© 2002 by The Society for Integrative and Comparative Biology
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Maneuvering Hydrodynamics of Fish and Small Underwater Vehicles1
1 Propulsion, Hydrodynamics and Silencing Division, Naval Undersea Warfare Center, Newport, Rhode Island 02841
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SYNOPSIS |
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The understanding of fish maneuvering and its application to underwater rigid bodies are considered. The goal is to gain insight into stealth. The recent progress made in NUWC is reviewed. Fish morphology suggests that control fins for maneuverability have unique scalar relationships irrespective of their speed type. Maneuvering experiments are carried out with fish that are fast yet maneuverable. The gap in maneuverability between fish and small underwater vehicles is quantified. The hydrodynamics of a dorsal fin based brisk maneuvering device and a dual flapping foil device, as applied to rigid cylindrical bodies, are described. The role of pectoral wings in maneuvering and station keeping near surface waves is discussed. A pendulum model of dolphin swimming is presented to show that body length and tail flapping frequency are related. For nearly neutrally buoyant bodies, Froude number and maneuverability are related. Analysis of measurements indicates that the Strouhal number of dolphins is a constant. The mechanism of discrete and deterministic vortex shedding from oscillating control surfaces has the property of large amplitude unsteady forcing and an exquisite phase dependence, which makes it inherently amenable to active control for precision maneuvering. Theoretical control studies are carried out to demonstrate the feasibility of maneuverability of biologically inspired bodies under surface waves. The application of fish hydrodynamics to the silencing of propulsors is considered. Two strategies for the reduction of radiated noise are developed. The effects of a reduction of rotational rate are modeled. The active cambering of blades made of digitally programmable artificial muscles, and their thrust enhancement, are demonstrated. Next, wake momentum filling is carried out by artificial muscles at the trailing edge of a stator blade of an upstream stator propulsor, and articulating them like a fish tail. A reduction of radiated noise, called blade tonals, is demonstrated theoretically.
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INTRODUCTION |
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The engineering community generally believes that man made vehicles and machines are matured in development for steady state operation; biology-inspired further improvements might not be cost effective. However, the following two products are recent examples where induced drag of wing-tip vortices has been reduced with devices inspired by winglets of soaring eagles: the Spiroid Wing Tip of Gulfstream II aircraft developed at Boeing, and the propellor with tip-less round blades, by Bannasch (2000). Moreover, the following developments have opened up opportunities in unsteady operations, namely in maneuvering, in addition to, perhaps, in propulsion: our new understanding of the mechanisms of lift enhancement via unsteady vortex dynamics, the advances in digital control, control theory and active materials technology (Ellington, 1984
; Dickinson et al., 1999; Bandyopadhyay, 1999; Bar-Cohen, 2001; Madden et al., 2001). This paper reviews the related progress made at NUWC. The science of maneuvering and stealth in swimming and flight in nature is distilled and applied to conventional rigid bodied and generic underwater vehicles. To name a few, the present work rests on the theoretical foundations of fish locomotion laid by Taylor (1952), Lighthill (1975) and Wu (1971a–d, 1972), the numerical studies by Webb (1975) and the scaling laws of Triantafyllou and Triantafyllou (1995). |
LOW-SPEED MANEUVERING DYNAMICS OF FISH AND SMALL UNDERWATER VEHICLES |
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TOP
SYNOPSISINTRODUCTION LOW-SPEED MANEUVERING DYNAMICS... FISH-INSPIRED CONTROL FOR...FISH-INSPIRED CONTROL FOR...FISH-INSPIRED CONTROL FOR...BIOMIMETIC PROPULSOR: ACTIVE...CONCLUDING REMARKSReferences |
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Morphology of control surfaces
While biologists tend to attach importance to variations of a theme, engineers have a need to simplify, that is to minimize the variations, to arrive at a robust application. This probably is a reflection of the fact that, in general, in a comparable environment, emerging materials, actuators and their control are far from being as dynamically competent as those in living animals. It is reasonable to assume that even the static morphology of a fish can provide clues to its locomotion, habitat and ecology (see among others, Lighthill, 1975
; Aleyev, 1977; Bandyopadhyay et al., 1997). As a starting point, the simplified morphology of various fish families is examined to determine what makes some families more maneuverable than others. Maneuverability is defined as the minimum or commonly observed turning radius at a given normal acceleration. As suggested here, internal Froude number may be interpreted as a measure of maneuverability. Ability to make a sudden stop or start is also a measure of maneuvering ability although this is not considered here. Length scales of the body and fins of a fish defined in Figure 1 are examined. Twenty-eight species of fish are considered. They are classified into three categories: low-speed highly maneuverable, high speed poorly maneuverable, and an overlapping category. The relationship between fin morphology and the characteristics of maneuvering is shown in Figures 2 and 3. Several definite trends are observed. The result in Figure 3 agrees with observation that a cylinder, whose length to diameter ratio is less than 10, tends to be unstable. In the next section, a dorsal fin device for brisk maneuvering based on the result in Figure 2, is examined in the context of a rigid cylinder having a large aspect ratio.
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Gap in maneuverability
An experiment was carried out comparing the maneuverability of fish and small underwater vehicles to quantitatively establish the gap. Bluefish and mackerel, which are oceanic fast swimmers and yet are maneuverable, were selected for an experiment on maneuverability. Swimming tanks with baffles, as shown in Figure 4, were designed to photograph their trajectories. The turning dynamics were determined from the digitized trajectories. The results of coefficient of normal acceleration Cg versus turning radius r/L were compared with two small underwater vehicles as in Figure 5a. Here, Cg = (V2/r)/g is acceleration perpendicular to the path, V is total velocity, r is the radius of curvature in trajectory, and g is acceleration due to gravity. There is a large gap between the maneuvering capability of fish and the vehicles. There is a universal trend, namely,
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which is followed but in large turn radii only. Compared to underwater vehicles, fish can make the same radius turn at a normal acceleration that is lower by a factor that can be as large as 10. Lower speed and laminar flow could allow a fish to make stealthy maneuvers.
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Froude number and maneuverability
The scaling law in Figure 5a can be improved so that the wide Reynolds number range between natural and man-made bodies is covered. Propose that the coefficient of normal acceleration during a turn is a function of inertia forces, viscous forces and gravity forces. Define Re = VL/
, and an internal Froude number Fr = V/
, where V is speed, L is length, is kinematic viscosity, and g is acceleration due to gravity. These two ratios can be combined as:
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This combined parameter can be used to rescale the coefficient of normal acceleration shown in Figure 5a, and the result is shown in Figure 5b. The solid lines in Figure 5b are:
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The inverse power trend (3) is now followed over a greater range of turn radius in both fish and vehicle data. The significance of internal Froude number is as follows. Rewrite Fr as:
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In other words, this internal Froude number is a ratio of the time period of natural oscillation T like a pendulum, and the time it takes for the body to travel a distance equal to its length. This is reminiscent of Froude number that is relevant to wave drag of surface ships. It is interesting that the jets produced by fish, dolphins or whales for propulsion is similar to the recent flow visualization of ‘fluttering’ of long and light strips in a liquid by Belmonte et al. (1998) (also see Aleyev, 1977; Videler, 1993). (Lightness can be considered equivalent to near neutral buoyancy for submerged bodies.) On the other hand, heavy and short strips "tumble." Note that they also similarly define an (internal) Froude number, which is the reciprocal of the one given above. According to Belmonte et al. (1998), Fr defined as in the present work, have a high value for long and light strips which flutter. Thus, fish, dolphins and whales have high Froude numbers. On the other hand, short heavy animals like the beetle of Fish (1999) have a low value of Fr. Thus, we reinterpret the conclusion of Fish that flexibility of a humpback whale gives it higher maneuverability than that of a whirligig beetle which is rigid. We conclude that internal Froude number is related to maneuverability.
The commonality of the wake pattern and the terms in the Froude number for fluttering objects suggests that a dolphin can be modeled as a simple pendulum. The dolphin swimming data due to Rohr et al. (1998) was examined (Bandyopadhyay et al., 2000a). Table 1 is a summary of the data sets based on lengths. It was proposed that the jet responsible for dolphin propulsion is analogous to the jet due to the predominantly side to side motion of fluttering long or light strips when dropped freely in air or water. The natural frequencies of dolphins calculated based on the following pendulum model is shown in Table 1:
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When dolphin-swimming data based on their length are extrapolated to zero swimming speed, they agree with the calculated values. This length consideration accounts for the seeming scatter in the data. Thus, (buoyancy or lightness), length (slenderness) and Froude number appear to play important roles in maneuvering.
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Strouhal number of dolphins
It is known indirectly that fish propulsion takes place predominantly in the Strouhal number range of 0.25 < St < 0.35, where St = fA/U (Fish and Rohr, 1999
). Here, f and A are frequency and amplitude of tail oscillation, and U is speed of fish motion. For dolphins, measurements due to Rohr et al. (1998) indicate that the amplitude of oscillation is given by A/L = 0.2 ± 0.02 and it is independent of speed. When speed is expressed as body length traversed per second, then frequency can be expressed as: f = 1.1(U/L). This is supported by Rohr et al.'s (1998) data, the mean trend of Kayan and Pyatetskiy (introducing f = 0 at U = 0) and the accounting of natural frequency (see Fig. 5 in Bandyopadhyay et al., 2000a). With the above two relationships, for dolphins, we then have
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Thus, dolphins also have a similar and constant Strouhal number as fish, although their Reynolds number is much higher. This provides another layer of evidence that fish, dolphins and whales have similar Froude numbers and mechanisms of maneuvering and propulsion, irrespective of their Reynolds numbers.
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FISH-INSPIRED CONTROL FOR MANEUVERING: DORSAL FIN DEVICE |
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The maneuverability of normally, stable cylinders (length/diameter
10) is considered here. Their maneuverability is a slow process when it involves the pitching of small fins near the boat tails and a subsequent large-scale separation of the entire cylinder. A 76 mm diameter and 754 mm long cylinder model with a dorsal fin device that is abruptly deployable, and shown in Figure 6, was constructed. The model was towed in a tank and a typical result of the production of side thrust is shown in Figure 7. It is clear that the dorsal fin when cambered abruptly produces large levels of side thrust practically immediately.
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The theoretical study of biologically inspired control surfaces described later showed that the above result is of general significance. While man made vehicles, like aircraft, have a moment-based control, biologically based maneuvering of engineering bodies would be force based. One consequence of the latter is the faster under water response allowing a greater agility.
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FISH-INSPIRED CONTROL FOR MANEUVERING: SMALL AGILE VEHICLES |
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Pectoral fins and turning
Several species of fast yet agile fish (Fig. 8) have large pectoral fins. They do not have a gas bladder and retract these fins to control lift force. A detailed hydrodynamic coefficients based modeling was carried out to determine the effectiveness of these pectoral fins in low speed maneuvering in an engineering context. The computational modeling of cruise and turn was carried out on three configurations of the cylinder shown in Figure 9: a reference plain cylinder and two others where pairs of wings, of the order of cylinder diameter, are attached. All three cylinders are provided with a pair of tail planes for stability. The results are shown in Figures 10 and 11. The wings allow the cylinder to be sustained at lower angles of attack. They also allow the cylinder to make lower radii turns.
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Damping due to pectoral fins near surface waves
The effect of the pectoral wings was examined in presence of travelling surface waves. This geometry and the forces on a body due to linear theory are shown in Figure 12. The model shown in Figure 13 was constructed, where the wing was offset from the cylinder—below it and not above. The maximum and minimum values of the periodic coefficients of axial force and pitching moments are compared in Figure 14 between the hydrofoil-cylinder and the plain cylinder case. Here, coefficient of axial force Cfx = Fx/(
gbAf), and coefficient of pitching moment CTy = Ty/(gbAp 
), where Fx is axial force, Ty is pitching moment, is fluid density, g is acceleration due to gravity, b is peak-to-trough wave height, Af is frontal area, Ap is planform area, and is the offset of the hydrofoil leading edge from the model axis (=11.43 cm for the hydrofoil-cylinder model and D/2 for the plain cylinder). The pectoral wings have a stabilizing damping effect.
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Theoretical control study of biologically-inspired control surfaces
The control system synthesis of a small cylinder equipped with a pair of dorsal and caudal fins, and in the presence of surface waves, as shown in Figure 15, was examined. Closed loop control laws were derived using the dorsal and caudal fins for depth and pitch control, respectively. The system is shown in Figure 16. For the typical cylinder geometries considered here, Figure 17 shows an example of a simulation where precise depth control and pitch regulation were achieved using the dorsal fin only.
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– 
= Translation of Inertial Frame (Origin at Geometrical Center). XB – ZB = Body Fixed Coordinate System
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= u; (d) Pitch angle |
FISH-INSPIRED CONTROL FOR MANEUVERING: DUAL FLAPPING FOILS |
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Unsteady vortex mechanism
The remaining gap in turning ability depicted in Figure 5 between fish and rigid bodies can be attributed primarily to the absence of sufficient control surfaces and perhaps to flexibility of the main body in the latter category (Fish, 1999
). Fish-inspired control surfaces can therefore be a partial solution for maneuverability for rigid bodies. The hydrodynamic mechanism of maneuvering and stealth of the moving surfaces of a fish was examined in an engineering experiment. The head and tail oscillations of a fish were simulated on a rigid cylinder that also removed the added complication of body undulation. Several experiments were carried out on the strut mounted floating cylinder shown in Figure 18. A six-component dynamic load cell, three actuators and two displacement sensors for measuring the phase of the tail actuators were housed within the model. Dye visualization and phase-matched laser Doppler velocimetry measurements of the vortex shedding due to the oscillation of the tail flaps, and time histories of the forces and moments on the entire model assembly were carried out for an uniform freestream and various tail oscillation Strouhal numbers. In the first configuration, only the pair of flapping foils attached to the tail was oscillated and the slider in the nose was absent. In the second, the interaction of both the nose and tail oscillations was examined. Figures 19 and 20 show the photographs of the instrumented model in these two configurations, respectively.
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Figure 21 shows the measurements of time integrated axial (thrust) force coefficient (ca) versus Strouhal number (St) of tail flap oscillation, They are defined as: ca = F/(1/2
U
2D2) and St = fA/U. Here, F is axial force, is fluid density, U is freestream velocity, d is length scale of flap, f and A are flap oscillation frequency and amplitude, repectively. The data asymptotes to Lighthill's two-dimensional inviscid theory at Strouhal numbers that are below the range of fish. We believe that Lighthill's line indicates the natural shedding symptote. As St is increased, axial force generated becomes oscillation frequency dependent, approaching the forced shedding asymptote of Bandyopadhyay (1996).
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The pairs of tail flaps (Figs. 18 and 19) were oscillated in two modes termed waving and clapping. Their phase is the same and opposite in them, respectively, mimicking the motion the names indicate. The respective vorticity-velocity vector maps of the vortex shedding process in the axial plane are shown in Figures 22 and 23, where t* = tU
/D is phase and the locations of the flap trailing edges are indicated by two small filled squares on the vertical axis. The large cross-stream forces in waving mode (Fig. 22), which have maneuvering and axial components, owe their origin to the formation of a staggered vortex train. On the other hand, the clapping mode which produces a pure axial thrust only (Fig. 23), owes its origin to symmetric vortex trains which are mirror images of each other. The resulting induced velocity between the successive vortices in the waving (maneuvering) mode is inclined to the streamwise axis. On the other hand, it is a perfectly aligned streamwise jet in the clapping mode. The general conclusion is that the mechanism of discrete and deterministic vortex shedding from oscillating control surfaces has the property of large amplitude unsteady forcing and an exquisite phase dependence, which makes it inherently amenable to active control for precision maneuvering.
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Decay of wake
The vorticity-velocity vector maps were used to compute the circulation in the shed vortices by two methods: velocity line and vorticity area integrals. The circulation distributions are compared immediately after formation and after a short travel in Figure 24, for the waving mode. The maximum value of circulation (–
) drops by a factor of 3 within a mere distance of half the body diameter or flap width (D). The effect is a rapidly dissipating wake. Such a rapid drop is attributed to a transverse-to-freestream orientation of the vortex, rather than a streamwise orientation.
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In the second configuration (Fig. 20), a small obstruction, a maximum of 3–4 mm, was alternately protruded at the port and starboard sides of the nose to generate small vortices and simulate the effect of the head swaying of a fish. The vortices interacted with those shed from the dual flapping foils in the tail. The phase between the nose and tail flap motion was varied and the axial force signature on the entire cylinder assembly was measured. The time-integrated values are shown in Figure 25. It is remarkable that a fine thrust regulation within ±5% can be achieved by phased seeding of vortices between spatially distributed control surfaces.
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A biologically-inspired maneuvering vehicle
Measurements with the flapping foil cylinder model shown in Figure 19 indicate the following. At say, 20 cm/sec of flow speed, the steady drag levels are 1/100th of the peak unsteady forces and 1/50th of the time mean values due to the dual flapping foils. Thus, a remarkable feature of flapping foil locomotion is the production of large unsteady forces. However, in man-made systems, unsteady mechanisms are rare.
While their peak and mean values are large, the forces produced by flapping foils are inherently periodic with large differences between the minimum and maximum values. To generate large forces practically at all phase, pairs of flapping foils may therefore be operated out of phase. Figure 26 shows such a small vehicle. The dual flapping foils are arranged in a crucifix form. For pure thrust, the horizontal pairs operate out of phase with respect to the vertical pairs. Independent operation of the dual flapping foils could provide precision maneuvering. The main body has a laminar low drag profile. The low-speed swimming of the tethered neutrally buoyant vehicle in Figure 26 has been demonstrated in a small tank (Bandyopadhyay et al., 2000b).
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BIOMIMETIC PROPULSOR: ACTIVE NOISE CONTROL |
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In a predator-prey environment, the ideal underwater animal or vehicle should produce no noise and leave no wake signature. Thus the primary motivation for biological studies might be stealth. Maneuvering involves operation at off-design condition. Two strategies for propulsor noise reduction are examined here: reduction of rotational speed and blade tonals, where science distilled from biology may be applicable to engineering problems.
Reduction of rotational speed
A modeling was carried out to determine the effect of reduction of rotational rate of a propulsor on these sources of radiated noise: blade rate tonal due to wake deficit; trailing edge singing; and, ingested turbulence. They are expressed as rotational rate (RPM) to the power of 4, 5 and 6, respectively. The result is shown in Figure 27. A RPM reduction of 5% can give a 3–5 dB reduction in noise.
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In principle, we can propose that the application of unsteady lifting mechanisms of fruit fly (Ellington, 1984
; Dickinson et al., 1999), namely rotation lift, delayed stall and wake capture, might be an avenue for lift enhancement, leading to reduction of RPM. However, the implementation of these mechanisms to practical propulsors is fraught with difficulties. An important step in this direction involves the development of programmable cambering of blades using electro-active polymeric artificial muscles. A small two-bladed propulsor, shown in Figure 28, was built. A controller was built that could power the muscle electrodes by means of square waveforms: positive, negative or bi-polar. The peak volts, frequency and duty cycle could be varied. Figure 29 shows the types of blade cambering achieved. A low frequency pulse (O(1 Hz)) led to cambering and oscillation, while a high frequency pulse (O(100 Hz)) led to cambering only. Figure 30 shows an example of thrust enhancement of about 15% at a RPM of 520 due to blade cambering.
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Reduction of blade tonals
A rotor blade traversing the wake of stator blade experiences a time-dependent load due to vertical gust, which gives rise to radiated noise, called blade tonals. Lighthill's equation relating the load derivative to fluctuating pressure describes this overall process. Now, if the trailing edge of the upstream stator blade is oscillated like a fish tail, then the momentum deficit in the wake can be filled. This can be expected to reduce the vertical gust on the rotor, and thereby the noise radiated. This flow, shown in Figure 31, was modeled hydrodynamically. The controller, of the dynamic inversion type, is shown in Figure 32, which produces the circulation necessary to cancel the derivative of the lift. Sears' function is used to account for the phase difference between gust and rotor lift. The result of a controller canceling the derivative of the lift is shown in Figure 33. The peak-peak amplitude of the radiated noise is reduced by at least 40 dB.
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in rotor blade is reduced
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CONCLUDING REMARKS |
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The progress made at NUWC in bridging biology and the hydrodynamics of small underwater vehicles is reviewed. Maneuvering hydrodynamics is examined, the goal being the understanding of mechanisms of noise reduction. The approach is not to build robotic replicas of swimming or flying animals. Instead, the scientific principle is distilled and applications are conceived for retro-fitting to, or for modification of existing engineering vehicles or components.
The general conclusion is that, swimming and flight in nature are characterized by oscillating control surfaces, which could sometimes include the main body. The mechanism of discrete and deterministic vortex shedding from oscillating control surfaces has the property of large amplitude unsteady forcing and an exquisite phase dependence, which makes it suitable for active control for precision maneuvering, stealth and lift enhancement. The knowledge base is new but advanced. Several examples of applications to rigid cylinders and propulsors are discussed. However, the applications are not widely explored yet. Rich possibilities remain. Advances in actuator technology, like artificial muscles, are required for new and matured application of the recently understood unsteady vortex dynamics principles of swimming and flight in nature.
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ACKNOWLEDGMENTS |
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The author gratefully acknowledges the sponsorship of the Office of Naval Research (Codes 342 and 333), Program Managers Dr. Teresa McMullen, and Mr. James Fein, for early support. Collaboration with the following are acknowledged: Professor Anuradha Annaswamy of MIT, William P. Krol, Jr., William H. Nedderman, John Castano, James Dick, James Q. Rice and Daniel P. Thivierge of NUWC, Dr. William Macy of URI, Professor Sahjendra Singh of UNLV and Dr. Mehran Mojarrad of BPI of New Mexico.
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FOOTNOTES |
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1 From the Symposium Stability and Maneuverability presented at the Annual Meeting of the Society for Integrative and Comparative Biology, 3–7 January 2001, at Chicago, Illinois.
2 Present address: Code 342 Cognitive & Neural S&T Division, Office of Naval Research, 800 N. Quincy St., Arlington, Virginia 22217-5660. E-mail: bandyop{at}onr.navy.mil
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