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First published online December 28, 2007
Journal of Experimental Biology 211, 206-214 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.012161
Research Article, Biomechanics of Swimming |
Biorobotic insights into how animals swim
Naval Undersea Warfare Center, Newport, RI 02841, USA
* Author for correspondence (e-mail: bandyopadhyaypr{at}npt.nuwc.navy.mil)
Accepted 15 October 2007
Summary
Many animals maneuver superbly underwater using their pectoral appendages. These animals range from sunfish, which have flexible, low aspect ratio fins, to penguins, which have relatively stiff, high aspect ratio wings. Biorobotics is a means of gaining insight into the mechanisms these animals use for maneuvering. In this study, experiments were carried out with models of abstracted penguin wings, and hydrodynamic characteristics – in particular, efficiency – were measured directly. A cross-flow vortex model of the unsteady force mechanism was developed that can compute instantaneous lift and drag forces accurately. This makes use of the steady characteristics of the fin and proposes that cross-flow drag vortices of bluff bodies in steady flow are analogous to dynamic stall vortices and that fin oscillation is a means for keeping the vortices attached to the fin. From what has been reported for sunfish with pectoral fins to our current measurements for single abstracted penguin wings, we infer that the maximum hydrodynamic efficiency has remained largely unchanged. A selection algorithm was used to rapidly find the fin oscillation parameters for optimum efficiency. Finally, we compared the measurements on the penguin-like relatively stiff fins and the reported flow visualization of flexible sunfish pectoral fins. The flexible pectoral fins of station-keeping sunfish exhibit a rich repertoire of capability such as the formation of dynamic stall vortices simultaneously on two leading edges during part of the cycle, changes in projected area in different planes, and the vectoring of jets. However, such fins may not be scalable to larger biorobotic vehicles and relatively stiff fins appear to be better suited instead, albeit with somewhat limited station-keeping ability.
Key words: biorobotics, flying, high-lift, swimming
Introduction
The beauty and performance of swimming and flying animals are fascinating
to man. If we consider biology as nature's design and engineering as man-made
design of competing functions, then these two applied disciplines may be
deemed akin (Bandyopadhyay,
2004
). However, there are some differences in approach that
reflect on interpretations. Biorobotics allows synthetic examination of
mechanisms and functions, complications are built over simplicity and
unsteadiness may be considered a special case of steady behavior. On the other
hand, in biology, steady flow, for example, is a special case of unsteadiness.
Here, we sought to conduct controlled and compartmentalized engineering
experiments on abstracted aspects of swimming animals to gain insight into
their mechanisms of propulsion. In particular, we focused on the pectoral
appendages of penguins and sunfish, which are relatively stiff and flexible,
respectively. We sought answers to several questions. (1) Why is it that the
wing and body forms of swimming animals in general are similar to those of
National Advisory Committee for Aeronautics (NACA) profiles but the former are
unsteady while the latter are developed for steady flows? In other words, how
unsteady is the hydrodynamics of the pectoral appendages of swimming and
flying animals? How did unsteady hydrodynamics evolve? How did animals
discover the seemingly counter-intuitive notion that stall vortices can be
harnessed for lift and thrust enhancement? (2) While engineers design based on
their understanding of inherent laws and principles, how do animals evolve to
optimize their efficiencies, albeit with constraints? (3) Is the dynamic stall
mechanism present in flexible fins? Do flexible fins have inherent abilities
of station keeping in disturbed streams? How do they compare with relatively
stiff fins?
The engineering implementation of biology-inspired fluid dynamics has
received much attention since the work of Ellington and Dickinson showing how
dynamic stall is used by flying animals to sustain themselves aloft in a
low-density medium such as air (Ellington,
1984a; Ellington,
1984b; Ellington,
1984c; Ellington,
1984d; Ellington,
1984e; Ellington,
1984f; Dudley and Ellington,
1990; Ellington,
1991; Dickinson,
1996; Dickinson et al.,
1999; Ellington et al.,
1996; Sane and Dickinson,
2002; Birch et al.,
2004; Triantafyllou et al.,
2004; Bandyopadhyay,
2004; Bandyopadhyay,
2005). Similar investigations of the mechanisms of the appendages
of swimming animals, however, are not as advanced
(Lauder and Drucker, 2004),
although progress is now being made
(Lauder et al., 2007). The
engineering significance of these largely model-based `biorobotic'
investigations is that it is indeed possible – after accounting for
penalties – to sustain an entire platform based on the high-lift
mechanism of dynamic stall, a phenomenon that has been known to engineers for
a long time (Bandyopadhyay,
2005; Ellington,
1999). It is frequently cited that animals have wings and bodies
that closely match NACA profiles, although the design and selection of the
NACA profiles are based on steady-state flow. This led Bandyopadhyay, while at
the Office of Naval Research, to commission a medical imaging survey of the
cross-sections of the appendages of swimming animals. The close match with
NACA profiles was found to be widely prevalent (for a collection of medical
cross-sectional digital images of common dolphins, contact Prof. F. Fish,
Westchester University, Weschester, PA, USA), but the role of the foil
cross-section derived from studies on steady-flow section in the unsteady
high-lift mechanism is unclear. The performance of swimming animals greatly
depends on the hydrodynamic efficiencies of their active appendages, such as
the pectoral fin. Estimates of performance based on hydrodynamic models tend
to have large uncertainties and pertain to the animal as a whole, as opposed
to their individual appendages (Fish,
1993). The hydrodynamic efficiencies of pectoral appendages for
cruising and maneuvering still remain unknown. The position and number of
appendages and their morphological scaling depend on whether the animals are
dexterous in cruising or in maneuvering
(Bandyopadhyay et al., 1997).
In this study, we explored how swimming efficiency could be rapidly optimized
without any a priori knowledge of the hydrodynamic mechanisms.
Finally, intrigued by the fact that large, open-water cruising animals (such
as whales, dolphins and penguins) have fairly stiff wings, while smaller and
more maneuverable and station-keeping animals (such as sunfish) have flexible
pectoral fins, we examined the question: is dynamic stall universally present
in both flexible and rigid high-lift appendages? What are the functional
differences between relatively stiff and flexible fins?
Materials and methods
The apparatus used to determine the hydrodynamic characteristics of rolling and pitching fins, which are similar to the pectoral fins and wings of swimming animals, is shown in Fig. 1. Three rigid fins with chord (c) of 10 cm in most of the span (s) and span/chord (s/c) ratios of 1, 2 and 3 were tested in a low-speed tow tank. The fin is imparted pitching and rolling motions using a set of two orthogonally placed motors. The pitch motor and fin assembly is attached to the roll motor shaft. The six forces and moments are read by the load cell that connects the roll motor to the tow carriage. Optical encoders on the motors give rolling and pitching positions and angular velocities, while torque sensors attached to each motor shaft used to measure the efficiency. The chord Reynolds numbers of the hovering and towing tests are in the 20 000 to 150 000 range, based on total speed Ut (the symbols are defined below, see List of symbols and abbreviations).
|
 | (1) |
(t) is the instantaneous roll position, 0
is the roll amplitude, and 
is the flapping frequency in radians
s–1. The pitch motor, which rides on the roll motor,
oscillates as:
 | (2) |
(t) is the instantaneous pitch position,
0 is the pitch amplitude, 
is the phase angle between
roll and pitch, and Bias is the pitch bias (which is 0 for
zero-mean lift). Roll and pitch torque are measured from the output at each
motor's gearbox. In conjunction with motor velocity data, these give the power
applied by the motors to the fluid:
 | (3) |
is the instantaneous torque for the roll and pitch axes. This is a
measurement of power independent of the specific actuators used in this
experiment. To estimate inertial uncertainties, power time traces were
measured in air. They were compared with the measurements made in water. The
power trace in air leads that in water by 90°, because the resistance to
motion in air is predominantly inertial, whereas in water it is predominantly
hydrodynamic with velocity squared. The mean power in air calculated using
torque and angular velocity is near zero, because power expended during
acceleration is absorbed during deceleration. At the moment when the inertial
effect in air is at maximum, it is about 15–25% of the hydrodynamic
power in water for the cases shown in Fig.
8. At the moment of peak hydrodynamic power, the inertial
component is zero, because the fin is at a point of peak velocity and zero
acceleration.
|
Hydrodynamic efficiency 
hydrodynamic is defined as:
 | (4) |
X is the
cycle-averaged force in the forward direction,
hydrodynamic is the
cycle-averaged hydrodynamic power, and U is either the forward
velocity of the tow carriage U
or, if the carriage
speed is zero, an estimate of the induced velocity through the swept area
Uind. The induced velocity is defined as:
 | (5) |
is the fluid density and As is the area swept
by the wing (Wakeling and Ellington,
1997).
Cross-flow vortex modeling of unsteady stall hydrodynamics
We propose that the dynamic stall vortices are analogous to cross-flow drag
vortices of a fin placed normal to a steady uniform stream as a bluff body and
as shown in Fig. 2. At
shallower angles of attack, the vortices in
Fig. 2 are simply the leading
and trailing edge vortices resulting in the fin-normal component of force due
to the cross-flow, of which lift and drag are resolved representations. If the
angle of attack is constantly changing such that these drag vortices can be
retained over the fin surface during the cycle, then stall can be prevented
and lift enhanced at large angles of attack. Thus, unsteadiness does not do
away with the basic steady lift and drag characteristics of the fin, but
merely delays the occurrence of stall.
|
Consider a fin undergoing rolling and pitching motions in the
Y–Z plane and moving forward in the direction X, as
shown in Fig. 3. Forward
velocity U and rolling velocity
Uwing result in the total velocity
Ut. The fin forces (lift
Lfin and thrust Tfin) are transverse
and in line with Ut, respectively, and N is
normal to the fin surface and is the resultant of lift and thrust. Viscous
drag is ignored. The angles are defined as follows: (t) is the
roll angle, 
(t) is the instantaneous angle of attack between
the fin and Ut, and (t) is the pitch
angle; t is time. For a steady fin at small angles of attack, lift
changes much faster with angle of attack than drag does. Therefore, we assume
that the coefficient of force normal to the fin surface is given by the same
slope:
 | (6) |
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 | (12) |
 | (13) |
|
Method of optimizing fin efficiency
We implemented a downhill simplex method
(Flannery et al., 1988), using
frequency, roll amplitude, pitch amplitude, pitch bias, and phase between the
roll and pitch sinusoids as the dimensions to be searched, and a user-chosen
optimization parameter to minimize. This optimization parameter could be
efficiency while hovering, defined from Wakeling and Ellington
(Wakeling and Ellington, 1997)
and subtracted from one:
 | (14) |
X,
Y and power, such as:
 | (15) |
When the search algorithm outputs a new set of motion parameters to try,
the fin will change its motion to the new set and flap a user-defined number
of times. The fin control program returns the mean forces, power and
efficiency recorded during the routine, which are then sent back to the search
algorithm, where they are translated into the optimization parameter of
interest. A simulated annealing term was added to the method
(Flannery et al., 1988), with
a gradually reducing `temperature', to avoid becoming stuck at a point where
the repeatability level of the data resulted in an anomolous local minimum due
to noise in data.
|
Cross-flow vortex model
The cross-flow vortex model is compared with measurements in Figs
4 and
5 at a tow speed of 1.34 m
s–1. The agreement is good. In Figs
6 and
7, a similar comparison is made
but at a lower speed of 0.46 m s–1 and a higher pitch
amplitude of 35° to retain a similar maximum angle of attack, but with
other fin oscillation parameters remaining the same as in the case of the 1.34
m s–1 tow speed. At the lower speed, the model agrees with
the time signatures of the force measurements as well. However, the
measurements, and not the model, indicate a seeming hysteresis. This
hysteresis is a direct measurement and is reminiscent of a Wagner effect as
per which there is a time delay in the mechanism of force production in
impulsively started lifting surfaces. Due to viscous effects, there is a delay
in the development of the asymptotic value of the circulation around the
lifting surface, that is, a delay in the establishment of the Kutta condition.
The proximity of the starting vortex near the trailing edge in the early
stages also affects this delay.
|
|
|
)
– is constant in time and space, which will lead to errors in the
magnitude and direction of Ut. To resolve whether there
truly is hysteresis in the low-speed fin behavior, the time signatures of fin
torque were examined. This is because torque is an independent and direct
diagnostic of the fin response to the roll and pitch oscillations where
induced velocities do not have to be estimated. The torque versus
roll velocity response is shown in Fig.
8 at tow speeds of 1.34 and 0.46 m s–1. The
torque versus roll velocity is linear and `elliptic', respectively,
in Fig. 8 at the higher and the
lower speeds. While there is no hysteresis at the higher speed, some is seen
at the lower speed, but it is not as dramatic as the reduced data in
Fig. 6 suggest. At 0.46 m
s–1, for peak–peak roll torque of 8 N m, there is a
maximum deviation from the mean value of about ±1 N m – a 12%
effect. A more accurate method of estimating induced velocities during hover
or low speeds of tow is needed.
The universal validity of the cross-flow vortex model for lift and drag over all oscillation parameters such as frequency, roll angle, pitch bias and pitch amplitude during hovering and towing, as well as for varying fin spans, is shown in Figs 9 and 10, respectively (roll 30°, 40°; pitch bias 0°; pitch 25°, 35°, 45°, 55° and 65°; roll–pitch phase –90°, 90°; frequency 0.75, 1, 1.25 and 1.5 Hz; span 20, 30 cm; cruising speed –0.46, 0.46, 0.83 and 1.34 m s–1). The steady fin measurements are included for reference. The unsteady data indicate lift and drag forces averaged for bins at a given angle of attack for all cases. For low tow speeds, the averaged value of the hysteretic behavior is shown. The cross-flow vortex model describes the unsteady measurements well.
|
|
Rapid optimization of hydrodynamic efficiency
The magnitude of the mean lift and thrust forces produced by the fin are
functions of the flapping frequency, roll amplitude, pitch amplitude, phase
difference between roll and pitch, and pitch bias, as well as the incoming
flow speed and direction. To make the force production robust to disturbances
in the incoming flow, such as that created by cross-flows and large-scale
vortices, one would either have to study the fin in all sorts of flow fields
and then measure those in practice, or enable the system to react to changing
flows in such a way as to produce the desired forces nonetheless. As an
initial step to a flow-adaptable fin, we demonstrated the fin searching
through its parameter space – with no a priori knowledge of its
hydrodynamic capabilities – in order to optimize between force
production and power cost.
The optimization method was successful in converging to optimized points within 40–50 cycles, in approximately 4 min, depending on the initial random set of motion parameters. This is shown in Fig. 11. The optimized motion parameters were within the range seen in a previous matrix study of the fin performance versus input parameters. This is shown in Fig. 12.
|
|
The measurements of efficiency in Fig.
12 contain two sets of data – one for hovering and one for
cruising. The thrust was normalized as
CX,wing=X/(1/2U2wingAplanform),
and the efficiency was defined as
hydrodynamic=XU/hydrodynamic
for U>0 and
hydrodynamic=XUinduced/hydrodynamic
for U=0, where the induced velocity was estimated
using the disk method of Wakeling and Ellington
(Wakeling and Ellington,
1997), meaning that the hydrodynamic values during
hovering have higher uncertainties than those during cruising. All data with
CX,wing>1.5 are for hovering, and most of
the data for CX,wing<1.5 are for cruising.
Using the best means available for comparison, the best efficiency for
hovering is nearly half that during cruising. This suggests that propulsive
efficiency when maneuvering is less efficient than when cruising. Note that
the highest efficiency of about 0.6 is similar to the 0.5–0.6 measured
for two-dimensional fins (Read et al.,
2003). Thus, increasing the aspect ratio of fins beyond the
maximum of 3 in the present work does not offer an increase in efficiency. A
private communication with G. V. Lauder and P. G. A. Madden in 2006 indicated
that the highest efficiency of a sunfish with a pair of flexible pectoral fins
during station keeping in a stream is 0.42. If we allow that the efficiency of
a fish is lower than that of its appendages, then the sunfish pectoral fin
efficiency is probably higher than 0.42 and may be closer to 0.6. This
suggests that the efficiency of pectoral appendages from penguins to sunfish
is similar.
|
|
) have
argued that the sunfish pectoral fin has two leading edges because, due to the
cupping of the fin during outstroke, both the leading and trailing edges
become positioned at the same axial station during station keeping in a
laboratory flow tunnel. If so, then, where are the trailing edges? A still
frame showing the formation of two clear vortices at the fin edges is given in
Fig. 13 and a synthesis is
shown in Fig. 14. If both
edges are leading, then the contrarotating vortices in
Fig. 13 must be from the
leading edges of two symmetric fins. We then attach importance to the fact
that similar contrarotating vortex pairs away from the leading edge vortices
are not produced, as would be expected had there been two trailing edges. This
implies that there are no trailing edges at all, but only two leading edges.
Accordingly, we synthesize these arguments to arrive at the hypothesis that
the sunfish pectoral fin acts as a conjoined symmetric biplane during station
keeping in a uniform stream during the outstroke half of the fin beat. The
flexibility is a means of making two fins out of one in order to balance
unsteady forces in the vertical direction. The formation of the two attached
edge vortices shows that the dynamic stall mechanism is present in sunfish
pectoral fins although the projected fin area is much reduced from the maximum
possible.
To achieve station keeping, it is necessary to be able to promptly produce
forces and moments along stream and also normal and cross-stream in amplitudes
that are just enough to cancel the perturbations. The flexible fin has the
ability to control the projected surface area in all planes as indeed we see
in the movie frames in Lauder et al.
(Lauder et al., 2007). The fin
folding also changes the local angles of attack. The twist of the fin can also
be controlled at the root, thereby controlling the vector of the co-flowing
jet (Fig. 14). These three
traits, with the aid of sensitive lateral line sensors and an instantaneously
deployable controller, could dynamically cancel force and moment perturbations
in all directions and axes. The rigid fin measurements of force vectors at
each instant were examined in three-dimensional fields. Rigid fins produce
large transverse periodic forces, which may be undesirable when holding
position (Beal and Bandyopadhyay,
2007). To achieve station keeping in the vertical plane, two
symmetric fins are needed to balance the instantaneous vertical (Y-)
forces. Such data show quantitatively that station keeping can be achieved at
all times by symmetric biplanes, conjoined or not. The symmetric fins would
not have to be mirror images; their angles of attack to the flow and flapping
speed could be different as long as they cancel undesired unsteady forces. A
sample of the sunfish pitch angle time history from Lauder et al.
(Lauder et al., 2007) is
compared with a similar sinusoidal history of the rigid fin in
Fig. 15, where a qualitative
similarity is seen to exist. The sum of the pitch angles between the dorsal
and ventral rays is within 0° to 20° during abduction, while it is
closer to zero during the adduction phase. The pitch angles in the biorobotic
rigid fins, of course, sum exactly to zero at all times. In the absence of
detailed kinematics of the sunfish pectoral fin, this zero sum is taken as an
indication of real-time station keeping.
|
Discussion
What can we learn by comparing the force production in sunfish due to their
flexible pectoral fins and the present relatively stiff penguin wing-like
biorobotic fins? Lauder et al. (Lauder et
al., 2007) have given the time sequences of the flexible fin
contortions and the distributions of local velocity variations about the
freestream velocity for a sunfish holding station in a stream. We propose the
following mechanism to be in play in the flexible fin. During the outstroke,
while the spanwise edges of the fin are folding inward, due to separation and
the ensuing pressure difference, two symmetric leading-edge vortices are
formed (Fig. 14). The
projected area of the fin is much reduced from its maximum value during the
cupping process, which suggests that its thrust production role is smaller.
The two leading-edge vortices coalesce to form a downstream-pointing jet. The
spanwise bone structure and the chordwise fin corrugations would assist this
spanwise jet flow. The jet is inclined rearward to the body and is a source of
thrust. The outstroke takes the fin to a position just short of being normal
to the body. The fin subsequently expands, which causes the twin leading-edge
vortice jet to expand into a diffuser, allowing pressure recovery, at the end
of the outstroke. During the return stroke, the fin concaves with the fin tip
facing upstream. Due to stiffness, the spanwise edges of the fin this time do
not cup inward and no significant leading-edge vortex is probably produced,
although a stall vortex could still form at the tip. Conceivably, this tip
vortex inclined cross-stream and parallel to the body could interact with the
caudal fin downstream, providing a means for precision streamwise control of
the fish body to station itself in the face of perturbations, because it is
sinusoidal and small in amplitude compared with net thrust
(Bandyopadhyay et al., 1998).
During the sweeping motion of the return stroke, the fin acts like a row. The
sweep motion is analogous to rolling in rigid fins and the cupping of the fin
changes pitch – the angle of attack constantly changes during both
motions. In Lauder and colleague's (Lauder
et al., 2007) experiment, pitch bias may be zero because the fish
is not steering (Fig. 15).
Because rowing is inefficient, and the fin edges would break at higher thrust
levels, flexible fins are not seen in larger animals. The flexible pectoral
fins are probably more suitable in low speed and smaller swimming animals.
They would be difficult to scale up for biorobotic application. Because flying
insects need to produce steady lift as well as thrust – unlike fish,
which are nearly neutrally buoyant – they do not use similar flexible
fins, which are optimal for station keeping. Although the cupping motion
during the outstroke produces a lower thrust peak than the return stroke
(Lauder et al., 2007), it can
nevertheless cancel vertical perturbations to allow exquisite station keeping.
Therefore, the outstroke cupping motion may assist in station keeping with
regard to vertical perturbations while the return stroke sweeping motion is
important for thrust production because the projected fin area approaches its
maximum value. What is fascinating about the flexible fin is that it can
conceivably cancel force and moment perturbations in all directions and axes
and produce thrust. Clearly, there is a need to investigate how in a sunfish
the controller uses lateral lines to actuate this unsteady hydrodynamics of
the flexible fin. At the very least, the control and sensing of the flexible
fin hydrodynamics are shaping up to be far richer than those in rigid
fins.
This work identifies the common underlying principles in steady and unsteady fin dynamics and in rigid and flexible fins. The effectiveness of the cross-flow vortex model helps to place the role of unsteadiness in the context of classical steady-foil theories, which have a firm theoretical foundation – unsteadiness, while being novel, is inclusive rather than an entirely exclusive phenomenon, from an engineering view point. On the other hand, in nature, unsteadiness is not novel and steadiness is rare. The validity of the model points to intriguing notions; for example, would it be possible to infer the habitats of ancient species from the fossils of their pectoral fins?
Quasi-steady modeling has been attempted in the past to explain the force
production mechanism of flying insects that enables them to hover, by
comparing them with what would be produced had the actuators been entirely
steady (Ellington, 1984a;
Wilkin and Williams, 1993).
However, early works suffered from a lack of measurements on isolated insect
wings and of time histories of forces produced by the lifting appendages. Sane
and Dickinson (Sane and Dickinson,
2002) generated such data and developed a semi-empirical `revised
quasi-steady' model. Their measurements of lift and drag show that the time
histories are fairly steady but for brief transients during the beginning and
end of the up- and downstrokes. Their model agrees uniformly well with
measurements when the forces have reached a steady level, which is about 75%
of the duration of the cycle. It does not agree well during the beginning of
the down- and upstrokes when the forces rapidly rise – which are about
25% of the cycle time – but does agree at those times when the force is
dropping near the ends of the strokes. In their data, the force spikes from
zero or negative values during the beginning of the strokes, while it spikes
from a steady level near the end of the strokes. The gradient of unsteadiness
is higher during the beginning of the stroke, probably causing Wagner and
added mass effects to spike as well. Because we have a sinusoidal oscillation,
which is always smooth in contrast to that of Sane and Dickinson, we did not
have any such force transients. But, we do have some Wagner effect at speeds
approaching zero. It would be useful to know whether the force transients in
the biorobotic experiments of Sane and Dickinson, where the fin kinematics are
strongly non-sinusoidal, are truly present in insects. Their quasi-steady
model involves the budgeting of the effects of added mass, translation,
rotation and wake capture. This requires empirical data for translational
effects (including wing tip vortex effects), as well as simplifying
assumptions of quasi-two-dimensionality for added mass effects and small
angles of rotation for rotational effects. The model is useful for a
qualitative understanding of the kinematics that influences the components of
the mechanism. However, it does not clearly state what the relationship is
between the steady and unsteady characteristics and why the animal fin
cross-sections are so similar to NACA profiles which have been developed for
steady flow application (Fish, 1999). On the other hand, the present work
addresses these questions clearly. The general conclusion is that biorobotics
produces data of low uncertainty but that the data's fidelity to the animals
simulated requires closer scrutiny. On the other hand, controlled measurements
with animals have fidelity, but they do not allow synthetic studies of the
mechanisms of the appendages.
Swimming and flying animals and their biorobotic renditions are reported to
have efficiencies that vary widely. Direct measurements have tended to be on
the low side. On the one hand, Fish (Fish,
1993) has modeled bottlenose dolphin efficiency to be 81%. On the
other, Wakeling and Ellington (Wakeling
and Ellington, 1997) have reported the measurements of dragonfly
mechanical efficiencies to be 9–13% based on heat production after
flight. Anderson et al. (Anderson et al.,
1998) have reported the measurements of peak hydrodynamic
efficiencies of 87% for two-dimensional heaving and pitching foils. However,
later measurements from the same laboratory have reported an efficiency
plateau of 50–60% (Read et al.,
2003). The present measurements are in this range, as are the
estimates for sunfish by G. V. Lauder and P. G. A. Madden (personal
communication, 2006).
The efficiency optimization work shows that the process is remarkably rapid and seemingly effortless. This approach makes a cumulative accounting of past experience in the sense of `learning' and is akin to the spirit of evolution. This suggests that once the species discovered the high-lift mechanism, the optimization may have evolved quickly. As a consequence, it may be that rolling and pitching appendages of all species are of generally high efficiency.
Neuroscience posits that mobility demands richness in neuronal abilities, and the higher density of neurons is a precursor of an ability to perform complex motions. If we look at specialized maneuvering as more complicated than cruising, then was the evolution of species endowed with such richness the result of a step increase in neuron density? Our experience indicates that the roll and pitch motions of the fin are useful for both cruising and maneuvering, whereas pitch bias is useful only for maneuvering. These classes of behaviors might give us a framework for understanding the relationship between hydrodynamics and control in different species of swimming animals because they are proving crucial in the development of biorobotic underwater vehicles (Menozzi et al., 2007).
Conclusions
The following conclusions can be drawn from our controlled hydrodynamic experiments in a laboratory with biorobotic abstracted penguin pectoral wings.
List of symbols and abbreviations
Ut2Aplanform)
Ut2Aplanform)
Ut2Aplanform)
X
Y
near =0
electric
hydrodynamic
or
Uind
hydrodynamichydrodynamic
electricelectric
0
Bias
Roll
Pitch
0
Acknowledgments
The authors acknowledge the sponsorship of the Cognitive and Neurosciences Program of the Office of Naval Research, Program Officer Dr Thomas McKenna and NUWC ILIR Program, Program Officer Mr Richard Philips. Discussions with Professor George Lauder are appreciated.
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