Integrative and Comparative Biology
Integrative and Comparative Biology, volume 55, number 4, pp. 753–764
doi:10.1093/icb/icv053
Society for Integrative and Comparative Biology
SYMPOSIUM
Stability versus Maneuvering: Challenges for Stability during
Swimming by Fishes
Paul W. Webb1,* and Daniel Weihs†
*School of Natural Resources and Environment, University of Michigan, Ann Arbor, MI 48109, USA; †Department of
Aerospace Engineering and Autonomous Systems Program, Technion—Israel Institute of Technology, Haifa 32000, Israel
From the symposium ‘‘Unsteady Aquatic Locomotion with Respect to Eco-Design and Mechanics’’ presented at the
annual meeting of the Society for Integrative and Comparative Biology, January 3–7, 2015 at West Palm Beach, Florida.
1
E-mail: [email protected]
Synopsis Fishes are well known for their remarkable maneuverability and agility. Less visible is the continuous control of
stability essential for the exploitation of the full range of aquatic resources. Perturbations to posture and trajectory arise
from hydrostatic and hydrodynamic forces centered in a fish (intrinsic) and from the environment (extrinsic).
Hydrostatic instabilities arise from vertical and horizontal separation of the centers of mass (CM) and of buoyancy,
thereby creating perturbations in roll, yaw, and pitch, with largely neglected implications for behavioral ecology. Among
various forms of hydrodynamic stability, the need for stability in the face of recoil forces from propulsors is close to
universal. Destabilizing torques in body-caudal fin swimming is created by inertial and viscous forces through a propulsor
beat. The recoil component is reduced, damped, and corrected in various ways, including kinematics, shape of the body
and fins, and deployment of the fins. We postulate that control of the angle of orientation, , of the trailing edge is
especially important in the evolution and lifestyles of fishes, but studies are few. Control of stability and maneuvering are
reflected in accelerations around the CM. Accelerations for such motions may give insight into time-behavior patterns in
the wild but cannot be used to determine the expenditure of energy by free-swimming fishes.
Introduction
The remarkable maneuverability and agility of aquatic vertebrates has impressed biologists for generations, and more recently caught the attention of
engineers seeking to emulate such performance in
human-constructed underwater vehicles. Maneuvering is defined as intentional changes in an existing
state of a body (i.e., posture, position, velocity) as a
result of linear and/or rotational accelerations. Maneuvering appears to have been a major factor in the
evolution of the chordate body-plan. The apomorphies of an incompressible notochord and muscular
tail are adaptations for more powerful swimming
(Clark 1964), especially fast-starts and mechanically
similar powered turns (Weihs 1972, 1973). In many
chordate larvae these adaptations facilitate rapid settlement from the water column onto appropriate
substrate (e.g., Clark 1964). Subsequent evolution
of maneuverability in vertebrates was undoubtedly
facilitated by the high density of water, w, similar
to that of the body, b, resulting in a density ratio,
b/w close to unity (Daniel and Webb 1987). Then
the body is largely supported by the medium and all
appendages can be deployed for propulsion, rather
than for support, and each propulsor is also a control-surface. The many propulsors permit numerous
maneuvers as well as providing substantial redundancy for the execution of any given maneuver
(Webb 2006). In addition to the diversity of maneuvers, the rate of maneuver (agility) can also be large
because the relatively high w permits propulsors to
generate large reaction forces relative to the bodymass to be accelerated.
Maneuvers are readily observed and are often dramatic (fast starts). However, organisms also must be
stable, where stability is defined as the tendency of a
body to return to its original state after a displacement. In contrast to maneuvering, obvious motions
meant to control stability are difficult to recognize, a
testimony to the effectiveness of morphological and
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754
sensory-motor controls (Sefati et al. 2013). Dramatic
maneuvers and less visible stability-control have suggested that many aquatic vertebrates are both highly
maneuverable and stable (Jing and Kanso 2013), a
goal that largely has evaded human-engineered,
rigid, aquatic vehicles. In practice, fishes often must
continuously expend substantial energy for stability,
such that control of posture and of swimming trajectories was undoubtedly critical in their evolutionary success.
The evolutionary importance of control of stability
versus maneuverability per se can be illustrated for
swimming using the body and caudal fin (BCF
swimming). The fast-starts and rapid powered turning maneuvers that are considered ancestral can be
launched successfully and executed without active
feedback control, although control during these maneuvers is common in modern forms (Eaton 1984;
Domenici and Blake 1997). Thus, the maneuvers
mentioned above can be driven by simple unilateral
contraction of the muscle on each side of the body.
Indeed, the resulting yawing instability can be incorporated into the behavior, potentially generating
uncertainty in escape trajectories. Fast-starts and
powered turns were probably important in early
fishes (thought to have radiated along currentswept shorelines) while holding position on the
bottom and capturing passing food using fast-start
strikes. These same behaviors would have come into
play when avoiding predators, especially other predatory species of fishes following the ascent of the
gnathostomes (Moy-Thomas and Miles 1971).
However, the domination of the aquatic environment by fishes has been contingent upon continuous,
‘‘steady’’ (i.e., not varying in time) swimming, over
distances L, where L is the total length of the body.
Swimming is essential for exploiting the whole watercolumn of both lentic and lotic waters, for example
locating widely distributed pelagic food and mates,
and avoiding risky habitats. In contrast with the
simple BCF maneuvers, the head must be controlled
in steady swimming so that the center of mass (CM)
can translocate as closely as possible to a rectilinear
path, an essential condition for minimizing the
energy-costs of locomotion. Indeed, costs of steady
swimming may represent over 90% of daily budgets
of energy (Brett 1995), partly as a result of loss of
energy associated with the stabilization of posture
and of trajectories against recoil (Schultz and Webb
2002).
Thus, the evolution of morphological and sensorymotor systems to control yaw during steady
BCF propulsion may have been critical in the evolution and radiation of fishes. Here, we focus on
P. W. Webb and D. Weihs
functional-morphological
mechanisms
whereby
fishes have successfully achieved stability during
steady locomotion, concentrating on motions at
high Reynolds numbers (103) where inertial effects
dominate.
Types of stability
In considering stability in the locomotion of fishes,
several different types of stability are recognized
(Fig. 1) depending on whether perturbations from
the steady state arise from hydrostatic or hydrodynamic forces and whether these destabilizing forces
arise in the organismic frame of reference (intrinsic)
or from the environment (extrinsic). Torques result
from misalignment of force vectors as well as separation of the locations of the CM and buoyancy and
so may arise during both steady and time-dependent
swimming motions.
It is often thought that destabilizing hydrostatic
and hydrodynamic forces and torques are undesirable. There is, for example, an extensive literature
on the benefits of neutral buoyancy (e.g., Aleyev
1977). In practice, neutral stability is the worst-case
scenario for any body at rest or in motion, leading,
for example, to the tumbling motion of neutrally
buoyant bodies moving through water (Schultz and
Webb 2002). Instead, some intrinsic instability is
functionally desirable, providing a reference upon
which control systems can work.
Hydrostatic stability
Although many fishes reduce or eliminate net hydrostatic forces, torques arise because of separation
between the center of mass, CM, and the center of
buoyancy, CB. CB is typically above CM because
low-density tissues are predominantly ventral, associated with viscera as well as the abdominal buoyancy-control systems of gas bladders and liver. In
contrast, higher density tissues such as myotomal
muscle are typically dorsal. In addition CM and
CB often are separated horizontally (Aleyev 1977).
Vertical separation of CM and CB creates rolling
torques and the longitudinal separation causes pitching torques, thereby affecting stability.
It has been estimated that interventions to control
horizontal posture whenever fishes swim off the
bottom may be as large as 10% of the total energycosts of swimming (Weihs 1987). Potential costs of
stabilizing rolling associated with the vertical separation of CM and CB have not been estimated. In
general, the locations of CM and CB impact the behavior, ecology, and evolution of fishes, such that
their locations should be subject to adaptation.
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Stability versus maneuvering
Stability
Hydrostatic stability
Intrinsic torques due
to distribution of
materials of different
densities along
dorsoventral and
longitudinal body
axes.
Hydrodynamic stability
Static hydrodynamic
(SH) stability.
Perturbations during
constant speed,
steady.
Intrinsic perturbations
Shape-based:
Perturbations arising
from asymmetries in
body shape, and fin
form and distribution.
Dynamic
hydrodynamic (DH)
stability.
Perturbations due to
maneuvers.
Extrinsic perturbations
Propulsor-based:
Perturbations
arising from
periodic motions
of propulsors.
Perturbations from
unsteady and
turbulent motion in
the environmental
frame-of-reference.
Fig. 1 A classification of types of stability for which control systems are required in fishes. For simplicity, the addition of control during
maneuvers in unsteady flows has been omitted as the same principles apply as for the categories of stability that are included.
Categories are based on aeronautical technology (e.g., von Mises 1945; Boiffier 1998) as previously applied to biological systems (Weihs
1993, 2002).
Such questions have been neglected. Systematic differences should be expected (Aleyev 1977). For example, elasmobranchs usually have compressed
bodies that probably reduce the costs of yawing
recoil (Table 2). This may also result in smaller vertical separation of CM and CB. The shape of the
elasmobranch liver and associated pitching torques
could contribute to postural control in partnership
with lift forces during swimming by these negatively
buoyant animals (Aleyev 1977). Longitudinal locations of CM and CB in teleosts also are affected by
the shape of the swimbladder (Webb and Weihs
1994). The shape of the liver in elasmobranchs and
that of the swimbladder of teleosts affect stability in
pitch and hence might be subject to selection.
Pitching torques can vary over short time-scales.
For example, contents and fullness of the gut, defecation, and parasite-loads will affect the locations of
CM and CB. Similarly, pitching torques will vary
with time for fish such as amphidromous migrants
moving through waters of different density (Weihs
1987) and those crossing the thermocline or making
other diurnal migrations crossing large temperature/
density gradients.
Static and dynamic hydrodynamic stability
Hydrodynamic instabilities arise from the motion of
an organism through the water, including unsteadiness of swimming trajectories, torques due to spatial
asymmetries in body-shape and the distribution of
the fins during translocation, time-dependent propulsor motions, and externally driven flows of the
surrounding water. We use the terms ‘‘static’’ and
‘‘dynamic’’ originating from aeronautical technology
(e.g., von Mises 1945; Boiffier 1998), following
common practices when engineering principles have
been applied to biological questions (Weihs 2002).
Thus, static hydrodynamic (SH) stability deals
with the time-dependence of excursions arising
from both intrinsic and extrinsic perturbations for
fishes moving in a steady, time-independent,
straight-line state in which displacements are
damped out. In contrast, dynamic hydrodynamic
(DH) stability covers intrinsic unsteady motions of
linear and/or angular accelerations in maneuvers, including turning, fast-starts, and braking, essentially
the major determinants of behavior (Webb and
Gerstner 2000) and its control.
DH stability—maneuvering
Voluntary swimming is rarely rectilinear at steady
speeds. Instead, changes in state are normal. For
much of the time-activity budget, changes in state
are small and the principles of the control of stability
during rectilinear propulsion apply, as in that for
yawing discussed below. As noted above extreme maneuvers may be executed without closed-loop control, such as some fast-starts. Consequently, stability
756
during maneuvers appears to add little, if anything,
to the questions of unsteady motions compared with
executing the maneuvers themselves.
Unsteady swimming in maneuvers underlies most
behavior (Webb and Gerstner 2000) and there is
substantial interest in finding ways to record behavior in free-swimming fishes. There is currently a
rapid growth in the use of measurements from accelerometers in this role (Gleiss et al. 2011). Current
work focuses on the absolute sum along the orthogonal axes of accelerations above background gravitational acceleration, and is defined as the overall body
dynamic acceleration (OBDA). OBDA is associated
with propulsor motions plus signals from maneuvers,
although in swimmers, signal-averaging methods
may remove accelerations due to maneuvers. It
should be possible to create libraries correlating
accelerometer signals along x, y, and z axes with validated behaviors. However, as recognized by
Freadman (1981), accelerometer records of swimmers cannot provide measures of absolute expenditure of mechanical energy. For example, OBDA
measures only undamped recoil and indeed multiple
propulsors (Hove et al. 2001) could reduce OBDA to
zero; imagine OBDA from a walking millipede. In
addition, in contrast to terrestrial locomotion, resistance and added mass effects arising from the density
and viscosity of water during swimming result in a
continuous and substantial drag force that is not
included in accelerometer records. Furthermore, calibrating accelerometer records with the energy-costs
of metabolism is only possible during steady swimming in a flume (Wilson et al. 2013a). Such correlation is not portable to maneuvers or situations
requiring the control of stability, as these increase
metabolic rates for aquatic and terrestrial locomotion, and by an order of magnitude, or more, for
the former (Boisclair and Tang 1993; Wilson et al.
2013b).
Intrinsic shape-based SH stability—form of the body
and fins
Intrinsic perturbations arise because body-form and
the distribution of fins usually are not symmetrical
about any axis, except laterally. This aspect of SH
stability has been studied in great detail for
human-engineered systems such as ships, submarines, torpedoes, and aircraft. Much of the early
work on stability during swimming in fishes derives
from such principles (e.g., Harris 1937; Aleyev 1977).
Analysis of shape-based SH stability is facilitated for
fishes because the lateral symmetry of the body
allows motions in the vertical plane (pitch, heave,
P. W. Webb and D. Weihs
and surge) to be treated independently from those
in the horizontal plane (slip or sway, yaw, and roll).
Stability is closest to the shape-based SH case when
fishes swim using their median and/or paired fins
(MPF swimming), especially when using the pectoral
fins at high frequency and/or when multiple fins
work together.
Explorations of stability associated with perturbations arising from asymmetries in body-shape and
the form and distribution of fins are few but this
area is especially amenable to analysis by physical
models (Harris 1937, 1953; Fish and Lauder 2006;
Bartol et al. 2008).
Intrinsic propulsor-based SH stability—recoil from
propulsors
Shape-based SH stability of a stretched body is only a
rough approximation of actual locomotion by fishes.
A major source of usually destabilizing intrinsic SH
perturbations arises from oscillating propulsors or
their elements. Such destabilizing forces and torques
are especially large in BCF swimming. In most cases,
there is surge fore and aft, so that the forward speed
varies periodically during a propulsor’s beat-cycle
(Freadman 1981). As such, swimming at a true
‘‘steady’’ speed is non-existent, and steady swimming
is understood as that with constant average speed
over successive similar propulsor-beats. In addition
to surge forces, each propulsor generates destabilizing torques causing recoil motions in slip and yaw in
BCF swimming (anguilliform, carangiform, thunniform modes) while heave and pitch recoil are usual
with short-based paired fins (e.g., labriform and
chaetodontiform modes) (Weihs 1993, 2002).
Among the many types of challenges to stability
faced by fishes, intrinsic propulsor-based SH stability
is especially pervasive because swimming is a necessity for exploiting the full range of aquatic habitats.
Forces and torques are generally small for intrinsic
shape-based SH and hydrostatic stability so that control of posture is readily achieved with a portion of
the large propulsive forces. Similarly, many maneuvers involve small rates of acceleration, again readily
met by comparably large forces generated by propulsors. As noted above, the highest levels of unsteady
performance can be executed without feedback control. Among extrinsic perturbations, forces and torques associated with turbulence are probably among
the largest sources of all disturbances (Tritico and
Cotel 2010; Cotel and Webb 2012), so much so
that the major response by fishes is avoidance.
Avoidance of intrinsic propulsor-based SH stability
would require eschewing steady swimming, which is
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Stability versus maneuvering
not a permanent option for most fishes. Thus, issues
of intrinsic propulsor-based SH stability are unavoidable, and hence we focus on them here. In particular
we explore challenges during motions of the body
and tail that generate the largest perturbations
during BCF propulsion.
Stability of yawing motions during BCF steady
swimming
Recoil forces and torques occur due to asymmetric
movements of propulsors in space or time, especially
for yawing displacements during BCF swimming.
Pitching torques also can be produced by spatial positioning of the caudal axis, for example when the
caudal peduncle is inclined to the longitudinal axis
or dorsal and ventral tail lobes are inclined at different angles to the flow (Harris 1953; Lauder 2000).
Periodic displacements along the heave axis are
common during use of the pectoral fins in swimming (Webb 1973; Drucker et al. 2006).
Destabilizing yawing torques vary over time
during the beat of a propulsor. Consider a full tailbeat cycle composed of four quartiles (Fig. 2 after
Lighthill 1969). The first (Q1) starts from the maximal deflection of the tail to the left when viewed
from above, head forward, and ending when the
center of force, situated approximately at the quarter
chord of the fin (Lighthill 1975) crosses the centerline of both the fish and its trajectory. During Q1,
the fin moves toward the centerline of the fish,
building up lateral velocity, W, with a geometric inclination, , to the direction of the fish’s motion,
both reaching maximal values when the fin’s center
of force crosses the centerline. Q2 starts as the center
of force crosses the centerline ending at the maximum displacement of the tail to the right. W and both decrease and go to zero at maximal lateral displacement. Q3 is antisymmetric to Q1 and finally Q4
is the mirror image of Q2, after which the fin returns
to the starting point, while the fish’s CM has been
advanced U/f along the swimming trajectory, where
U is the translational velocity of the fish and f is the
frequency of the tail-beat.
The origin of the yawing force can be illustrated
assuming; (1) a rigid, flat, tail fin as in Fig. 2, but
flexibility and thickness of the fin do not qualitatively change the analysis; (2) quasi-steady motion,
so that the forces produced at any instant are a result of the instantaneous orientation of the fin.
This assumption is relaxed later and its ramifications
explained.
The instantaneous yawing torque has three components, where drag is a force in the plane of the
motion of the fin, and lift is the force normal to that
plane of motion:
(1) Lateral (recoil) component of lift: The instantaneous force produced has a component of the
lift, L, on the fin that is:
L /ðU 2 þ W 2 Þ;
where A is the hydrodynamic angle of attack, in
turn the algebraic sum of and arctan (W/U).
The lateral component of this force is always
largest when the tail crosses the fish’s centerline
when W is also largest, but the large value of at that point orients the lift-force more posteriorly thereby reducing its magnitude.
(2) Lateral offset to thrust component of lift: The
thrust component of the lift-force is not
aligned with the CM, but has a lateral offset
in all positions except when the center of lift
crosses the centerline.
Fig. 2 Motions of the dorso-ventral axis of the caudal fin in the
lateral plane over a single tail-beat of a fish, viewed from above
(adapted from Lighthill 1969). The blue quartiles and red quartiles each represent mirror images. L illustrates the lift vector and
D the drag vector.
(3) Lateral offset to drag on the tail: The fin experiences drag originating in the viscosity of the
water. This force is affected by the changes in
speed between U and (U2 þ W2)0.5 during the
tail-beat cycle. The force vector follows the inclination, , of the fin. The viscous force is
758
P. W. Webb and D. Weihs
smaller than that associated with the lift-force
during most of the cycle, except when the tail
is at the extreme lateral position at the
larger Reynolds numbers considered here.
Observations on larval fish swimming in
water with varying viscosity suggest this applies
to Reynolds numbers of above approximately
450 (Fuiman and Batty 1977).
The lateral offset of the thrust component of lift
and the lateral offset to drag on the tail are not in
phase. For example, at the start of a beat (position 1
during Q1 in Fig. 2), the fin is oriented in the direction of motion and W and are both zero; zero
thrust is produced. Nevertheless, there is a counterclockwise yawing torque due to the viscous force because the tail is displaced laterally relative to the fish’s
centerline. (Note: this force is not considered in inviscid models such as those formulated by Lighthill
1975.) As the tail moves toward the centerline, both
W and are positive so that thrust generates a clockwise yawing torque. Thus, the two forces provide
some smoothing of the yawing torque, but because
the thrust component is much larger than the viscous
drag, they do not cancel each other.
Calculations based on the equations of Table 1
assume steady lift-forces. This is an overestimation, as
there is a reduction in both instantaneous lift and drag
due to the Wagner effect (Weihs 2002). Thus, the hydrodynamic force takes time to attain steady values
following a change in speed, as occurs through a finbeat. However, the time to reach the steady value depends on the fin-chord, C, and U, as C/U, with steady
state values reached at about C/U ¼ 0.2 s. As the caudal
fin-chord is 5% L and speeds at which fishes voluntarily swim are of the order of 1 L s1, C/U is small and
the Wagner effect negligible (Lighthill 1975). Even if
the Wagner effect were much larger, the qualitative
arguments are not changed.
Reducing yawing recoil intrinsic for propulsor-based
SH stability
Numerous mechanisms can be identified generally
reducing, or even eliminating, recoil that are associated with swimming motions, body-shape, and distribution of the fins (Table 2).
Swimming mode
Lateral oscillations of propulsors to produce thrust
create intrinsic torques around the CM, with the
largest disturbances being for yawing motions in
BCF swimming (Lighthill 1977). Steady BCF swimming varies along a spectrum from anguilliform to
thunniform modes in which propulsive motions are
increasingly concentrated toward the tail. This is associated with increasing efficiency in generating
thrust. However, the yawing torque also becomes
more pronounced; i.e., the increase in efficiency
from limiting the parts of the body oscillating is
potentially accompanied by an increase in yawing
torques.
The smaller yawing-torque component in anguilliform propulsion results from the center of force—for
both thrust and lateral components—integrated
along the body’s length, being closer to the CM
than in thunniform locomotion. Such reduction in
the yawing torque also will accrue to fishes with
‘‘double tails’’, that is fishes with dorsal fins located
posteriorly with a phase difference in their motions
and those of the tail.
Swimming with posterior propulsors is intrinsically destabilizing as displacements in the trajectory
due to external forces are amplified. Consequently
stability is generally facilitated by labriform modes
with pectoral-fin propulsors anterior to CM.
Yawing torques can be eliminated, although surge
and heave remain, but the magnitudes of these disturbances are small compared with BCF yawing
recoil (Webb 1973; Drucker et al. 2006).
Tail inclination, The relative magnitude of forward and lateral force
components depends on the orientation of the tail.
The lateral velocity, W, of the fin’s center is a function of position during a tail-beat, varying with the
amplitude of the tail’s motion (assumed to be 20%
of L, over a range of speeds in rectilinear swimming)
and frequency f. The average velocity can be obtained
by integration as 2W/, and as the tail crosses a
distance of 0.2 L during a time of ½f, W can be
written as:
W ¼ 0:2Lf :
Lift is also determined by the hydrodynamic angle of
attack, , the sum of þ arctan(W/U). Both W and increase as the lateral displacement of the fin decreases (see Fig. 2) so that forces are largest when
the fin is close to mid cycle. However, also increases. Therefore, the lift-force is oriented more
posteriorly, reducing the proportion of lift contributing to recoil.
The relationship between and W with reduces
variation in the yawing torque through a tail-beat
compared with that expected in the absence of any
variation in . While this is a feature of the oscillatory motion of a propulsor, the magnitude of also
depends on the wave-form. In general, the shorter
the length of the propulsive wave, , and in the
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Stability versus maneuvering
Table 1 The periodic variation of angles and forces on the caudal fin during a BCF propulsive cycle
Q1
Q2
Q3
Q4
Angle of inclination
¼ 0 sin 2ft 05t51/4f
¼ 0 sin 2ft 1/4f5t51/2f
Q1
Q2
Lateral velocity
W ¼ W0 sin 2ft 05t51/4f
W ¼ W0 sin 2ft 1/4f5t51/2f
Q1
Q2
Geometrical angle of attack
¼ 0 sin 2ft 05t51/4f
¼ 0 sin 2ft 1/4f5t51/2f
Q1
Q2
Hydrodynamic angle of attack
A ¼ þ arctan(W/U)
Notes: The angle of inclination is measured anticlockwise from the direction of motion, with 0 defined as the angle of inclination of the fin at the
end of Q1. The geometric angle of attack is the angular difference between and the angle of the caudal fin trajectory in the fish-centered frame
of reference while the hydrodynamic angle of attack produces the forces includes the instantaneous forward speed of the fish.
Table 2 Mechanisms currently recognized and proposed to damp and reduce recoil energy losses in BCF steady swimming
Kinematics
Body and
fin form
Alternative
propulsion
Character
Function
Yawing forces from viscous drag
and lift forces.
Phase difference between the thrust force with small offset and viscous resistance
with large offset.
and W
and W increase toward the centerline such orienting the normal force more
posteriorly, increasing the thrust component and reducing the lateral component.
Swimming Mode—multiple halfwaves within the length of the
BCF propulsor.
Distributes thrust and lateral forces over the length of the body, moving the center
of force closer to the CM. Applies to anguilliform swimmers and amiiform, gymnotiform, balistiform, and rajiiform swimmers using elongate fins.
Powered lateral head motions.
Oppose yawing recoil by contraction of muscles on contralateral side of motions of
the head.
Lateral compression
Increases acceleration reaction resisting lateral motions while reducing side-slip for
fishes with more fusiform and stiffer bodies. Especially common among bony fishes.
Anterior body compression
Drag costs associated with lateral motions are reduced while facilitating yawing
turns in fishes with more elongate, flexible bodies. Common in elasmobranchs.
Vortex generators
Create vortices whose strength varies with body angle generating self-correcting
stabilizing forces. Fishes with rigid carapaces, such as boxfishes and cowfishes.
Trailing edge stiffness
Caudal fin feathering, powered and passive as in self-cambering, varying trailing edge
inclination, and hence the magnitude of thrust and lateral forces.
Deployment of median and
pelvic fins.
Various functions creating trimming and/or powered correction stabilizing forces.
Deployment of pectoral fins
Most commonly deployed stabilizers, generating trimming and powered corrective
forces.
Finlets
Reduce separation and drag forces due to lateral motions over posterior of body/
caudal peduncle.
Counter directional propulsive
waves fins
Waves moving in opposite directions along long-based fins cancel lateral forces and
controls surge.
Multiple fin-pairs
Orthogonal placement of usually median and pectoral fins cancels recoil forces.
Notes: Many adaptations also facilitate changes in state in maneuvers.
absence of trailing-edge control, the larger is . This
contributes to the overall center of force in anguilliform swimming being closer to CM than in thunniform swimming, as noted above. Although mean
values of typically are reported in the literature,
changes in flexural stiffness along the length of the
body result in functionally shorter values as the wave
travels backwards over the body; i.e., increases toward the tail, reaching maximum values at the trailing edge where the lift-force is also largest. The
flexural stiffness varies with the axial skeletal morphology among species and is also modulated by
tension of the myotomal muscles (Wainwright
1988; Long 2012).
Furthermore, can be modulated separately from
wave-form. One adaptation for such fine-scale control of is ‘‘self-cambering’’ in which the curvature
of the fin depends of the hydrodynamic force acting
on the elasticity of the fin-web skeleton. Then varies such that when the lift-force is largest, increases, reducing the potential yawing perturbation
(McCutcheon 1970; Lucas et al. 2013). Clearly, the
degree of self-cambering depends on the mechanical
properties of fin-rays and associated soft tissues.
760
In addition, fin-webs contain muscle that alter a
fin’s shape, and could equally actively vary stiffness
thereby controlling (Lauder 2011). Direct control
of may be a key adaptation in the evolution of
thunniform swimming that mitigates the potential
increase in the lateral tail-force in this mode.
Analytic and computational models (e.g., Lighthill
1975; Borazjani and Sotiropoulos 2010) have shown
the importance of the relationship between heave
and pitch of the tail, hence the effect of on the
lift-force and the thrust and recoil components. In
thunniform swimmers, the body tapers posteriorly to
a relatively long and narrow caudal peduncle, with
amplitude growing as if the peduncle and tail
were hinged, thereby providing for remarkable control of .
Evaluating mechanisms modulating Many mechanisms have evolved that modulate ,
thereby providing some control on intrinsic propulsor-based DH stability. Exploration and discovery of
such mechanisms constitute a growing area of research. Indeed, we suggest this is a central, outstanding problem and an especially fertile area for future
study. However, suitable animal models appear to be
lacking. Experimental studies of animals may benefit
from species with substantial morphological variation, such as short-tailed and long-tailed morphotypes of goldfish (Blake et al. 2009) and propulsor
modification is possible using Botox and formalin
(P. W. Webb, unpublished observations). Observations on live fishes have shown systems of propulsion
to be plastic, which tends to confound interpretation
of results. Nevertheless, studies of animals clearly
show that a wide range of intrinsic control systems
affect recoil. In general, sensory-motor systems are
multifunctional, not only initiating the propulsive
waves, but also combining with anatomical and morphological structures to actively and passively control
flexural stiffness along the length of the body,
thereby affecting and hence yawing recoil. Furthermore, fishes can exercise substantial control over the
area, span, camber, and angle of attack of the tail
during a beat (Fish and Lauder 2006), with differences within and among species.
In view of the plasticity inherent in animals, we
suggest that physical and numerical modeling are
essential for addressing questions of control of the
trailing-edge’s properties, shape, and kinematics. For
example, experiments with physical models are being
performed that generate traveling propulsive waves
along elongated plastic strips. So far, elastic modulus
of the material and the shapes of strips have been
varied and the forces or speeds that are attainable
P. W. Webb and D. Weihs
have been measured (Quinn et al. 2014b). In these
experiments, anterior driving-amplitudes have been
large compared with those of equivalent-sized
fishes, while amplitudes at the tail are much smaller
(Xiong and Lauder 2014). Nevertheless, these experiments have shown that elasticity and shape affect
kinematics, forces, and swimming speed (Daniel
1988). Not surprisingly there are complex relationships among variables of shape, elasticity, and swimming speed that can be attained with a given driver,
many not seen in real fishes. We suggest these methods will come to fruition when length-wise stiffness
is varied both in space and time to explore factors
that determine the shape of the propulsive wave, the
importance of trailing-edge stiffness, and perhaps
why fishes in general cluster around a maximum
amplitude of the order of 0.2 L. The potential of robotic models to explore SH-stability has not been
pursued.
Damping recoil with body form
Body-form and fin-distribution can help stabilize
recoil by damping displacements for a given perturbing yawing force. Among these is the anterior form
of the body (Lighthill 1975). A common adaptation
is lateral compression, especially in bony fishes, resulting in roughly elliptical cross-sections with a vertical major axis. This concentrates the fish’s mass
anteriorly, thereby increasing the inertial resistance
to yawing displacements. The inertial resistance is
amplified by added mass, which is proportional to
the square of the dorso-ventral span. Consequently,
span is often further increased by dorsal fin(s).
Spines, especially in acanthopterygians stiffen these
fins thereby resisting bending due to pressure
acting on the fin that could reduce the span.
Indeed, the fringing fulcra, an apomorphy of the
actinopteri, is a row of scales strengthening fins
(Lauder and Liem 1983) that would have contributed
both to the production of thrust and to DH-stability.
Reducing loss of energy due to recoil
Elliptical cross-sections of the body can affect drag due
to separation of flow on the rear side, increasing the
energy costs associated with recoil. Such drag could be
especially large caudally where lateral amplitudes
become large. Highly specialized fast swimmers such
as thunnids have developed finlets along the posterior
part of their body that reduce separation (Aleyev 1977;
Weihs and Webb 1983; Fish and Lauder 2006). Some
elasmobranchs have gone in opposite different direction, flattening the head, such that the large, lateral,
recoil-motions produce less drag while enabling better
maneuverability (Weihs 1981).
761
Stability versus maneuvering
Active countering of recoil
The pectoral fins also can help reduce yawing recoil.
The inclination and sweep of the pectoral fins can
vary on each side of the body to produce a net lateral
force in the horizontal plane, countering yaw, as described in the early literature (e.g., Breder 1926; Gray
1933; Harris 1953; Aleyev 1977). This use of the pectoral fins may be associated with a vertical force creating destabilizing pitching and rolling torques.
Asymmetries in body form, deployment of the posterior median fin, and lift from differing inclination
of dorsal and ventral lobes of the tail all may counteract pitching (Weihs 2002). The latter vertical
forces can be produced both by heterocercal and homocercal tails (Lauder 2000). Rolling forces can be
counteracted by pelvic fins, as well as being damped
by large body-depth and by extension of the median
fins.
Recoil motions could potentially be counteracted
by powered lateral motions of the head that could be
more effective than moving other fins and nonlocomotor body parts. The distance from the head
to the CM is generally larger, thereby producing
greater correction of yawing torques. EMGs driving
the traveling propulsive waves caudally along the
body show alternating contractions of myotomal
muscle on each side of the body along its length
(Altringham and Ellerby 1999). Propulsive wavelengths are of the order of g to 1L, when anterior
muscle contractions could contribute to stabilizing
the head, either as their primary function or as a
result of initiating the propulsive wave.
Avoiding recoil
Perturbations on posture and on swimming trajectories may be distributed spatially around the CM by
long-based or multiple fins spanning CM. Then perturbing forces can cancel out and provide SH and
intrinsic DH stability (Sefati et al. 2013). Indeed,
such dynamic stability control is advantageous in
maneuvering, which can be rapidly initiated by modulating the expression of these forces.
The use of multiple propulsive elements is perhaps
the best-recognized mechanism that promotes
stability by cancelling recoil-forces. This is especially
well-developed in boxfishes and cowfishes (Hove
et al. 2001). Their pectoral and dorsal/anal fins occur
in orthogonal pairs that beat out of phase and
thereby balance forces and torques, resulting in
extremely smooth trajectories during swimming.
Similarly, some long-finned fishes can generate
traveling waves that move in opposite directions
along the fin, whose phase relationships can cancel
recoil-forces during hovering and swimming (RuizTorres et al. 2013). The role of body-shape working
with recoil-forces generated by propulsors is also well
illustrated for fishes with rigid tests, such as boxfishes. The keels on the body create vortices that
oppose yawing and pitching displacements (Bartol
et al. 2003, 2008; Van Wassenbergh et al. 2015).
Bartol et al. (2002, 2003) measured forces and moments in a water tunnel on whole bodies including
tails of several species of boxfishes and trunkfishes
and propose these contribute to stability (Bartol
et al. 2008). Van Wassenbergh et al. (2015) disagree,
based on computational fluid dynamics (CFD) and
measurements on carapaces, claiming vortex-related
forces are too small to be a significant factor in stabilization. The two studies used multiple but different methods that could affect the results and hence
conclusions. Differences in methods include: CFD
and particle image velocimetry (PIV), pressure measurement, source materials and verification in
making boxfish models, inclusion of the tail, location
of the CM, and sting location and shape suspending
models in flows. Clearly additional work is necessary
to understand the apparent conflicts between these
studies. In addition, we recommend that such studies
focus on rolling and pitching torques. Because the
effects of stability are manifest at small displacement
while maneuvers are characterized by large displacements (Weihs 1993), measurements should include
small angles of attack.
Unsteady motions during swimming
BCF swimming is far more complicated than the
motions at constant speed and the quasistatic
forces discussed so far. In practice, swimming is
based on cyclic motion with a dominant frequency
f such that the ground state is actually an oscillating,
curved fish. Classical analysis of stability seeks to
determine the effect of the insertion of a perturbation of given frequency into such an assumed-time
independent ground state. As fishes swim with periodic motions, some perturbations will have frequencies that positively reinforce the perturbing forces.
Consequently, extrinsic perturbations with harmonics of the swimming frequency may be most
challenging for stability, e.g., eddies with frequencies
as harmonics of f. These would be damped quickly
by the fish’s control-mechanisms until a threshold is
approached when the momentum and energy of an
eddy approach that of the fish (Cotel and Webb
2012). There can be positive effects of eddies when
fishes adjust to externally forced frequencies and are
able to harvest energy, as in the Kármán gait (Liao
762
2008). Nevertheless, this gait reflects a stability issue,
as motions that are out of phase would cause additional perturbations that need to be corrected to stay
in position, with negative impacts on the use of
energy or on the kind of habitat that could be occupied (Liao 2008).
Swimming in shallow water
Fish often swim in shallow water. Shallow water is
defined in various ways depending on the critical
part of the body and fins interacting with a solid
or air/water surface. In terms of yawing stability, relevant forces act on the body and tail, which tend to
have the same depth. Then shallow water is defined
as having a depth less than three body spans (Webb
et al. 1991; Quinn et al. 2014a). Such shallow water
enhances damping of yawing displacements, mainly
due to the channeling effect on drag. Thus, the
boundary-condition of zero velocity at the bottom
makes it ‘‘harder’’ to move sideways while production of waves at a free surface results in dissipating
recoil energy. This aspect of edge (wall) effects is
among several aspects of swimming near surfaces
needing exploration (Quinn et al. 2014a).
Extrinsic SH stability
Unsteadiness arises in the environment wherever flow
interacts with a solid surface, protuberances such as
rocks, coral reefs, large woody debris, the air/water
interface, and other organisms. The resulting turbulence affects habitat-choices, behavior, energetics, and
functional morphology (Liao 2008; Webb et al. 2010;
Heatwole and Fulton 2013). Unsteadiness in the
water-column can reduce performance and increase
the cost of swimming, but shear-zones at surfaces
and protuberances, and the eddies they generate, can
improve station-holding. Of critical importance for
eddies is the scale of disturbances relative to the size
of a fish, but adequate measurements have not always
been made, thereby resulting in apparently confounding results (Webb et al. 2010). Measuring this source
of unsteadiness requires PIV.
A major unknown aspect of fishes’ interactions
with unsteady flows is the postulated ability of
fishes to choose pathways through turbulent fields
of flow, behavior that is thought to reduce the cost
of migration (Weihs 1987; Webb et al. 2010) and
facilitate passage through fish-ladders (CastroSantos et al. 2009). For example, energy levels in
subsamples of populations before and after migration
often suggest energy-costs of swimming are lower
than expected, given the mean current speeds along
the route of migration (Weihs 1987). Discriminatory
P. W. Webb and D. Weihs
data specifically testing these ideas are largely lacking.
Fishes might do little more than swim near the
bottom, where shear-forces result in speeds of flow
below the average. Fish-ladders may work by providing resting sites in back eddies. Explicit tests are
needed to determine the ability of fishes to anticipate
eddy-patterns and thereby choose energy-minimizing
paths over distances of multiple fish-lengths.
Selection of energy-minimizing paths will require
distance sensors and analytical abilities for both diurnal and nocturnal navigation. Learning the geography of the habitat presumably plays a role when
migrations are repeated over relative short timescales (e.g., diurnal migrations). Such learning is unlikely in seasonal migrations when storms can
quickly rearrange streams, rivers, and shorelines
and are not possible for once-in-a-lifetime migrations. Thus, we suggest determining how fishes
choose pathways through unsteady flows is an important candidate for future studies, especially germane to behavioral and reproductive ecology and
habitat-improvement. Methods for creating and
quantitatively describing flow in the laboratory are
already available. Laboratory and field studies
should become more conclusive as equipment such
as telemetry with accelerometers, and sensors of pressure and flow become smaller, battery life becomes
longer, and recovery of data more certain.
Conclusions
Studies of stability and maneuverability have a long
history. Our review suggests mechanical issues associated with stability may be especially important to
understanding the evolution and ecology of fishes,
and other aquatic vertebrates. Several areas that
have been neglected are identified, especially in the
areas of hydrostatic stability and control of recoil
during steady swimming. In the latter area, we suggest, modulation of is especially important, and
will require collaboration among mathematicians,
engineers, and biologists, such as those that underscore much historical progress in understanding the
swimming of fishes. A major area of concern is
energy-costs both for stability and for various maneuvers, and in both static and turbulent habitats.
Stability and maneuverability are especially important for understanding ecology. The ability to stabilize posture and the trajectory of swimming
influences habitats chosen by fishes. Correlates between behavior and accelerations of CM that can
be recorded in free-swimming fishes have potential
to construct time-behavior budgets as such
Stability versus maneuvering
accelerometer records reflect control of stability and
maneuvers.
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