 1992). The
internal moment due to inertia of the thigh and shank was calculated as
follows:
 | (6) |
is the segment's angular acceleration, summed
over i=1 to n joint segments distal to the joint of interest
(see Appendix B for further details). The external moment required at each
joint to resist hydrodynamic forces at the foot was calculated and the total
moment at each joint was then obtained:
 | (7) |
 | (8) |
), R is the perpendicular distance between
the Ffoot vector (acting at the center of pressure on the
foot) and the joint of interest, and Mhydrodynamic is the
moment due to hydrodynamic forces acting directly on the segments (see
Appendix B). Instantaneous power required to overcome the total joint moment
was modified from Roberts and Scales
(Roberts and Scales, 2004):
 | (9) |
is the angular velocity of the joint of interest. To follow, the
net work required to move a given joint is:
 | (10) |
Hypothetical performance space
The effects of varying the relative magnitude of translational and
rotational foot velocity on swimming performance (e.g. peak stroke velocity)
were explored by running simulated power strokes across the entire observed
range of translational and rotational velocity in 33 increments, generating a
33x33 matrix of unique input conditions (i.e. 1089 simulated swimming
strokes). Partial least squares regression was used to evaluate the relative
contributions of translational or rotational velocity on swimming performance
(Richards and Biewener,
2007).
Simulated foot kinematics
The range of input conditions in all swimming simulations was bounded by
maximum translational and rotational velocities of 0.8 m s–1
and 60 rad s–1, respectively, obtained from Xenopus
laevis foot velocity measurements of 35 swimming strokes spanning the
entire range of performance of frog 1. Similar velocity ranges were found in
the other three individuals used in this study. The power stroke of anuran
swimmers is often followed by a period where the joints are held in fixed
positions while the frog glides. Since the motion of the feet is impulsive
(rather than periodic), simple sine/cosine functions are inadequate to
describe the translational and rotational foot motion patterns. Hyperbolic
tangent functions (as opposed to sine or cosine functions) were therefore used
to approximate time-varying patterns of translational and rotational
displacement of the foot:
 | (11) |
 | (12) |
i, was derived such that the foot angle was always at 90
deg. to the swimming direction at peak rotational and translational velocity.
The phase angle between rotation and translation, 
, was set to 0 deg. for
all of the simulations in this study. Translational and rotational velocities
were modulated by varying the amplitudes of foot translation and angular
displacement while maintaining a constant stroke period. |
RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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Xenopus laevis foot kinematics
In both representative slow and fast swimming strokes, foot velocity peaked
prior to COM velocity (Fig.
3A,B). For the slow swimming stroke, translation and rotation were
out of phase for the duration of the power stroke, with peak foot
translational and rotational velocities occurring at 38% and 81% of the power
stroke duration, respectively (Fig.
3A). During fast swimming, in contrast, translational and
rotational foot velocity peaked in phase
(Fig. 3B). For these
representative strokes, peak translational and rotational foot velocities
increased 1.7- and 2.1-fold, respectively, from the slow to fast swimming
speeds.
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Kinematic components of thrust: translational and rotational thrust
In all swimming strokes observed, propulsion was predominantly powered by
rotational thrust. In all strokes observed, rotational impulse accounted for
93±24% of total thrust impulse (mean ± s.d., pooled data from 23
swimming strokes, N=4 frogs; Table
2). As exemplified by the two representative strokes chosen, the
time-varying patterns of translational vs rotational thrust produced
by the foot differed markedly in both slow and fast swimming. This variation
accounted for differences in the relative contributions of rotational
vs translational thrust to total thrust
(Fig. 4A,B). Independent of
swimming speed, translational thrust developed earliest, reaching a peak prior
to rotational thrust. Subsequently, translational thrust of the foot rapidly
diminished, becoming negative for the remainder of the stroke, resulting in a
net negative translational impulse in the representative slow stroke. In
contrast, rotational thrust of the foot was a significant component of total
thrust, peaking later in the stroke and remaining positive for the duration of
the power stroke. Because of the net negative translational impulse, total
thrust impulse (translational + rotational impulse) was sometimes less than
rotational impulse (Table 2).
From slow to fast swimming, peak translational thrust increased 2.0-fold from
0.04 to 0.08 N and peak rotational thrust increased 4.4-fold from 0.11 to 0.48
N. In contrast, net translational impulse decreased 3.0-fold from –1.9
to –6.3 N ms, whereas rotational impulse increased 2.5-fold from 11.0 to
27.0 N ms.
|
Hydrodynamic components of thrust: added mass and drag
Similar to the temporal pattern of translational thrust, added mass-based
thrust only contributed to propulsion early in the stroke
(Fig. 4C,D), rising to a peak
in the first half of the propulsive period then decreasing to become negative
at the end of limb extension, resulting in a net positive impulse of 1.1
vs 6.1 N ms in representative slow vs fast swimming strokes.
Similarly, the pattern of drag-based thrust differed only slightly between
slow and fast swimming strokes, peaking after mid-stroke, but becoming
negative in the last 10% of each stroke
(Fig. 4C,D). During slow
swimming, added mass-based thrust peaked 28% of the stroke earlier than
drag-based thrust. However, in fast swimming added mass-based thrust shifted
later and drag-based thrust shifted earlier in the stroke, being nearly in
phase for the representative fast stroke. Drag-based thrust dominated for both
representative swimming speeds, producing an impulse of 7.8 N ms in slow and
15.1 N ms in fast swimming and accounting for 86% and 70% of total thrust
impulse, respectively. Among all four animals observed, however, the relative
contributions of drag-based and added mass-based thrust to total thrust
impulse were highly variable. For three individuals (frogs 2, 3 and 4) net
added mass-based impulse was not significantly different from net drag-based
impulse (P>0.05; Table
3).
|
Kinematic components of added mass-based and drag-based thrust
Added mass-based thrust produced by translational motion gave a net
negative impulse during the representative slow swimming stroke. Rotational
added mass, however, was sufficient to overcome the negative translational
added-mass impulse, causing the net added mass-based impulse to be positive.
In contrast, translational and rotational motion contributed equally to the
observed added mass-based impulse during fast swimming
(Fig. 4E,F). Another notable
difference was the phase offset of 31% vs 5% of translational and
rotational added mass-based thrust in slow vs fast swimming,
respectively.
Both representative slow and fast swimming strokes showed a high contribution of rotational motion to total drag-based thrust (Fig. 4G,H). In both slow and fast strokes, rotational drag was positive for the duration of the propulsive phase. From slow to fast swimming, net drag impulse due to rotation increased from 9.1 N ms to 24.2 N ms. However, translational drag was mostly negative, resulting in negative net impulses that decreased from –1.31 N ms to –9.18 N ms in slow vs fast swimming.
Simulated anuran swimming: modeling stroke-to-stroke modulation of swimming velocity
Modulating the relative magnitudes of translational and rotational velocity
in the numerical model, as described above, caused marked differences in
simulated swimming performance among power strokes
(Fig. 5). The model predicted a
maximal swimming velocity of 1.2 m s–1 with maximum
translational and rotational velocities (0.8 m s–1 and 60 rad
s–1, respectively; Fig.
6). A simulation with pure rotational velocity
(Fig. 5A) reached a peak
swimming velocity of 0.54 m s–1 (45% of maximal velocity),
whereas the simulation driven by pure translation only reached 31% of maximal
velocity (Fig. 5C). In the
intermediate case, with 50% maximum translational and rotational velocity,
simulated swimming velocity peaked at 35% of maximal velocity
(Fig. 5B).
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Simulated variation of translational and rotational velocity also strongly affected the thrust profile and the underlying components of thrust (added mass and drag). In each power stroke model, thrust increased in the first half of the stroke, peaked prior to peak COM velocity and fell to a negative peak near the end of the stroke (Fig. 5). Comparing the pure translation to the pure rotation case (Fig. 5A vs Fig. 5C), added mass thrust contributed most significantly to overall thrust during pure translation, with a peak added mass to peak drag thrust ratio of 1.7, as opposed to a ratio of 0.5 in the pure rotational case. Since peak added mass thrust preceded peak drag-based thrust in all simulations, this change in the relative contributions of these hydrodynamic components caused a corresponding shift in the timing of peak thrust from 0.33 to 0.36 s in the pure translation vs the pure rotation model, respectively. Additionally, the ratio of peak positive thrust to peak negative thrust decreased from 2.9 to 2.0 to 1.5 as the ratio of translational to rotational velocity was increased from 0:1 to 1:1 to 1:0, corresponding to the increased negative added mass thrust incurred by translational motion (Fig. 5).
Hypothetical anuran swimming performance space
The dependence of two performance parameters, peak stroke velocity (the
peak swimming velocity reached in the stroke), and glide velocity (the final
stroke velocity at the end of the power stroke), was tested against the
relative magnitude of translational vs rotational foot velocity. As
reported above, stroke velocity peaked before the end of limb extension.
Consequently, in all simulations the velocity entering the glide phase was
less than peak stroke velocity. Both of these parameters depended strongly on
the magnitudes of translational and rotational foot velocity, with maximal
performance predicted at the highest translational and rotational velocity
(Fig. 7). Peak stroke velocity
increased linearly (as indicated by the parallel straight diagonal contour
lines) with peak translational velocity, but more strongly with rotational
velocity. Partial least squares regression indicated that 60% of the variation
in peak stroke velocity (among the 1089 simulated trials) was modulated by
changes in rotational velocity alone. The swimming stroke powered by maximum
foot translational velocity (zero rotation) reached 31% of the maximal
velocity achieved with full rotation and translation, whereas maximal pure
rotation produced a peak stroke velocity of 45% maximum
(Fig. 7A). In contrast, pure
translational velocity produced a glide velocity of only 15% maximum compared
with 44% maximum with pure rotational velocity
(Fig. 7B). Moreover, 76% of the
variation in glide velocity (compared with 60% of variation in peak stroke
velocity) was explained by modulation of rotational velocity.
|
50% maximum
translational velocity (Fig.
8C). Moreover, percentage deceleration was mainly dependent on
rotational velocity, with the highest values predicted for simulations lacking
rotational foot motion (Fig.
8D).
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|
DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
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;
Johansson and Lauder, 2004;
Nauwelaerts et al., 2005),
hydrodynamic thrust was hypothesized to be produced mainly from foot
translational velocity and acceleration early in the extension phase.
Therefore, stroke-to-stroke increases in translational velocity were expected
to cause increases in peak swimming speed. Data from this study do not support
either hypothesis. In all of the stroke cycles observed, the thrust impulse
produced by rotational motion was significantly higher than the translational
impulse (P<0.05 for each frog;
Table 3) and translational
thrust impulse did not correlate with total thrust impulse
(r2=0.01, P>0.05, N=23 strokes pooled
from four frogs). I propose that these disparities between the present
findings and previous work may be attributed to differences in joint
kinematics patterns as well as leg and foot morphology among anuran
species.
A generalized model for anuran swimming performance
A generalized model for anuran swimming can be used to explore interactions
between hydrodynamics and aspects of performance (e.g. swimming efficiency and
speed). Recent studies have reported low hydrodynamic efficiency for rowing
swimmers at high Reynolds number (Fish,
1996; Walker and Westneat,
2000). Consistent with these previous findings, simulations of
anuran swimming from the current study predict that the total net work summed
over all hind limb joints is high compared with the work required to move the
COM in a single stroke (Fig.
8A,B). This reflects the fact that additional work is required to
overcome hydrodynamic resistance, as well as any work done by muscles to move
body segments. As a result, efficiency is predicted to be low for anuran
swimming over the range of kinematic conditions explored here
(Fig. 8C).
The model was also used to address how swimming speed may be modulated by
variation of kinematic patterns of the feet. Frog hind limbs have a wide range
of joint configurations (Kargo and Rome,
2002) enabling a large repertoire of potential foot motion
patterns. Using a forward dynamic model, one can map the relationship between
foot kinematics and swimming speed by prescribing input joint kinematics and
simulating the frog's swimming velocity output. This allows the examination of
swimming hydrodynamics in the context of kinematic patterns that are
anatomically possible, but not realized in actual X. laevis behavior.
For example, simulations were bounded by two extreme hypothetical cases: (1)
maximal foot translational velocity with no foot rotation (with minimal ankle
action) and (2) maximal foot rotational velocity (with no hip or knee action).
Surprisingly, swimming speed in the pure translation model was lower than in
the opposite case of pure foot rotation (0.38 vs 0.54 m
s–1; Fig.
5A,C). There are two explanations for this result. Firstly,
because the foot rotates very rapidly in X. laevis the maximal
tangential rotational velocity (foot length x foot angular velocity) was
much higher than the translational foot velocity. The highest values observed
in X. laevis swimming (thus the values used as maximum input values
for the simulations) were 2.3 and 0.8 m s–1, for tangential
rotational and translational velocities, respectively (based on frog 1).
Accordingly, peak thrust was higher in the pure rotational vs
translational simulation (0.83 vs 0.50 N, respectively). Secondly,
added mass-based and drag-based thrust are out of phase in the translational
case, whereas they are nearly coincident in the rotational case, enhancing
their cumulative contribution to total thrust
(Fig. 5A,C).
In addition to peak stroke velocity, predicted glide distance was also considered an important performance parameter. Since there is no propulsion during the glide, distance is limited by glide velocity (defined as the swimming velocity at the end of the power stroke). In all power stroke simulations, the thrust impulse was positive for the first half of the stroke and negative for the second half. Because positive thrust always exceeded negative thrust, the net impulse was always positive, resulting in forward swimming velocity throughout each stroke. In the pure foot translational simulation (Fig. 5C), peak negative thrust reached 70% of peak positive thrust. During this stroke, aft-directed translational foot velocity exceeded forward COM velocity, so that the net translational foot velocity (foot translational velocity – COM velocity) was positive throughout limb extension and no negative drag was produced. Therefore, importantly, foot orientation 90 deg. to the flow did not cause drag retarding the forward movement of the body during the power stroke. In this case, negative thrust was produced entirely from added mass effects resulting from foot deceleration (i.e. positive, but decreasing translational velocity). In contrast, the simulation with sub-maximal foot translational and rotational velocities (Fig. 5B) reached a forward COM velocity that exceeded the rearward foot translational velocity, resulting in negative drag-based thrust (due to negative net translational velocity) in addition to negative added mass-based thrust (due to foot deceleration).
Searching the hypothetical performance space between the extremes of foot
motion provides additional insights into the control of swimming performance.
Fish increase swimming speed by increasing their stroke frequency
(Brill and Dizon, 1979;
Rome et al., 1984;
Altringham and Ellerby, 1999;
Swank and Rome, 2000).
Although variation in stroke frequency also occurs in frogs, the relationship
between power stroke period and performance is unclear
(Nauwelaerts et al., 2001). To
avoid potentially confounding effects of stroke duration, power stroke
simulations were run at a constant duration. As expected, simulations with
proportional increases in translational and rotational amplitude (i.e. moving
upwards and rightwards through the parameter space;
Fig. 7A) predict a linear
increase in peak stroke swimming velocity, as indicated by the diagonal
contour lines. However, predicted glide performance did not follow the same
trend (Fig. 7B). Glide velocity
was disproportionately lower than peak stroke velocity in
translation-dominated strokes, especially in the upper left quadrant of the
performance space. In these strokes, rotational motion is too minimal to
counteract the retarding thrust (from relatively large negative force due to
foot translational deceleration; see above). Therefore, the highest COM
deceleration during the power stroke is predicted to occur in the absence of
foot rotation (being largely independent of translational velocity;
Fig. 8D).
By predicting the hydrodynamic roles of translational vs rotational foot motion, this forward dynamic simulation provides a framework for understanding the kinematic determinants of thrust observed in frog swimming.
Dissecting the propulsive mechanism of a generalized Xenopus laevis swimming stroke
Despite observed variation in the temporal patterns of all components of
thrust across 23 strokes (N=4 frogs), most propulsive strokes show
two main phases. In the initial phase (Fig.
3, stages I, II and III), acceleration of the COM is driven mainly
by both net translational velocity and foot acceleration (both translational
and rotational). Propulsion in this phase, therefore, is dominated by
translational drag and total added mass-based thrust. In the final phase
(Fig. 3, stages IV and V),
propulsion is enhanced and sustained by rotational velocity (generating
rotational drag-based thrust), which usually peaks later than translational
velocity. In all strokes observed, net translational velocity peaked in the
first phase, but rapidly decreased to negative values in the second phase of
the stroke as the forward velocity of the COM exceeded the backward
translational velocity of the foot. This has two effects: (1) negative net
translational velocity produces negative drag-based thrust and (2)
translational deceleration (caused by the slowing translational foot motion
toward the end of the power stroke) results in negative added mass-based
thrust. Therefore, the kinematic components of thrust have unique roles in
propulsion: early translational and rotational motion accelerate the frog at
the onset of swimming. As foot rotational velocity increases later in the
stroke, drag-based rotational thrust counteracts and overcomes the negative
components of thrust, causing propulsion to continue until the end of the
power stroke.
Linking kinematic plasticity to hydrodynamics: a proposed mechanism for modulating swimming performance from stroke to stroke
Xenopus laevis hind limb kinematics are highly variable, even
within the behavioral subset of forward, straight and synchronous swimming. In
contrast with the reported `stereotypic' nature of Hymenochirus
boettgeri (Gal and Blake,
1988b), X. laevis modulate time-varying
flexion–extension patterns of the hind limb joints between sequential
kicks of a single swimming burst (C.T.R., unpublished observations). Because
foot motion is the sum of motion produced at the hip, knee, ankle and
tarsometatarsal joints, the relative phases and magnitudes of translational
and rotational velocity vary greatly from stroke to stroke in X.
laevis (Table 1). Despite
this variability, trends emerge. Most notably, peak stroke translational and
rotational velocity are positively correlated across all swimming speeds and
individuals (r2=0.71, P<0.0001, pooled data
from 23 swimming strokes, N=4 frogs). Further, peak COM velocity
correlates with peak translational velocity (r2=0.81,
P<0.0001) and peak rotational velocity
(r2=0.66, P<0.0001), as well as with net
thrust impulse (r2=0.58, P<0.0001).
Surprisingly, although peak translational and rotational velocity are
correlated, the observed increase in thrust impulse among strokes is explained
only by increases in rotational thrust impulse (r2=0.71,
P<0.0001). Total thrust impulse is independent of translational
thrust impulse among the strokes analyzed (r2=0.01,
P=0.63). This suggests that correlative observations of kinematics
and performance can be misleading in the absence of a more detailed
hydrodynamic analysis. Despite the co-variation of translational, rotational
and COM velocity, the underlying drive that provides increased net thrust from
one stroke to the next is the increase of rotational impulse alone. For any
given X. laevis stroke, the time-varying pattern of thrust depends on
both translational and rotational foot motion. However, the shift between slow
and fast swimming appears to require only an increase in net rotational
thrust.
Hydrodynamic model verification and comments on Gal and Blake's model
Before applying the hydrodynamic model to resolve thrust components in
X. laevis swimming, the model was verified using a modified
implementation of Gal and Blake's approach
(Gal and Blake, 1988b) to
determine whether the model adequately estimated the temporal pattern of
propulsive hydrodynamic force during Xenopus laevis swimming.
Differing from that of Gal and Blake (Gal
and Blake, 1988b), the modified blade element model was coupled to
a forward dynamic model to simulate swimming velocity profiles for comparison
to actual velocity profiles from analyzed video sequences (see Materials and
methods). Although this approach generally underestimated the hydrodynamic
forces required to propel the animal, the average error between observed and
simulated swimming velocity was low (Fig.
2D,E). Although improvements to this model may be made for future
studies (see below), the current approach is sufficient to investigate the
hydrodynamic mechanisms for X. laevis swimming examined here.
Gal and Blake (Gal and Blake,
1988a; Gal and Blake,
1988b) used an elegant approach to verify their quasi-steady blade
element model. From analyzed video sequences they calculated the force balance
on the frog's COM required to generate an observed swimming acceleration
pattern. They found a considerable disparity between thrust estimated by their
blade element model and thrust predicted from the force balance on the frog
body. Thrust estimated from the blade element model peaked at the beginning of
limb extension and fell to a minimum (negative thrust) towards the end of the
power stroke. Conversely, COM thrust calculated from the force balance peaked
at mid-stroke and remained positive throughout the limb extension period.
Although Gal and Blake acknowledged potential error from their added mass
calculations, they proposed that a jet (produced as the feet move toward the
midline) may account for thrust not predicted from blade element theory
(Gal and Blake, 1988a;
Gal and Blake, 1988b). I
propose two additional explanations for their findings. Firstly, Gal and Blake
did not account for the rotational component of added mass-based thrust, which
contributes substantially to thrust in X. laevis (71±26% of
total added mass impulse) and, therefore, may be important in Hymenochirus
boettgeri. Secondly, Gal and Blake's method for measuring COM velocity
may not be accurate. They used the animal's vent to mark the COM. However, if
the mass of H. boettgeri hind limbs is a significant portion of whole
body mass, motion of the legs would affect the COM position on the body.
Consequently, as the legs extend backwards the COM would also shift back,
causing COM velocity to be lower compared with the velocity of a fixed point
on the body. In X. laevis, hind limb motion resulted in a 16% change
in COM position relative to snout–vent body length. Because of this,
small modifications to Gal and Blake's model were used to correct for these
potential concerns. Nevertheless, despite these limitations, Gal and Blake's
model is a highly useful tool for resolving the complex mechanism by which
anurans propel themselves through water.
Further modifications to Gal and Blake's model
Small discrepancies between simulated and observed time-varying swimming
velocity may be resolved by future modifications of Gal and Blake's model
(Gal and Blake, 1988b). For
example, foot shape was approximated as a flat plate, yet X. laevis
feet are thin extensible membranes supported by flexible digits. Consequently,
foot shape may be dynamically changed through the power stroke, possibly
allowing the foot to form a concave surface in flow, thus increasing the
foot's drag coefficient considerably. For example, fish pectoral fins show
impressive flexibility, affecting the time-varying hydrodynamic performance of
the hydrofoil (Lauder et al.,
2006). Additionally, controlled changes in the
adduction–abduction angle between digits may affect the foot's projected
area into the flow, possibly increasing area near mid-stroke (maximizing
drag-based thrust) then decreasing area at the end of the stroke (reducing the
negative added mass-based thrust). Measurement of detailed 3D foot kinematics
that better describe time-dependent hydrodynamic coefficients would improve
the accuracy of the model. Furthermore, inputs to the model (e.g. initial
joint positions, joint excursions and relative phases of joint motion) could
also be expanded to better describe the complex kinematic variation observed
both within and among anuran species.
Diversity of anuran propulsive mechanisms
Recent studies have used particular species as models to understand the
generalized principles of anuran swimming. However, findings in Rana
pipiens (Peters et al.,
1996; Johansson and Lauder,
2004) and Rana esculenta
(Nauwelaerts and Aerts, 2003;
Nawelaerts et al., 2005; Stamhuis and
Nauwelaerts, 2005) differ from observations made on pipid frogs,
such as Hymenochirus boettgeri
(Gal and Blake, 1988a;
Gal and Blake, 1988b) and
Xenopus laevis (this study). For example, flow analyses of frog
swimming, using digital particle image velocimetry
(Johansson and Lauder, 2004;
Nauwelaerts et al., 2005),
show no evidence for a central propulsive jet formed by hydrodynamic
interactions of the two legs, as proposed in Gal and Blake
(Gal and Blake, 1988b). Yet,
the kinematics of R. pipiens and R. esculenta differ
strikingly from those of H. boettgeri. Therefore, these species are
unlikely to show similar propulsive mechanisms. Likewise, the predominance of
rotational foot motion observed in X. laevis need not negate earlier
findings (Peters et al., 1996;
Johansson and Lauder, 2004;
Nauwelaerts et al., 2005) that
thrust is powered mainly by translational foot motion (vs rotational
motion) in other species. Each of these species has a different limb
morphology and employs unique kinematics patterns during swimming. These
differences motivate continued exploration of the diversity of hydrodynamic
mechanisms evolved in anuran swimming related to their morphological and
ecological diversification.
|
APPENDIX A |
|---|
|
TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
|---|
) to resolve
added mass coefficients for translational, rotational and coupled
translation–rotation force components. Each added mass coefficient,
mij, represents a component of added mass in the
ith direction of translation (i=1 or 2 for cranio-caudal or
medio-lateral translation, respectively) or rotation about the z-axis
(i=6) causing a force in the jth direction. For example,
m61 represents the added mass coefficient describing
rotation about the z-axis causing an aft-directed force. Because the
limb was assumed to move only in the frog's frontal plane (1–2 plane),
only two components of translation i=1 (cranio-caudal axis) and
i=2 (medio-lateral axis) and a single rotational component
i=6 (ankle flexion–extension axis) were required to give the
added mass tensor:
 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
f is the angle of the foot (with respect to the body
midline), is water density, r is the distance from the ankle
joint and a, b and c are dimensions of the foot
(Fig. 1). |
APPENDIX B |
|---|
|
TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
|---|
):
 | (B1) |
) and
added mass of each segment was estimated as the volume of the cylindrical
segment (Newman, 1977):
 | (B2) |
is the water density, rs is the cylindrical
segment's radius and l is the segment's length.
The hydrodynamic center of pressure (COP) on the foot was estimated as the
weighted average of incremental forces (due to drag and added mass) occurring
along the length of the foot:
 | (B3) |
|
Acknowledgments |
|---|
|
References |
|---|
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BReferences |
|---|
Altringham, J. and Ellerby, D. (1999). Fish swimming: patterns in muscle function. J. Exp. Biol. 202,3397 -3403.[Abstract]
Andersen, A., Pesavento, U. and Wang, Z. J. (2005). Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541,65 -90.[CrossRef]
Biewener, A. and Baudinette, R. (1995). In vivo muscle force and elastic energy storage during steady-speed hopping of tammar wallabies (Macropus eugenii). J. Exp. Biol. 198,1829 -1841.[Medline]
Biewener, A. and Full, R. J. (1992). Force platform and kinematic analysis. In Biomechanics (ed. A. Biewener), pp. 45-73. New York: Oxford University Press.
Blake, R. W. (1979). The mechanics of labriform
locomotion i. Labriform locomotion in the angelfish (Pterophyllum
eimekei): an analysis of the power stroke. J. Exp.
Biol. 82,255
-271.
Blake, R. W. (1985). Hydrodynamics of swimming in the water boatman, Cenocorixa bifida. Canadian J. Zool. 64,1606 -1613.[CrossRef]
Brill, R. W. and Dizon, A. E. (1979). Red and white muscle fibre activity in swimming skipjack tuna, Katsuwonus pelamis (L.). J. Fish Biol. 15,679 -685.[CrossRef]
Daley, M. A., Felix, G. and Biewener, A. A.
(2007). Running stability is enhanced by a proximo-distal
gradient in joint neuromechanical control. J. Exp.
Biol. 210,383
-394.
Daniel, T. L. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24,121 -134.
Dutto, D. J., Hoyt, D. F., Clayton, H. M., Cogger, E. A. and
Wickler, S. J. (2006). Joint work and power for both the
forelimb and hindlimb during trotting in the horse. J. Exp.
Biol. 209,3990
-3999.
Fish, F. (1996). Transitions from drag-based to lift-based propulsion in mammalian swimming. Am. Zool. 36,628 -641.
Fish, F., Baudinette, R., Frappell, P. and Sarre, M. (1997). Energetics of swimming by the platypus Ornithorhynchus anatinus: metabolic effort associated with rowing. J. Exp. Biol. 200,2647 -2652.[Abstract]
Gal, J. M. and Blake, R. W. (1988a).
Biomechanics of frog swimming: I. Estimation of the propulsive force generated
by Hymenochirus Boettgeri. J. Exp. Biol.
138,399
-411.
Gal, J. M. and Blake, R. W. (1988b).
Biomechanics of frog swimming: II. Mechanics of the limb-beat cycle in
Hymenochirus Boettgeri. J. Exp. Biol.
138,413
-429.
Johansson, L. and Lindhe Norberg, U. M. (2001). Lift-based paddling in diving grebe. J. Exp. Biol. 204,1687 -1696.[Abstract]
Johansson, L. C. and Lauder, G. V. (2004).
Hydrodynamics of surface swimming in leopard frogs (Rana pipiens).
J. Exp. Biol. 207,3945
-3958.
Johansson, L. C. and Norberg, R. A. (2003). Delta-wing function of webbed feet gives hydrodynamic lift for swimming propulsion in birds. Nature 424, 65.[CrossRef][Medline]
Kargo, W. K. and Rome, L. C. (2002). Functional
morphology of proximal hindlimb muscles in the frog Rana pipiens.J. Exp. Biol. 205,1987
-2004.
Lauder, G. V., Madden, P. G., Mittal, R., Dong, H. and Bozkurttas, M. (2006). Locomotion with flexible propulsors: I. Experimental analysis of pectoral fin swimming in sunfish. Bioinspir. Biomim. 1,s25 -s34.[CrossRef]
McGowan, C. P., Baudinette, R. V. and Biewener, A. A.
(2005). Joint work and power associated with acceleration and
deceleration in tammar wallabies (Macropus eugenii). J.
Exp. Biol. 208,41
-53.
Nauwelaerts, S. and Aerts, P. (2003).
Propulsive impulse as a covarying performance measure in the comparison of the
kinematics of swimming and jumping in frogs. J. Exp.
Biol. 206,4341
-4351.
Nauwelaerts, S., Aerts, P. and D'Août, K. (2001). Speed modulation in swimming frogs. J. Mot. Behavior 33,265 -272.[Medline]
Nauwelaerts, S., Stamhuis, E. J. and Aerts, P.
(2005). Propulsive force calculations in swimming frogs I. A
momentum-impulse approach. J. Exp. Biol.
208,1435
-1443.
Newman, J. N. (1977). Marine Hydrodynamics. Cambridge: The MIT Press.
Pace, C. M., Blob, R. W. and Westneat, M. W.
(2001). Comparative kinematics of the forelimb during swimming in
red-eared slider (Trachemys scripta) and spiny softshell (Apalone
spinifera) turtles. J. Exp. Biol.
204,3261
-3271.
Peters, S. E., Kamel, L. T. and Bashor, D. P. (1996). Hopping and swimming in the Leopard Frog, Rana pipiens: I. Step Cycles and Kinematics. J. Morphol. 230,1 -16.[CrossRef][Medline]
Richards, C. T. and Biewener, A. A. (2007).
Modulation of in vivo muscle power output during swimming in the African
clawed frog (Xenopus laevis). J. Exp. Biol.
210,3147
-3159.
Roberts, T. J. and Belliveau, R. A. (2005).
Sources of mechanical power for uphill running in humans. J. Exp.
Biol. 208,1963
-1970.
Roberts, T. J. and Scales, J. A. (2004).
Adjusting muscle function to demand: joint work during acceleration in wild
turkeys. J. Exp. Biol.
207,4165
-4174.
Rome, L. C., Loughna, P. T. and Goldspink, G.
(1984). Muscle fiber activity in carp as a function of swimming
speed and muscle temperature. Am. J. Physiol. Regul. Integr. Comp.
Physiol. 247,R272
-R279.
Stamhuis, E. J. and Nauwelaerts, S. (2005).
Propulsive force calculations in swimming frogs II. Application of a vortex
ring model to DPIV data. J. Exp. Biol.
208,1445
-1451.
Swank, D. and Rome, L. (2000). The influence of temperature on power production during swimming. I. In vivo length change and stimulation pattern. J. Exp. Biol. 203,321 -331.[Abstract]
Van Wassenbergh, S., Strother, J. A., Flammang, B. E.,
Ferry-Graham, L. A. and Aerts, P. (2008). Extremely fast prey
capture in pipefish is powered by elastic recoil. J. R. Soc.
Interface 5,285
-296.
Vogel, S. (1994). Life in Moving Fluids. Princeton: Princeton University Press.
Walker, J. A., Westneat, M. W. (2000).
Mechanical performance of aquatic rowing and flying. Proc. R. Soc.
Lond., B, Biol. Sci. 267,1875
-1881.
Walker, J. A. and Westneat, M. W. (2002).
Performance limits of labriform propulsion and correlates with fin shape and
motion. J. Exp. Biol.
205,177
-187.
Walter, R. M. and Carrier, D. R. (2002).
Scaling of rotational inertia in murine rodents and two species of lizard.
J. Exp. Biol. 205,2135
-2141.
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