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P A P E R

Robotic Models for Studying UndulatoryLocomotion in Fishes
A U T H O R SGeorge V. LauderJeanette LimRyan SheltonMuseum of Comparative Zoology,Harvard University
Chuck WittErik AndersonDepartment of MechanicalEngineering, Grove City College

James L. TangorraDepartment of MechanicalEngineering, Drexel University

A B S T R A C T
Many fish swim using body undulations to generate thrust and maneuver in
three dimensions. The pattern of body bending during steady rectilinear locomotionhas similar general characteristics in many fishes and involves a wave of increasingamplitude passing from the head region toward the tail. While great progress hasbeen made in understanding the mechanics of undulatory propulsion in fishes, theinability to control and precisely alter individual parameters such as oscillation fre-quency, body shape, and body stiffness, and the difficulty of measuring forces onfreely swimming fishes have greatly hampered our ability to understand the funda-mental mechanics of the undulatory mode of locomotion in aquatic systems. In thispaper, we present the use of a robotic flapping foil apparatus that allows these pa-rameters to be individually altered and forces measured on self-propelling flappingflexible foils that produce a wave-like motion very similar to that of freely swimmingfishes. We use this robotic device to explore the effects of changing swimmingspeed, foil length, and foil-trailing edge shape on locomotor hydrodynamics, thecost of transport, and the shape of the undulating foil during locomotion. Wealso examine the passive swimming capabilities of a freshly dead fish body. Finally,wemodel fin-fin interactions in fishes using dual-flapping foils and show that thrustcan be enhanced under correct conditions of foil phasing and spacing as a result ofthe downstream foil making use of vortical energy released by the upstream foil.Keywords: fish, robot, swimming, biomechanics
propulsive and maneuvering forces
using midline fins (dorsal and anal)

Introduction

Fish moving through the water arecapable of using a variety of locomotormodes. Some species swim nearly ex-clusively using their fins and generate

or paired fins (pectoral and pelvic)(Drucker & Lauder, 1999, 2001;Hove et al., 2001; Standen, 2008;Standen & Lauder, 2007). Otherspecies generate thrust primarily byactivating body musculature to bendthe body and generate waves passingfrom the head toward the tail (Gillis,1996; Jayne & Lauder, 1994, 1995c;Lauder & Tytell, 2006; Rome et al.,1993). The caudal or tail fin is gener-ally considered as an extension of thebody in most analyses of fish loco-motion, but the tail also possesses asubstantial array of intrinsic musclesthat appear to stiffen the tail duringsteady forward swimming and gener-ate a variety of complex tail conforma-tions during maneuvering (Flammang

& Lauder, 2008; Flammang &Lauder, 2009).

Despite the large number of stud-ies of fish undulatory locomotionover the last 20 years, there are stillmany unanswered questions abouthow the pattern of body deformationis generated, the effect of body stiff-ness on locomotor performance, whateffect different tail shapes have on lo-comotor function, and how hydro-dynamic interactions among differentfins might influence the generationof swimming forces. Studies of freelyswimming fishes have contributedenormously to our understanding ofthe mechanics of aquatic locomotion,but such an approach is necessarilylimited to the behaviors voluntarily

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executed by living fishes. And mea-suring locomotor forces on freelyswimming fishes is a challenging prop-osition. Furthermore, a wide array ofinteresting experimental manipula-tions, including changing the flexuralstiffness of the body, altering theshape of the tail, and changing thespacing between fins, are clearly im-possible to conduct in living animals.

Robotics offers a complementaryapproach to studies of living fishes byallowing manipulation of variety ofparameters such as flexural stiffness,aspect ratio, tail shape, and spacing be-tween adjacent fins. A simple roboticflapping foil device can be used to gen-erate undulatory locomotion in flexi-ble fish-like materials, and forces can

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be measured and the effect of changesin body length, stiffness and tail shapecan be quantified.

The focus of this paper will be onundulatory locomotion in the waterand the use of a robotic flapping appa-ratus to produce swimming in bothflexible plastic foils and a passivefreshly dead fish body. After a briefoverview of undulatory locomotionin fishes, we discuss the design of a ro-botic controller for flapping flexiblefoils that allows measurement of self-propelled speeds (SPS), forces, andtorques and the cost of transport asso-ciated with foils of different shapes andstiffnesses. In addition, we addressan important technical issue in studiesof aquatic propulsion: the effect ofswimming at non-SPS on body wave-form, patterns of force production andwake flow patterns. Finally, we presentdata on the swimming performanceof passive fish bodies and discuss thefuture for studies of robotically con-trolled undulatory locomotion. Ouroverall aim is to introduce a numberof case studies with data that showthe utility of this approach for studyingunderwater propulsion and to presentan overview of how such studies canprovide new ideas and tests for currentviews of how fish swim.

Overview of UndulatoryPropulsion in Fishes

When fish swim using their bodiesand tail fin as the primary thrust gen-erators, they pass a wave of bendingdown the body that increases in am-plitude from the head toward the tail(Donley & Dickson, 2000; Gillis,1996; Jayne & Lauder, 1995a, 1995b;Liao, 2002; Long et al., 1994). Thisbending wave is produced by a waveof muscular activity that also moves

42 Marine Technology Society Journa

posteriorly and is created by spinalcord and hindbrain pattern generators(Bone et al., 1978; Fetcho & Svoboda,1993; Fetcho, 1986; Shadwick &Gemballa, 2006). Muscular powerto generate thrust is produced pri-marily by the segmented myotomalmusculature in the posterior region offish, especially during slow swimming( Johnson et al., 1994; Rome et al.,1993; Syme, 2006), and as speed in-creases more anterior body musclesare recruited to power locomotion.The division of the segmented bodymusculature into superficial “red” fi-bers and deeper and more complexlyarranged “white” fibers also plays animportant role in understanding themultiple “gaits” used by fishes; the rel-ative roles of these two types of musclefibers can change dramatically as fishchange swimming speed and executerapid locomotor behaviors such asfast-start escapes ( Jayne & Lauder,1993; Tytell & Lauder, 2002).

When fish swim slowly using bodyundulations, oscillation of the frontthird or so of the body is quite smalleven in species as diverse as eels, trout,and tuna (Donley & Dickson, 2000;Gillis, 1998; Lauder & Tytell, 2006),and a primary role of this reduced os-cillation appears to be drag reductionby minimizing the frontal area of thefish that encounters oncoming flow.As swimming speed increases, thefront region of fish shows increasinglylarge side-to-side oscillations, which isin part a reflection of the recruitmentof body musculature in this more an-terior region of the fish.

Even when fish swim by undula-tory propulsion using the productionof traveling waves, other fins oftenare used too, and it is incorrect to sug-gest that fish locomotor modes repre-sent completely distinct patterns ofmotion. For example, when trout or

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bluegill sunfish swim using body un-dulations, they are also actively usingtheir dorsal and anal fins, which playa key role in balancing roll torquesand generating thrust (Drucker &Lauder, 2001, 2005; Standen &Lauder, 2005, 2007). In addition,the pelvic fins play an important rolein controlling body stability duringlocomotion in fishes (Harris, 1936,1938; Standen, 2008, 2010). Fishesalso vary in tail shape, and the distinc-tion between the externally symmetri-cal (homocercal) tail of teleost fishesand the asymmetrical tail in sharksand fish such as sturgeon (Lauder,1989, 2000) is well known. The hetero-cercal tail shape induces torques aroundthe body center of mass that requirescompensatory changes in body posi-tion to allow steady horizontal swim-ming (Liao & Lauder, 2000; Wilga& Lauder, 2002, 2004b).

Although considerable progress hasbeen made in studies of fish locomo-tion by investigating the kinematics,muscle activity, and hydrodynamicsof live fishes swimming steadily andmaneuvering, there are many limitsto studies of this kind. Perhaps thetwo greatest limitations to researchon live fishes are (1) the considerabledifficulty in measuring locomotorforces and torques produced by thebending body as fish swim freely and(2) the inability to manipulate keyvariables such as body length, aspectratio, tail shape, and stiffness. Withoutan ability to alter such key componentsthat govern locomotor dynamics, wewill be limited in our understandingof the factors that influence undula-tory propulsion in the water.

The purpose of this paper is to pre-sent a number of new case studiesusing a robotic flapping device forgenerating undulatory locomotion inengineered materials as a means of

better understanding how fish swimand the dynamics of undulatory pro-pulsion. We discuss different examplesto show the utility of this approach andhow use of simple flexible foils informsstudies of fish locomotion and pointsto new avenues of research.

Simple Robotic Models ofFishUndulatory Locomotion

We have designed a robotic appara-tus that produces controlled heave andpitch motions of flapping foils. Themost important features of this deviceare (1) that it can be set up to be self-propelling (allowing locomotion byflapping foils at their natural swim-ming speed and not only at imposedspeeds) and (2) that forces and torquescan be measured on flapping foilsduring self-propelled swimming sothat within-cycle patterns of force andtorque oscillation can be comparedamong foils with different shapesand stiffnesses at different swimmingspeeds. This apparatus was designedwith two sets of flapping foils in seriesin order to be able to model, with foils,the interactions that can occur be-tween fins of fishes that are arrangedin series such as the dorsal and analfins and the tail fin (discussed furtherin the section on fin-fin interactionsbelow).

A general description of the firstgeneration of this apparatus is pre-sented in Lauder et al. (2007). Briefly,heave and pitch motors and rotaryencoders that allow readouts of foil po-sition are mounted on a carriage abovea recirculating flow tank. This carriageis supported on low friction air bear-ings, which allow the carriage to movein response to foil thrust and drag forcesgenerated during flapping motions.The second generation version of thisapparatus has an ATI Nano-17 six-

axis force/torque sensor mounted onthe shaft supporting the foils (Fig-ure 1A). This permits measurementof three forces and three torques duringself-propulsion at sample rates thatallow quantification of within-cyclepatterns of force production even dur-ing self-propulsion. In addition, a lin-ear encoder mounted on the carriageallows a readout of carriage position,

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and this is used by a Labview programto calculate SPS from data generatedby a series of swimming tests at arange of speeds. Synchronizing signalsfrom the LabView program control-ling the heave and pitch motors areused to trigger data acquisition fromthe ATI sensor and also to triggerimage acquisition from three syn-chronized Photron high-speed video

FIGURE 1

Images of a variety of flexible foils and freshly dead trout (Oncorhynchus mykiss) attached to arobotic flapping foil apparatus (see Lauder et al., 2007, for details on the basic design). (A) Flexiblefoil actuated at its leading edge in heave and pitch, suspended in a recirculating flow tank. The redarrow points to an ATI Nano-17 6-axis force/torque transducer on the foil shaft. (B, C) Flexible foilswith different trailing edge shapes are used to study the effect of tail shape on swimming performance.(D, E, F) Images of a trout held behind the head to allow imposition of heave and pitch motions tostudy passive body properties. (E) An image from a trout self-propelling under an imposed heavemotion; the body waveform produced is very similar to that generated during swimming.


cameras. Images of the flapping foils toquantify both foil motion and hydro-dynamic flow patterns are thussynchronized with force and torquemeasurements on the foil and withthe imposed heave and pitch motionson the foil.

The overall goals of using a flap-ping foil robotic device are to simplifyand control as much as possible pat-terns of motion imposed on flexibleand rigid foils that swim through thewater and to allow direct testing ofthe effect on propulsion of a varietyof key factors relevant to understand-ing fish locomotion: tail shape (Fig-ures 1B and 1C), foil flexibility, andinteractions among fins. This roboticapparatus can also be used to examinethe passive swimming capabilities of afreshly dead fish body (Figures 1D, 1E,and 1F ), and below we present data onthe ability of fish bodies to passivelypropel and generate propulsive wave-forms under imposed motions. Foilsof various kinds are attached to a stain-less steel sandwich bar (to hold theleading edge of flexible materials; Fig-ures 1A, 1B, and 1C) to a solid 8-mmshaft (for rigid NACA 0012 foils; seeLauder et al., 2007, and data shownin Figure 11), and flexible fish bodiesare mounted in a holder that is at-tached behind the head (Figures 1Dand 1F). Holding systems are designednot to flex in response to imposedheave and pitch motions and to allowforces and torques generated by swim-ming foils to be transmitted to theforce/torque sensor on the shaft. Hold-ing systems such as the sandwich barsystem do have their own drag, andthis drag is time-dependent due tothe heaving and pitching motion ofthe holding system as the foils self-propel. It is thus not possible to givea single value for the drag of the foilholding system. Since all foil compari-


sons within a single experimental typeused the same holding system andweretreated identically, we do not presentdata on the performance of the holdingapparatus alone.

Sample data from a self-propelledflapping foil actuated in heave onlyare shown in Figure 2. Monitoringfoil shaft position andmeasuring forcesin the X (upstream-downstream) andY (side to side) directions allows calcu-lation of the heave velocity and otherderived quantities such as the instan-taneous power required by the foil toswim and the coefficients of thrustand power. The cost of transport is cal-culated by measuring the foil mass anddividing the cost/meter by this valuefor each foil (see Table 1). Typicalpeak thrust coefficients for highly flex-ible foils (flexural stiffness in the rangeof 10−4 to 10−6 N m2) are in the rangeof ±0.2, lower than typical for rigidfoils, but these flexible foils none-theless are capable of self-propulsionat speeds of 10-30 cm s−1.

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A key technical issue arises in stud-ies of flapping foil propulsion thathope to imitate the self-propelled con-dition achieved by swimming fishes:foils that are not self-propelling mayexhibit patterns of thrust oscillationthat are not centered around zero.Any foil that is truly self-propelling(and not being dragged through thewater at speeds slower or faster thanit would naturally move) should gener-ate thrust in an oscillatory pattern, andthrust integrated over a single flappingcycle should equal zero. Graphs in theliterature of thrust coefficients duringfoil-based locomotion that are notcentered around zero indicate thatthe foil was being towed above orbelow the SPS and do not reflect theself-propelled condition. Data fromrecordings of foil forces generatedduring self-propulsion and at speedsbelow and above the SPS are shownin Figure 3. During self-propelledswimming, the thrust coefficient hasa mean of zero over each flapping

FIGURE 2

Sample data from a self-propelling flexible plastic foil (flexural stiffness = 9.2 × 10−5 N m2) actu-ated in heave at the leading edge at 2-Hz frequency. Heave position is monitored by rotary encoders,and X andY forces are measured by a force transducer mounted on the foil shaft. From the data onfoil position, force, and velocity, we make calculations of the instantaneous power, and dimension-less thrust and power coefficients. The dashed lines indicate zero for each trace.

cycle. When foils are forced to swimabove their SPS, the thrust coefficientcurves shift above the zero baseline,and when forced swimming occurs atspeeds below the SPS, data are shiftedbelow the baseline.

Change in foil swimming speedabove and below the SPS can alsohave dramatic effects on the kinemat-ics of the foil (Lauder et al., 2011) andon the hydrodynamic wakes displayedby swimming foils. Figure 4 shows

the shape observed during swimmingof a flexible foil (flexural stiffness,3.1 × 10−6 N m2) and the hydro-dynamic wakes that result from swim-ming at, below, and above the averageSPS. During swimming at an imposedspeed below the SPS, the trailing edgeof the foil has a large amplitude andirregular motion that produces a widebifurcating wake with separate mo-mentum jets to each side (Figure 4A).At the SPS where the foil is allowed to

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swim freely with no imposed con-straints on speed, the foil bends intoa regular ribbon-like pattern witha fish-like body wake (see Nauen &Lauder, 2002a, 2002b) and alternatingcenters of vorticity with a fluid jet thatmeanders in between these vorticalcenters (Figure 4B). Above the SPSwhere the external free-stream flowis increased above SPS and the foil isforced to swim against the increasedflow, the foil shape exhibits largeamplitude wave-like motion and asubstantial drag-like wake with fluidvelocities below that of the free streamin the wake (Figure 4C). These chang-ing patterns of foil kinematics andhydrodynamics during swimmingthat are not under conditions of self-propulsion are most easily seen in highlyflexible materials (flexural stiffnesses inthe range of 10−3 to 10−6 N m2) wherethe fluid-structure interaction is mostevident visually. These flexible materi-als which have flexural stiffnesses similarto those of fish (McHenry et al., 1995)show fish-like propulsion and deforminto wave-like patterns with a fish-likewake (Figure 4B) when allowed to self-propel in a low-friction system.

The structure of the wake behindflapping flexible foils has a significantthree-dimensional component dueto the finite chord and span, and we

TABLE 1

Locomotor properties of flexible plastic foils of three lengths while self-propelling.

Foil length
SPS (m/s) Reynolds number Strouhal number Work/cycle (mJ) Cost/meter (mJ/m)
gust 2011 Vo

Cost of transport (mJ/g/m)

20
0.105 21,114 0.831 2.248 42.59 106.47
25
0.101 25,250 0.775 2.257 44.69 89.39
35
0.091 31,850 0.349 2.233 49.08 70.11
The three foils were made of the same material with a flexural stiffness of 3.1 × 10−6 N m2. These highly flexible foils propel at relatively high Strouhal numbers atshorter lengths.Reynolds and Strouhal numbers are dimensionless.Foils were actuated at their leading edge with ±1 cm heave, no pitch, at 2 Hz.Foil span was 6.8 cm for all three foils.Standard errors for all parameters ranged from 0.3% to 1.5% of the mean values in each column.

FIGURE 3

Graph showing the dimensionless thrust coefficient versus time for a flexible plastic foil (flexuralstiffness = 9.2 × 10−5 N m2), 20 cm long, 6.8 cm high, actuated in heave ±1 cm at the leadingedge at 2-Hz frequency. The red curve shows data for the self-propelled condition (18.8 cm/s),during which the thrust coefficient integrated over a single flapping cycle equals zero. Note thatwhen experiments are done under non-self-propelling conditions (green and blue curves) theplots shift up or down so that the integrated coefficient over a flapping cycle is no longer zero.Data shown have been digitally filtered with a bandpass filter. Strouhal number for this exper-iment = 0.3 at the SPS (red curve). (Color versions of figures available online at: http://www.ingentaconnect.com/content/mts/mtsj/2011/00000045/00000004.)

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quantified this aspect of foil locomotordynamics using the volumetric flowvisualization system described byFlammang et al. (2011a, 2011b) andTroolin and Longmire (2010). Fig-ure 5 shows how each of the centers


of vorticity in the ribbon-like patternshown in Figure 4B actually representvortical columns on each side of thefoil, which connect to each other aboveand below the foil. These inter-columnconnections occur both to upstream

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and downstream columns on thesame side and also across the foil tovortical columns on the opposite side(Figure 5B). To date no other studieshave provided three-dimensionalwake snapshots during self-propulsionin highly flexible foils, but such studiesin the future may reveal interestingpatterns of wake interaction amongcomponents of the foil and may con-tribute to explaining the dynamics ofpropulsion in the flexing bodies.

These results also suggest that atleast some of the diversity of fishwakes reported in the literature resultsfrom situations in which the fisheswere not swimming steadily, eitherin still water or against imposedflows, as wakes that look like thosein Figures 4A and 4C are frequentlypresented. Fish commonly accelerateand execute small maneuvers duringlocomotion and considerable effort isneeded to ensure that kinematics andwake flow patterns are taken at mo-ments when the fish is swimmingsteadily. Even small accelerations cansubstantially change fish wake flows(Tytell, 2004), and data from theflapping foil robot here illustrate thatthese kinematic and hydrodynamic al-terations can be reproduced with flex-ible foils.

One of the most straightforwardquestions that could be asked aboutundulatory propulsion and one thatis difficult to study with live fishes isthe effect of changing length alone onlocomotor performance.We clearly can-not alter fish length experimentallyand expect reasonable swimming per-formance, and comparing fish ofdifferent lengths (while useful for stud-ies of scaling) does not account for themany other changes in the muscula-ture and skeleton that occur as fishgrow. Table 1 shows data obtainedfrom three flexible foils of different

FIGURE 4

Hydrodynamics of propulsion in a flexible foil (flexural stiffness = 3.1 × 10−6 N m2) swimmingbelow, at, and above its SPS (8.55 cm/s). The foil was actuated at the leading edge with amplitude±0.5 cm, no pitch, at 3Hz, and is 20 cm long; Strouhal number at the SPS is 0.83 (see Table 1). Thebottom margin of the foil is marked in white. Yellow arrows indicate water velocity; red color in-dicates counterclockwise vorticity; blue color indicates clockwise vorticity. The panels on the leftshow the whole foil swimming, while the matched panels on the right show a close-in view of thewake structure. The effect of swimming at a non SPS is dramatic, both on foil shape and on wakestructure. In (A) the foil swam at an imposed speed of 4.75 cm/s, and in (C) the foil swam at animposed speed of 25.7 cm/s.

lengths made of the same material andactuated in heave only at the leadingedge. Altering the length of this flexibleswimming foil from 20 to 35 cm pro-duces only minor changes in the SPS

(from 10 to about 9 cm s−1) andhence in Reynolds number but dra-matically lowers the Strouhal number(from 0.83 to 0.34; Table 1) due tochanges in foil trailing edge amplitude.

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The work per cycle stays nearly con-stant, as does the cost per meter(Table 1), but the cost of transportdecreases substantially as the increasedmass of the longer foils does not resultin increased energy requirements forpropulsion. Foils such as these thatare composed of a very flexible materialself-propel at shorter lengths with aStrouhal number that is quite largerelative to most self-propelling bodies.At longer lengths, the Strouhal numberapproaches that of many swimmingfishes or rigid foils (Table 1).

Figure 6 shows changes in shapeof self-propelling foils made of thesame material (flexural stiffness, 3.1 ×10−6 N m2) that occur due to change

FIGURE 5

Volumetric flow visualization using the V3V technique (see Flammang et al., 2011a, 2011b) toimage the 3D wake structure behind the flexible foil shown in Figure 4B. The trailing edge ofthe foil is located at the −60 mm position on the x-axis, and the fluid structures shown are allin the wake of the foil. This foil was self-propelling and was actuated using the same parametersshown for Figure 4B. (A) The flexible foil achieved a ribbon-like shape and columns of vorticityextend vertically on either side of the foil. Vorticity is isosurfaced at a value of 3.1 (blue surface),and a horizontal slice through the wake is shown to correspond to that shown using 2D piv inFigure 4B (green plane with velocity vectors). (B) The vortical columns on each side of theribbon-like foil motion connect to each other across the top and bottom of the same side (yellowarrows) and opposite sides (red arrows).

FIGURE 6

Graphs to show the effect of length on theshape (amplitude envelope) of a flexible foil(flexural stiffness = 3.1 × 10−6 N m2) swim-ming at its SPS. This foil was 6.85 cm inchord, and of varying length, given in thecolor-coded legend. The leading edge was ac-tuated at 2 Hz and with amplitude of ±1 cm.Each plot shows the shape of the foil duringself-propulsion as indicated by the peak-to-peak amplitude of the sideways flapping mo-tion. (A, B) Foil shapes as the absolute distancealong the foil and as percentage of the total foillength, respectively.


in length. Longer foils show peak ex-cursions at the same locations as short-er foils (Figure 6A) but amplitudes thatare lower. Each of the foils of this ma-terial generates a ribbon-like shapewhen self-propelling with a consistentwave-like pattern down its length (seeFigure 4B). When the amplitude ofeach foil as a percentage of foil lengthis plotted (Figure 6B), changes inamplitude are more evident withside-to-side excursion amplitudes de-creasing as length increases while thewave-like pattern is retained. Theshortest foils have high amplitudesnear the trailing edge and an amplitudeenvelope that grows along the foilwhile the longer foils display a taperingamplitude envelope (compare blue andblack curves in Figure 6B). These datashow that length alone can have signif-icant effects on locomotor efficiencyand foil kinematics and that, all otherthings held constant, foil length in-creases on the order of 100% can pro-vide reduced costs of transport.

The Effect of TrailingEdge Shape onSwimming Performance

The shape of the trailing tail edgeof swimming fishes has been the sub-ject of considerable discussion in theliterature, with various authors con-sidering the advantages or disadvan-tages of fish tails with symmetrical,asymmetrical, or forked shapes (Affleck,1950; Aleev, 1969; Lauder, 1989;Plaut, 2000; Thomson, 1971, 1976;Wilga & Lauder, 2004a). One advan-tage of a robotic flapping foil approachis that the trailing edge of a flexibleflapping foil can be altered and a vari-ety of configurations constructed thatallow the effect of trailing edge shapealone on locomotor performance to


be investigated. We designed five dif-ferent flexible foils (material flexuralstiffness = 3.1 × 10−4 N m2) that con-trol for total length and foil area andallow different tail shapes to be com-pared for the effect of these changeson swimming speed (Figure 7). Foil 1corresponds to a highly abstracted“trout-like” body shape with a mostlyvertical trailing tail edge, while foils 3and 4 present a shark-like tail trailingedge shape. Many fish have forkedtails, and this shape is represented byfoil 5.

The fastest swimming foil (Fig-ure 7, foil 4) is the one with themost area near the axis of actuation,even though it has the same area asseveral of the other foils. This resultcorresponds with our previous resultsshowing that foils with higher aspectratios and hence more material nearthe actuator swim significantly faster(Lauder et al., 2011). Interestingly,the foil with the angled trailing edgeand the shark tail shape (Figure 7,foil 3) swims significantly faster (P <0.003) than a foil with the same

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area but a straight trailing edge (Fig-ure 7, foil 1). The foil shape withthe significantly lowest swimmingspeed was the notched shape (Fig-ure 7, foil 5) even though it possessesthe same area as foils 1 and 3. The re-duced performance of the notchedshape may be due to bending of theupper and lower “lobes” of the tailduring the flapping motion, and kine-matic data obtained for these foilsdoes show that each lobe of this shapetwists during the flapping cycle. Twist-ing of the tail could be reduced by in-troducing stiffening elements, and thismay be one reason that fishes withhigh-performance tail shapes such astuna possess substantial stiffening ofboth the upper and lower tail lobes(Fierstine & Walters, 1968; Westneat& Wainwright, 2001).

Although the angled foil shapeswims significantly faster than a foilof the same area with a straight trailingedge (Figure 7: compare foils 1 and 3),more power is required for this foil toswim at this (self-propelled) speed(Figure 8A). The foils were made of

FIGURE 7

Propulsion by flexible foils of different shapes (material flexural stiffness = 3.1 × 10−4 N m2) swim-ming at their SPS. Each foil was actuated at ±1 cm heave at 2 Hz. Error bars are ±2 SE. Strouhalnumbers for these experiments range from 0.2 to 0.3. Foils were constructed to be of differingshapes and areas as follows: foil 1= square trailing edge, area = 131.2 cm2, dimensions = 6.85 cm ×19.15 cm; foil 2 = angled trailing edge, same length as foil 1, area = 107.71 cm2; foil 3 = angledtrailing edge, same area as foil 1; foil 4 = angled trailing edge: same length and area as foil 1; foil 5 =forked trailing edge: same area as foil 1.

the same material and have the samearea, so the cost of transport can becalculated in mJ/m. Figure 8B showsthat there is no significant difference(P = 0.07) between the foils in cost oftransport, although there is a trend inthe data toward the angled foil edgecosting less per meter resulting in themarginally non-significant difference.Further experiments on both foilswill be needed to extend this resultand to determine if in fact there is asmall difference in cost of transport be-tween these two foil types.

Undulatory Locomotionof a Passive Fish Body

Although flapping flexible foilsprovide a reasonable and simplemodel for undulatory propulsion in

fishes that minimizes the complexityof the foil and maintains constant ma-terial properties along the foil length,fish bodies are clearly different inshowing changing material propertiesfrom head to tail (McHenry et al.,1995). To better understand the loco-motor properties of the passive fishbody alone, we used freshly dead rain-bow trout (Oncorhynchus mykiss) andattached the body to the robotic flap-ping foil apparatus (Figures 1D, 1E,and 1F). We took care to ensurethat rigor mortis had not set in duringthe experiments and to ensure thatthe body had thus not stiffened dur-ing the time the flapping trials wereconducted. By actuating the passivetrout body in heave and pitch in var-ious combinations just behind thehead, we were able to construct aswimming performance surface forthe passive fish body (Figure 9).Body waveforms that are remarkablylike those occurring in live troutwere generated by the passive fishbodies. These data show that heaveamplitude overall has the greater ef-fect on swimming performance than

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changes in pitch alone. At any givenheave value increases in pitch furtherincrease swimming speed, but athigher heave amplitudes near ±2 cm,changes in pitch only produce modestincreases in SPS (Figure 9). Thesedata provide an interesting compari-son to our previous results showingthe effects of heave and pitch actuationon flexible foil propulsion (Lauderet al., 2011). In that study addingpitch motions to baseline heave actu-ation for foils of varying flexural stiff-ness did not produce significantincreases in foil SPS but did allowstiffer foils to maintain a relativelyhigh swimming speed that wouldhave declined with heave actuationonly.

As driving frequency increases, thetail beat amplitude of the passive flap-ping trout body remains relativelyconstant from 0.1 to 2.0 Hz beforeincreasing steadily to a peak at3.5 Hz (Figure 10). These tail beatamplitude values are very similar tothose observed in live trout swimmingunder a variety of locomotor con-ditions (Liao et al., 2003a; Webb,

FIGURE 8

Graphs of power consumed (A) and the cost oftransport (B) of foils 1 and 3 (see Figure 7) self-propelling. Both foils have the same surfacearea (131.2 cm2) and were actuated with±1 cm heave at 2 Hz. Power values are signifi-cantly different at P = 0.003. Cost of transportvalues are not significantly different at P = 0.07.

FIGURE 9

Performance surface of a freshly dead trout (Oncorhynchus mykiss) attached to a robotic control-ler (see Figures 1D, 1E, 1F) driven at 2 Hz at a variety of heave and pitch amplitudes. This trout was25.3 cm in total length. The graph shows how SPS of the passive trout body varies with differentpitch and heave actuation parameters.


1971; Webb et al., 1984) and in-creases in tail beat amplitude of themagnitude shown here for the passivetrout body are similar to those ob-served previously as fish increaseswimming speed and alter both fre-quency (primarily) and amplitude(to a lesser extent) of the tail beat.

Although during routine undula-tory swimming fish bodies are notpassive and are certainly stiffened bybody musculature as fish swim, forall but the fastest swimming speedslocomotion is powered by red musclefibers, which can make up a rathersmal l percentage of body mass(around 1.5% in largemouth bass;see Johnson et al., 1994). The bulkof the locomotor musculature is com-posed of white fibers that are not ac-tivated until the fastest swimmingspeeds are needed (Jayne & Lauder,1994, 1995a). Thus, the red musclefibers can be considered as acting tobend a mostly passive fish body com-posed of white muscle fibers and asso-ciated skeletal tissues that may not bemuch different in flexural stiffnessfrom the passive bodies studied here.


Also, data on the propulsion of pas-sive fish bodies are relevant to fishesswimming in turbulent flows. Troutswimming in a vortex street have beenshown to greatly alter body kinematicsand to utilize vortical energy shed fromobjects in the flow (Liao, 2004; Liaoet al., 2003b). The amplitude of thecenter of mass oscillation by troutswimming in the Karman gait is up to±2 cm for a 10-cm-long trout, whichcorresponds on a length-specific basisto themiddle region of the performancesurface in Figure 9 (Liao et al., 2003a).By greatly reducing muscle activitywhen trout enter a vortex street, thebody becomes largely passive and de-forms in response to the oncomingflows. Trouts are able to maintaintheir position in such flows entirely pas-sively and allow their bodies to extractenergy from the oncoming flow andgenerate thrust. This phenomenonwas further demonstrated in a studyusing passive foils and freshly deadfish bodies to show that the passivetrout body alone in a vortex wake cangenerate sufficient thrust to maintainposition in flow (Beal et al., 2006).

l

Fin-Fin HydrodynamicInteractions

One of the most intriguing aspectsof fish functional design is the arrange-ment of two fins in series that couldallow enhanced locomotor efficiencythrough hydrodynamic interactionsbetween the fins. For example, thedorsal fin and the anal fin in most fishesare located upstream of the caudal fin,and the wakes shed from these finscould interact with the tail fin duringlocomotion (Drucker & Lauder, 2001,2005; Tytell, 2006). Undulatory loco-motion using the body also causes theattached median fins to oscillate fromside to side, and in the fish species stud-ied so far these fins have been shown alsoto be actively oscillated by intrinsic finmusculature (Jayne et al., 1996) andthus can generate thrust on their own.

The possibility of fin-fin hydrody-namic interactions during locomotionhas been explored in a number ofexperimental papers on living fishes(Drucker & Lauder, 2001, 2005;Standen, 2008; Standen & Lauder,2007; Tytell, 2006; Webb & Keyes,1981) as well as using computationalapproaches (Akhtar et al., 2007; Weihset al., 2006). Akhtar et al. (2007) useddata on the kinematics of the bluegilldorsal fin and tail from Drucker andLauder (2001) and performed a two-dimensional computational analysis ofthe effect of having the fins in seriesand found that vortex shedding fromthe dorsal fin can increase thrust of thetail and that the amount of this thrustincrease depends on the phasing ofdorsal fin and tail motion.

In order to experimentally evaluatehydrodynamic fin-fin interactionsusing an apparatus in which phasingand the distance between fins can beexperimentally manipulated, we usedour robotic flapping foil apparatusin the dual-foil configuration with

FIGURE 10

Graph of tail-tip amplitude versus heave actuation frequency for a freshly dead trout (Oncorhynchusmykiss) attached to a robotic controller driving the passive body at a variety of frequencies. This troutwas 25.3 cm in total length and was actuated with a constant heave (±2 cm) and pitch (±5°). Errorbars are ±1 SE of the mean.

upstream and downstream foils. Foilswere rigid aluminum plates in aNACA 0012 airfoil shape, and eachfoil could be moved in heave andpitch, and both foils were driven froma common carriage mounted on airbearings as described above. Foils weremoved according to the parametersmeasured for bluegill sunfish dorsaland caudal fins (Drucker & Lauder,2001) and used by Akhtar et al.(2007) for their computational studyof this problem: the upstream foil wasmoved with heave amplitude of2.5 cm, the downstream foil at3.5 cm heave amplitude. Pitch ampli-tude for both foils was ±20°, and thefrequency for both foils was 1.7 Hz,corresponding to the frequency of finflapping in the bluegill sunfish modelcase (Drucker & Lauder, 2001).

Increases in SPS as a result ofchanging foil phase and distance re-flect thrust enhancement beyond thethrust achieved by the two foils oper-ating separately (assessed by offsettingthe foils from each other so that thedownstream foil was no longer inthe wake of the upstream foil). A gen-eral description of the dual-foil con-figurat ion and images of flowsaround and between the foils are pre-sented in Lauder et al. (2007). Datafor SPS were plotted over a range ofphase differences between the up-stream and downstream foils andpolynomial fits to these data pointswere used to determine the changein SPS with phase (e.g., Figure 11).

Here we present the results of ex-periments measuring the SPS of dualfoil flapping. Figure 11a shows the ef-fect of changing the phase of sinu-soidal motion of the downstreamfoil relative to the upstream foil onSPS for three different spacings be-tween the foils. In each case, there isa clear peak in swimming speed, and

as the distance between the foils is in-creased the peak swimming speedshifts toward a larger phase lag ofthe downstream foil. Interestingly,the maximal swimming speed of thetwo foils together does not change sig-nificantly as the distance between foilschanges, and alterations in phasingbetween the foils can thus be usedto compensate for changes in spacing.

July/Au

For each interfoil spacing, however,there are phase relationships that sig-nificantly reduce swimming perfor-mance, indicating that thrust of theentire two-foil system is sensitive tophase relationships between two foilsflapping. These data correspond verywell to the computational results ofAkhtar et al. (2007), which showedpeak thrust enhancement at a phase

FIGURE 11

Robotic model of fish fin hydrodynamic interactions. Dual NACA 0012 flapping foils in series aredriven by a robotic flapping foil apparatus. Details of this device and of the experimental setup aregiven in Lauder et al. (2007). (A) SPS plotted against the phase difference between the upstreamand downstream foils. Foils were arranged with 0 side-to-side offset (in cm) and are thus movingin line with each other with the downstream foil at varying chord length separations from theupstream foil: 0.5, 1, and 2 chord lengths separation; foil chord length was 6.85 cm. Furtherexperimental details are provided in the text. (B) Effect of changing the foil offsets (in cm) sothat the downstream foil is not moving in the direct wake of the upstream foil. The one chord lengthspacing with zero offset curve (dark blue) is the same as in panel (A) and is shown for reference.The other three plots show the effect of a 3.5 cm offset of the midline motion of the downstreamfoil relative to the upstream foil, removing it from the upstream foil wake.


of about 40°, very similar to our data(Figure 11A, blue curve) where theplateau around the SPS peak includesthe 40° value.

Figure 11B illustrates the changesin propulsion that result from movingthe downstream foil to the side by anoffset of 3.5 cm to move it out of thewake of the upstream foil. This ef-fectively creates a tandem foil con-figuration in which fluid dynamicinteractions between the foils areminimized, although not completelyeliminated. Comparison of the threeoffset curves to the zero offset curveshows that offsetting the foils reducesboth the SPS and the effect of foilphasing. Offset plots show reducedeffects of foil phasing (with lowermaxima and higher minima) and lessdistinct overall peaks in SPS, showingthat the downstream foil was not ableto improve swimming speed of thetwo foils together when removedfrom the upstream foil wake.

Computational work and prelimi-nary flow visualization (Lauder et al.,2007) data indicate that the behaviorof tandem foils can be explained interms of how the downstream foil inter-acts with fluid structures generated bythe upstream foil. The shifting of thepeak SPS with increased spacing ofthe tandem foils can be explained bythe fact that, for larger spacings, the con-vection of those structures to the down-stream foil takes longer. That time, tc, issimply the spacing between the foils, s,divided by the convection velocity, Uc.The increase in phase lag, Δϕ, neededso that the downstream foil meets thefluid structures at the right time is simply

Δϕ ¼ 2πfS2 � S1ð ÞUc

where f is the frequency of flappingand s2 and s1 are two different spacings.


If we assume that Uc is close to theSPS, this equation estimates the Δϕbetween the peak SPS well for thethree spacings we used. The equationpredicts Δϕ = 0.648 radians, or 37°,for a spacing difference of 0.5 chordlengths, and 74° for a difference ofone chord length. The Δϕ from ourdata are approximately 30°and 75° forthe corresponding spacing differences.The nearness of the foils and/or higherconvection velocities at the smallerspacings may explain the lower thanpredicted Δϕ for the peaks in SPS inthese cases. Overall, the agreement isvery good considering that the peaksare somewhat broad and the equationfor Δϕ is a simplification of the fluiddynamics.

The Future ofUndulatory Biorobotics

In this paper, we use a robotic toolfor investigating a variety of phenom-ena relating to undulatory propulsionin fishes and present experimentaldata that would be difficult if not im-possible to obtain from studying liveanimals. The promise of robotic mod-els for studying the biomechanics oflocomotion in fishes has just begunto be realized (Curet et al., 2011;Long et al., 2006, 2010; Tangorraet al., 2010, 2011), and fundamentalquestions relating to the mechanics ofundulatory propulsion remain to beaddressed. In particular, key unre-solved issues are the extent to whichchanges in body stiffness during pro-pulsion affect locomotor performance(see Long & Nipper, 1996) and howactive modulation of stiffness duringan undulatory cycle and acrosschanges in swimming speed areachieved and affect propulsive speedand efficiency.

l

To address these questions, a newgeneration of robotic undulatory de-vices will be needed that allow forcontrolled modulation of body stiff-ness and the phasing of stiffnesschanges with undulatory cycles ofcompression and tension on thebending fish or foil body. An addi-tional arena that is key to makingprogress in understanding undulatorymechanics is the ability to perturb thelocomotor system to assess how stiff-ness of the body relates to the abilityto recover from perturbations. Therehave been very few studies of pertur-bations of undulatory locomotor sys-tems (see Webb, 2004), and yet fishesoften swim in challenging hydrody-namic environments in which theyare forced to recover from impul-sive challenges to their undulatorypattern.

AcknowledgmentsThis work was supported the Of-

fice of Naval Research grant N00014-09-1-0352 on fin neuromechanicsmonitored by Dr. Thomas McKennaand by the National Science Foun-dation grant EFRI-0938043. Wethank the members of the Lauderand Tangorra labs for many helpfuldiscussions on fish fins and flexibleflapping foil propulsion and NateJackson for his assistance with thedual-flapping foil experiments. Manythanks to Brooke Flammang, TysonStrand, and Dan Troolin for assis-tance with the V3V volumetric flowimaging experiments on the undulat-ing plastic foil.

Lead Author:George V. LauderThe Museum ofComparative Zoology

26 Oxford Street, Harvard UniversityCambridge, MA 02138Email: [email protected]

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