Biorobotic insights into how animals swim

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?

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 U_t (the symbols are defined below, see List of symbols and abbreviations).

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.

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.

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.

A clear hysteretic unsteady behavior has not been directly evidenced in the literature on animal-inspired flying and swimming. In our measurements, such hysteresis is present in the cases of hovering and low speeds of tow, and this is eliminated as the speed is raised. The hysteresis increases at the extreme values of the angle of attack (Fig. 6), which coincide with the extreme roll positions of the fin. Our estimation of induced flow velocity – from an actuator disc method(Wakeling and Ellington, 1997)– is constant in time and space, which will lead to errors in the magnitude and direction of U_t. 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 versusroll 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.

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 C_X,wing=F̄_X/(1/2ρU²_wingA_planform),and the efficiency was defined asη _hydrodynamic=F̄_XU_∞/P̄_hydrodynamicfor U_∞>0 andη _hydrodynamic=F̄_XU_induced/P̄_hydrodynamicfor U_∞=0, where the induced velocity was estimated using the disk method of Wakeling and Ellington(Wakeling and Ellington,1997), meaning that the η_hydrodynamic values during hovering have higher uncertainties than those during cruising. All data with C_X,wing>1.5 are for hovering, and most of the data for C_X,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.

A convergence between the above results of the rigid biorobotic fin and those of the sunfish flexible pectoral fin was sought to explore the force production mechanism of sunfish pectoral fins. An explanation of how sunfish maintain station within close tolerance has been lacking. Lauder et al.(Lauder et al., 2007) 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.

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).

The following conclusions can be drawn from our controlled hydrodynamic experiments in a laboratory with biorobotic abstracted penguin pectoral wings.

1. A cross-flow vortex model of unsteady lift and drag forces has been proposed that is reasonably accurate, instant to instant. It follows that the rolling and pitching motions of unsteady pectoral appendages may be viewed as a means of `trapping' what are essentially inviscid drag vortices manifested at the fin leading and trailing edges over the fin surface for the augmentation of lift. From an engineering viewpoint, unsteady hydrodynamics is seen as a manifestation of the steady-state characteristics in the regime studied and not as a totally new phenomenon. However, unsteadiness in nature is old and steadiness is rare.

2. Measurements of hydrodynamic efficiency have been given for hovering and for forward motion. The peak efficiency for cruising is similar to that for two-dimensional foils and in the neighborhood of that for the pectoral fins of sunfish. Maneuvering is less efficient, hydrodynamically speaking, compared with cruising, and in the best cases it is twice as costly.

3. An optimization method has been shown to rapidly determine the fin oscillation parameters required for highest efficiency at a given forward speed, or during hovering for a given coefficient of force. The method does not require any prior knowledge of any steady or unsteady characteristics of the fin.

4. The biorobotic measurements of the rigid fin were used to explore the station-keeping mechanism of sunfish pectoral fins in a stream. It is proposed that a dynamic stall mechanism is present in the flexible fins of sunfish pectoral fins and that such fins fold to act essentially as pairs of symmetric biplanes that are conjoined at their trailing edges in order to balance unsteady force byproducts. The flexible fin appears to have a rich portfolio of control schemes for station keeping in streams of low speeds. Scaling up of rigid fins appears to be more feasible than scaling up of flexible fins.

List of symbols and abbreviations

 
• A_s

  fin swept area

 
• A_planform

  fin planform area

 
• c

  fin chord

 
• C_D,model

  coefficient of drag force; C_D,model=D/(1/2ρU_t²A_planform)

 
• C_L,model

  coefficient of lift force; C_L,model=L/(1/2ρU_t²A_planform)

 
• C_N,model

  coefficient of normal force; C_N,model=N/(1/2ρU_t²A_planform)

 
• C_X,wing

  coefficient of cycle-averaged force in X-direction

 
• f

  frequency of flapping (Hz)

 
• F_opt

  optimization function

 
• F̄X

  cycle-averaged force in direction of U

 
• F̄Y

  cycle-averaged force in horizontal direction transverse to U

 
• L_fin

  fin lift relative to instantaneous inflow angle

 
• m

  slope of N with α near α=0

 
• N

  normal force on fin

 
• P̄ _electric

  cycle-averaged electrical power

 
• P_hydrodynamic

  power put into the fluid by the fin

 
• P̄ _hydrodynamic

  cycle-averaged hydrodynamic power

 
• R_avg

  average radius of the fin swept area

 
• r_i

  inner radius of wing span

 
• r_o

  radius of wing tip

 
• s

  fin span

 
• St

  Strouhal number

 
• t

  time

 
• T_fin

  fin thrust relative to instantaneous inflow angle

 
• U

  flow velocity, defined as U_∞ or U_ind

 
• U_ind

  induced velocity through A_s

 
• U_∞

  velocity of tow carriage

 
• U_t

  absolute wing inflow velocity at R_avg

 
• U_wing

  flapping-induced fin velocity at R_avg

 
• X, Y, Z

  coordinate system attached to tow carriage

 
• X₀, Y₀, Z₀

  coordinate system attached to R_avg

 
• α

  instantaneous angle of attack of U_t on fin

 
• ηhydrodynamic

  efficiency defined using P̄_hydrodynamic

 
• ηelectric

  efficiency defined using P̄_electric

 
• θ

  pitch position of fin

 
• \({\dot{{\theta}}}\)

  pitch velocity

 
• θ0

  pitch amplitude of the fin

 
• θBias

  fixed pitch amplitude of the fin

 
• ρ

  fluid density

 
• τRoll

  roll torque measured at the motor output shaft

 
• τPitch

  pitch torque measured at the motor output shaft

 
• ϕ

  roll position of fin

 
• \({\dot{{\phi}}}\)

  roll velocity

 
• ϕ0

  roll amplitude of fin

 
• ψ

  phase between roll and pitch sinusoids, defined as positive for pitch leading roll

 
• ω

  frequency of flapping (rad s^–1)

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.