PAPER Inspired by Sharks: A Biomimetic Skeleton for the Flapping, Propulsive Tail of an Aquatic Robot AUTHORS John H. Long, Jr. Department of Biology, Vassar College Tom Koob MiMedx Group, Inc. Justin Schaefer David Geffen School of Medicine, University of California Adam Summers Friday Harbor Labs, University of Washington Kurt Bantilan Department of Biology, Vassar College Sindre Grotmol Department of Biology, University of Bergen Marianne Porter Department of Biology, Vassar College ABSTRACT The vertebral column is the primary stiffening element of the body of fish. This serially jointed axial support system offers mechanical control of body bending through kinematic constraint and viscoelastic behavior. Because of the functional importance of the vertebral column in the body undulations that power swimming, we targeted the vertebral column of cartilaginous fishes—sharks, skates, and rays— for biomimetic replication. We examined the anatomy and mechanical properties of shark vertebral columns. Based on the vertebral anatomy, we built two classes of biomimetic vertebral column (BVC): (1) one in which the shape of the vertebrae varied and all else was held constant and (2) one in which the axial length of the invertebral joint varied and all else was held constant. Viscoelastic properties of the BVCs were compared to those of sharks at physiological bending frequencies. The BVCs with variable joint lengths were then used to build a propulsive tail, con- sisting of the BVC, a vertical septum, and a rigid caudal fin. The tail, in turn, was used as the propeller in a surface-swimming robot that was itself modeled after a biological system. As the BVC becomes stiffer, swimming speed of the robot in- creases, all else being equal. In addition, stiffer BVCs give the robot a longer stride length, the distance traveled in one cycle of the flapping tail. Keywords: biomimetics, robot, vertebral column, propulsion Propulsive Functions of Vertebral Columns I n sharks and other fi sh, the body’ s primary skeleton is the ver- tebral column, which runs from the head to the caudal fin (Summers & Long, 2006). The vertebral column is a jointed framework to which muscles attach and on which the muscles pull to create the traveling waves of flexure that transfer momentum from the body to the surrounding fluid. The vertebral column is composed of rigid elements, called vertebrae, connected by fl exible intervertebral joints (Grotmol et al., 2003; Koob & Long, 2000). The joints and their adjoining vertebrae limit the body’s ki- nematic degrees of freedom, constrain- ing bending primarily to the lateral direction in response to loads imposed by muscle, inertia, and external fluid forces (Grotmol et al., 2006; Porter et al., 2009; Symmons, 1979; Schmitz, 1995). In this way, the vertebral column functions to control dynamic reconfigurations of the self-propelling body. In roll-stable sharks and fish, lateral body bending is characterized by the overlay of harmonic and transient mo- tions that range from small traveling flexures to large-amplitude standing waves (Long et al., 2010). These bend- ing motions produce the propulsive forces that create forward swimming, turning maneuvers, and rapid accelera- tions. Across many different kinds of fish, the stiffness of the bending joints, measured as the apparent Young’s mod- ulus, E, ranges from 0.1 to 8 MPa (Long et al., 2002). The E is a me- chanical property, sometimes called the “material stiffness,” that measures the contribution of the material, inde- pendent of its geometric arrangement in the structure, to the structure’s resis- tance to changing shape when an exter- nal load is applied to it. July/August 2011 Volume 45 Number 4 119
Transcript
P A P E R
Inspired by Sharks: A Biomimetic Skeletonfor the Flapping, Propulsive Tail of anAquatic Robot
A U T H O R SJohn H. Long, Jr.Department of Biology,Vassar College
Tom KoobMiMedx Group, Inc.
Justin SchaeferDavid Geffen School of Medicine,University of California
Adam SummersFriday Harbor Labs,University of Washington
Kurt BantilanDepartment of Biology,Vassar College
Sindre GrotmolDepartment of Biology,University of Bergen
Marianne PorterDepartment of Biology,Vassar College
A B S T R A C T
The vertebral column is the primary stiffening element of the body of fish. This
serially jointed axial support system offers mechanical control of body bendingthrough kinematic constraint and viscoelastic behavior. Because of the functionalimportance of the vertebral column in the body undulations that power swimming,we targeted the vertebral column of cartilaginous fishes—sharks, skates, and rays—for biomimetic replication. We examined the anatomy and mechanical properties ofshark vertebral columns. Based on the vertebral anatomy, we built two classes ofbiomimetic vertebral column (BVC): (1) one in which the shape of the vertebraevaried and all else was held constant and (2) one in which the axial length of theinvertebral joint varied and all else was held constant. Viscoelastic properties ofthe BVCs were compared to those of sharks at physiological bending frequencies.The BVCs with variable joint lengths were then used to build a propulsive tail, con-sisting of the BVC, a vertical septum, and a rigid caudal fin. The tail, in turn, wasused as the propeller in a surface-swimming robot that was itself modeled after abiological system. As the BVC becomes stiffer, swimming speed of the robot in-creases, all else being equal. In addition, stiffer BVCs give the robot a longer stridelength, the distance traveled in one cycle of the flapping tail.Keywords: biomimetics, robot, vertebral column, propulsion
Propulsive Functions ofVertebral Columns
I n sharks and other fish, thebody’s primary skeleton is the ver-tebral column, which runs from thehead to the caudal fin (Summers &Long, 2006). The vertebral column isa jointed framework to which musclesattach and on which the muscles pullto create the traveling waves of flexurethat transfer momentum from thebody to the surrounding fluid. Thevertebral column is composed of rigid
elements, called vertebrae, connectedby flexible intervertebral joints(Grotmol et al., 2003; Koob &Long, 2000). The joints and theiradjoining vertebrae limit the body’s ki-nematic degrees of freedom, constrain-ing bending primarily to the lateraldirection in response to loads imposedby muscle, inertia, and external fluidforces (Grotmol et al., 2006; Porteret al., 2009; Symmons, 1979; Schmitz,1995). In this way, the vertebralcolumn functions to control dynamicreconfigurations of the self-propellingbody.
In roll-stable sharks and fish, lateralbody bending is characterized by theoverlay of harmonic and transient mo-
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tions that range from small travelingflexures to large-amplitude standingwaves (Long et al., 2010). These bend-ing motions produce the propulsiveforces that create forward swimming,turning maneuvers, and rapid accelera-tions. Across many different kinds offish, the stiffness of the bending joints,measured as the apparent Young’s mod-ulus, E, ranges from 0.1 to 8 MPa(Long et al., 2002). The E is a me-chanical property, sometimes calledthe “material stiffness,” that measuresthe contribution of the material, inde-pendent of its geometric arrangementin the structure, to the structure’s resis-tance to changing shape when an exter-nal load is applied to it.
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Joints with any appreciable stiffnessat all may at first seem to be a paradox:why not have low-stiffness joints thatcost little, in terms of mechanicalwork, to bend? The answer seems tobe two-fold: (1) Stiff joints increasetheir resistance, in terms of the abso-lute bending moment, M (in units ofNm), in proportion to the magnitudeof bending curvature, κ (m−1). This re-sistance, which can grow nonlinearlywith κ, serves as a brake, limitinglateral bending (Long et al., 2002).(2) Stiff joints store and release moremechanical work, so-called “elasticenergy,” than flexible joints (Long,1992). The amount of work releasedin elastic recoil is also in proportionto E and to the square of κ. Hence,the vertebral column functions bestas a spring when muscles have reachedtheir functional limits, at the end of a
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large-κ bend when connective tissuesare stiffest.
To explore the mechanical designspace of vertebral columns, we createdbiomimetic vertebral columns (BVCs).The BVCs are modeled after the ver-tebral column of sharks. We chosesharks’ vertebral columns since theyare structurally simple, compared tothose of bony fish, consisting of cylin-drical centra, small neural and hemalarches, and thin intervertebral joints(Figure 1). Together, a centrum andits arches are called a vertebra, and inthe cartilaginous sharks, skates, andrays, the vertebrae (plural form of‘vertebra’) are composed of mineral-ized cartilage. The compressive stiff-ness of these vertebrae, measured byE, ranges from 25 to 500 MPa, over-lapping the lower range of E for bone(Porter et al., 2006). Compared to the
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bone of mammals, for a given value of Ethe vertebrae of sharks are stronger, wherestrength is measured in terms of break-ing stress (Porter & Long, 2010). Thus,in some ways, vertebral columnsmade ofmineralized cartilage perform better thanvertebral columns made of bone.
In summary, vertebral columns serveat least three important propulsive func-tions during swimming: (1) they controldynamic reconfigurations of the bodyby limiting the kinematic degrees offreedom, (2) they brake high-amplitudebends by virtue of their stiffness, and(3) they integrate muscle work overtime by recoiling elastically.
To build a BVC that can function aspart of an aquatic propulsion system,we (1) characterized the morphology(size and shape) of the vertebral col-umns of sharks, (2) measured the me-chanical properties of those vertebralcolumns as they underwent sinusoidalbending, (3) used that informationabout morphology and mechanicalproperties to design BVCs, (4) testedBVCs as they underwent the dynamicbending characteristic of swimmingand propulsion, and (5) tested theBVCs as the primary skeleton in theflapping tail of an aquatic robot.
Morphology of Sharks’Vertebral Columns
In three individuals of the black-tip shark, Carcharinus limbatus, andthe bonnethead shark, Sphrynatiburo, we measured, from radio-graphs, the following features fromthe head to the beginning of the caudalfin (Figure 2): (1) the length of centrum,c, (2), the diameter of the centrum,d, (3) the length of the intervertebraljoint, j, and (4) the cone angle, Ξ,of the capsule of the joint. Black-tip sharks, members of the familyCarcharhinidae, were chosen because
FIGURE 1
Vertebral columns of two species of sharks. For scale, each centrum shown here is between 4- and6-mm long in the head-to-tail direction. Head is to the left; tail is to the right.
they are known to be fast swim-ming predators of fish. In contrast,bonnethead sharks, members of thehammerhead family Sphyrnidae, areknown less for their speed and morefor their maneuverability and abilityto find and eat crustaceans. Of sim-ilar adult body size, the two speciesrepresent contrasting swimming stylesand ecologies. Three individuals ofeach species were used for this study.
In the bonnethead shark, all themorphological features, except d,increased in size from the head to theend of the abdomen and then decreasedtowards the caudal fin (Figure 3). Inblacktip shark, only Ξ and c varied
from head to caudal fin. The sig-nificance (α = 0.05) of the morpholog-ical variation was determined with amultivariate analysis of covariance(MANCOVA), with species and posi-tion as main effects and individual asthe covariate (JMP 8.0.2., SAS Insti-tute, Cary, NC). Following an iden-tity model MANCOVA, univariateANCOVAs were also run.
The variation in the morphology ofthese vertebral columns was used toguide the construction of the BVCs.For each of the sharks’ morphologicalfeatures, we indicated which dimen-sions were used (see black arrows onthe ordinates, Figure 3).
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Mechanical Properties ofSharks’ Vertebral Columns
Using the same freshly dissectedvertebral columns from which mor-phology was measured, we conducted3-point dynamic bending tests usingan MTS model Mini Bionix 858(Eden Prairie, MN) with a 500 Nload cell. For blacktip sharks, eachvertebral column was cut into fivesegments of 19 vertebrae each. Forbonnethead sharks, each vertebralcolumn was cut into five segments of14 vertebrae each. The number of seg-ments in each species was varied tokeep the absolute length of each testsegment approximately equal. Eachsegment was subjected to sinusoidalbending at a frequency, f (Hz), andmaximum curvature, κ (m−1), variedto hold constant the time rate ofchange of κ, which is equivalent tothe strain rate (actuator displacementamplitude of 2 mm s−1).
To characterize the viscoelasticproperties of the vertebral column dur-ing bending, the apparent storage andloss moduli, E′ and E″ (MPa), respec-tively, were measured at each combi-nation of two species, five segmentpositions, and three κ. The E ′ mea-sures the purely elastic component ofthe stiffness; it is the force proportionalto the magnitude of the bending of thevertebral column. The E″measures thepurely viscous component of the stiff-ness; it is the force proportional to thevelocity of the bending of the vertebralcolumn. These properties were calcu-lated from the following formulae:E ′ ¼ E*cos δ a n d E ″ ¼ E*sin δ ,wherein E* ¼ FmaxL3
48Iymaxand δ i s the
phase lag (radians) between the dis-placement and load signals. Moreover,Fmax is the force (N) measured at theload cell, L is the gauge length of thespecimen (m), I is the specimen’s
FIGURE 2
Measuring vertebral morphology of blacktip and bonnethead sharks. Representative X-raysshow the heavily mineralized vertebral centra, which possess an “X” shape in this two-dimensionalview that is from cone-shaped joint capsules. The dark space between vertebrae is the intervertebraljoint. The morphology of each vertebra and intervertebral joint was measured from digitized land-marks (blue dots). (Color versions of figures available online at: http://www.ingentaconnect.com/content/mts/mtsj/2011/00000045/00000004.)
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second moment of area (m4), and ymax(m) is the distance from the presumedneutral plane of bending (transversecenter of specimen) and the lateral-most fibers of the specimen.
In blacktip sharks, the E ′ and E″values were of greater magnitude
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( p < 0.05) than those of bonnetheadsharks (Figure 4). In both species,E ′ increased towards the tail, an effectthat is amplified at higher values of κ,as indicated by a significant (p < 0.05)interaction term. The significance ofthe variation in E ′ and E″ was deter-
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mined using ANOVA, with species,position, and κ as main effects (JMP8.0.2., SAS Institute, Cary, NC).
Since the data blacktip and bonnet-head sharks were taken at a single am-plitude of strain rate (2.0 mm s−1), wesought additional information abouthow E′ and E″ vary with changes instrain rate. We also wanted to test thehypothesis that the intervertebral cap-sule, which contains liquid underabove-ambient pressure, uses its inter-nal fluid pressure to alter the apparentE′ and E″ of the vertebral column. Be-cause blacktip and bonnethead sharkswere not available for these tests,spiny dogfish, Squalus acanthias, wereused. Fresh 10-vertebrae segmentswere removed from the region of thefirst dorsal fin in three dogfish. Eachsegment was pressure-clamped at theterminal vertebrae and end-loadedwith bending moments, M (for exper-imental configuration, see Long et al.,2011). The bending motion was deliv-ered via moment arms attached to asingle-axis linear actuator using anMTSmodel Tytron 250 (Eden Prairie,MN) and a 50-N load cell. To test theeffects of both f and κ on E ′ and E″,each segment was bent sinusoidally ateach combination of five f values andthree κ values. In addition, to test theeffects of the integrity of the fluid-filledintervertebral joint capsule on E′ andE″, we repeated this suite of testsafter (a) puncturing a single joint cap-sule located in the middle of the seg-ment and (b) puncturing three jointcapsules, including the first one punc-tured and two adjoining capsules.
Increases in f increased only E′( p < 0.05) while increases in κincreased both E ′ and E″ (Figure 5).The only significant effect of punc-turing the intervertebral capsule waswhen three capsules were punctured,and even then only E″ increased. The
FIGURE 3
Vertebral morphology of sharks. Four dimensions were used to characterize the size and shape ofthe vertebral centra and the intervertebral joints. The means of three individuals for each speciesare shown; individuals ranged from 0.59 to 0.91 m in overall body length. The error bars indicatethe standard error of the mean. Black arrows show the specific dimensions represented in ourBVCs. MANCOVA, using the identity method, calculated a significant Wilkes λ (p < 0.0001),with a significant interaction of species and position and significant main effect of species; thecovariate, individual, was also significant. Partial correlations among the response variablesranged from a low of 0.38 between j and d to a high of 0.79 between Ξ and c. Significancelevel is indicated (*p < 0.05, **p < 0.01, ***p < 0.001).
significance of the variation in E′ andE″ was determined using ANCOVA,with puncture, f, and κ as main effectsand individual as the covariate (JMP8.0.2., SAS Institute, Cary, NC).
In summary, the vertebral columnsof sharks have mechanical propertiesthat are highly variable. As speciesand anatomical position change, so,too, do E ′ and E″. Within a given ver-tebral segment, the apparent storagemodulus, E ′, and the apparent lossmodulus, E″, can be altered by thebending that they undergo. Increasingthe segment’s curvature, κ, increasesbothE′ andE″; increasing the segment’sfrequency of bending, f, increases the
E′. Knowing the mechanical behaviorof shark vertebral columns under real-istic bending conditions creates specifi-cations for BVCs.
Designing BVCsTo begin to understand how to
control the mechanical behavior ofBVCs, we built two classes of shark-inspired BVC: (1) BVC with variablecone angle, Ξ (BVCΞ): vertebraewere created with variable Ξ and theBVC had constant joint length, j,and (2) BVCwith variable joint length,j (BVC j): vertebrae were created witha constant Ξ and the BVC had vari-
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able j. In addition to exploring the ef-fects of the structures Ξ and j, wealso varied the amount of cross-linkingof the hydrogel material forming thejoint. Thus, we explored the BVC“morphospace,” the variety of designsdescribed by three dimensions: Ξ, j,and cross-linking. Part of this explora-tion involved the challenge of makingcomposite structures that concatenateflexible and rigid elements. After fabri-cation and mechanical testing of bothclasses of BVC, we selected a singleclass, the BVCj, for performance test-ing in a tail-flapping aquatic robot.
In the BVCΞ , vertebrae were de-signed in software (SolidWorks,Dassault Systèmes SolidWorks Corp.,Concord, MA) to have the followingvalues of Ξ: 15°, 30°, and 45° (Fig-ure 6). These values correspond tolow, medium, and high values of Ξmeasured in sharks (see Figure 3).The diameter, d, and axial length, c,of the vertebrae were fixed at 1 cmfor both. The j of the column wasfixed at 0.25 cm.
Vertebrae were fabricated with arapid prototyper (Z-Corp, model310), which produced a porous, plas-ter part that was subsequently infil-trated with cyanoacrylate (EZ bond5cps, K&R International, DiamondBar, CA). This process yielded verte-brae with mean compressive moduli,E (MPa) of 43, 50, and 61 for verte-brae with values of Ξ at 15°, 30°,and 45°, respectively. These values ofE are within the range measured forshark vertebrae (Porter et al., 2006).
Vertebrae of a givenΞ were assem-bled into a BVCΞ in two stages. First,seven vertebrae were linked together,spaced at the fixed j, with eight horsehairs (E in tension of 900 MPa)arrayed axially and affixed to theouter circumference of the vertebrae.These horse hairs served as first
FIGURE 4
Mechanical properties of the vertebral columns of sharks in sinusoidal bending. Points are meansfrom three individuals. Error bars are the standard error of the mean. Size of the symbol indicatesthe relative magnitude of the curvature, κ. Significance level is indicated (n.s. = not significant,*p < 0.05, **p < 0.01, ***p < 0.001).
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approximations of the intervertebralligaments found in the vertebral col-umns of sharks. Second, a 10% por-cine gelatin solution was injected inbetween the vertebrae; the gelatin wassolidified at 4° C. Once solidified, eachBVCΞ was then subjected to oneof three fixation treatments: 0, 1%,or 5 % glutaraldehyde, a chemicalagent that cross-links the collagen inthe hydrogel. A total of nine differenttypes of BVCΞ were produced, witheach possible pairwise combination ofΞ and glutaraldehyde concentration.
In the BVCj, vertebrae were de-signed to have a Ξ of 90°, which cre-ated ring-shaped vertebrae (Figure 7).The d and c of the vertebrae werefixed at 0.5 and 1.0 cm, respectively.
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The overall length of the BVCj wasfixed at 8.4 cm. As the number ofnonterminal vertebrae were variedfrom 0 to 11, j varied from 720 to0.5 mm. These values of j created arange that extended below and abovethe range of j measured in sharks (seeFigure 3).
Ver tebrae were mi l l ed f romDelrin™, a polyoxymethylene thermo-plastic. Delrin has a compressive Eof 3.1 GPa (Delrin Design Guide,Module III, from DuPont), whichlies in the middle of the range ofE values reported for shark vertebrae(Porter et al., 2006).
The ring vertebrae had an inner di-ameter of 0.8 cm, which matched theouter diameter of hydrogels made from
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10% porcine gelatin fixed in 2.5%glutaraldehyde (Long et al., 2006).Vertebrae were slid onto the hydrogel,spaced evenly at the desired j, and af-fixed to the hydrogel with cyanoacry-late adhesive. A total of 12 differenttypes of BVCj were produced, onetype for each of the 12 different valuesof j. Three replicates of each type wereproduced and tested. Please note thatin the BVCj horse hairs were omittedbecause at all but the smallest valuesof j, the hairs cut into the hydrogel dur-ing bending.
Mechanical Propertiesof BVCs
The E ′ and E″ of the BVCs weremeasured in two different kinds ofsinusoidal bending test, which cor-responded to the tests performed onsharks’ vertebral columns. In theBVCΞ, 3-point bending tests wereconducted in a manner identical withthose on the blacktip and bonnetheadsharks. In the BVCj , end-loaded bend-ing tests were conducted in a manneridentical with those on the spiny dog-fish sharks. The E′ and E″ data for theBVCj have been analyzed previously(Long et al., 2011). In the analysishere, the data have been reanalyzedto calculate the mechanical work re-quired to bend the BVCj and the me-chanical work recovered as recoil.
In the BVCΞ , both E′ and E″ in-creased as the glutaraldehyde concen-tration increased, E ′ and E″ decreasedas the Ξ increased, and E ′ increasedand E″ decreased as κ increased (Fig-ure 8). The significance of the varia-tion in E ′ and E ″ was determinedusing ANOVA, with glutaraldehydeconcentration, Ξ, and κ. as maineffects ( JMP 8.0.2., SAS Institute,Cary, NC).
FIGURE 5
Mechanical properties of the vertebral column vary as a function of cycle frequency, f, and the in-tegrity of the intervertebral joint in the spiny dogfish, Squalus acanthias. Points are means fromthree individuals. Error bars are the standard error of the mean. Size of the symbol indicatesthe relative magnitude of the curvature, κ. Significance level is indicated (n.s. = not significant,*p < 0.05, **p < 0.01, ***p < 0.001).
Compared to the mechanical prop-erties of shark vertebral columns, theBVCΞ have values of E ′ that have awider range, overlapping the lowervalues and exceeding the sharks’ highervalues by an order of magnitude (com-
pare Figures 8 and 4). In contrast, theE″ values of the BVCΞ overlap onlywith those of the bonnethead shark;the BVCΞ has much lower values ofE″ than either the blacktip or spinydogfish shark. Moreover, the E ′ for
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BVCΞ decreases as κ increases; wemeasured the opposite trend in sharks(see Figures 4 and 5). Hence, theBVCΞ is not a good biological modelin this sense. Our hypothesis as to thesource of this strain softening is thatthe horse hairs force the column tobend primarily by compression, ratherthan by a combination of tension andcompression.
In the BVCj, both E ′ and E″ de-creased nonlinearly as j increased(Figure 9). Compared to the mechan-ical properties of dogfish vertebralcolumns, the BVCj span a nearly iden-tical range of E ′ and E″ values. Thegreatest sensitivity to changes in j oc-curred at the smallest values of j (Fig-ure 9) in the region that correspondsto the j measured in the vertebral col-umns of sharks (Figure 3). In datashown elsewhere (Long et al., 2011),E ′ and E ″ of the BVC j increasedwith increasing κ, just as in sharks(see Figure 5 herein). Moreover the E′increased with increasing f, as like-wise seen in sharks (Figure 5).
The mechanical work to bend theBVCj increased with increasing κ andincreasing E′ (Figure 9). The mechan-ical work recovered as elastic recoil,Wrecoil , is a function of the resil-ience, R, which, over all the testingconditions and sizes of joints, averaged76%.
BVCs in Aquatic VehiclesThe flexible skeletons of sharks and
fish are inspiring the design of novelpropulsive systems and aquatic vehi-cles (for review, see Fish, 2006; Long,2007, 2011). Fins with life-like flexi-bility have been built to propel a 0.7 mlong robotic turtle (Long et al., 2006)and a 0.4 m long robotic knife-fish (Curet et al., 2011). Bodies withlife-like flexibility have been built to
FIGURE 6
BVCs (BVCΞ) with variable cone angles, Ξ, and constant joint length, j.
FIGURE 7
BVCs (BVCj) with variable joint lengths, j, and constant cone angle, Ξ.
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F
Taa
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propel a 0.7-m long robotic electricray (Krishnamurthy et al., 2010),a 0.5-m long robotic trout (Kruusmaaet al., 2011), a 0.12-m long mechan-ical sunfish (McHenry et al., 1995),and a 0.5-m long mechanical pick-erel (Conte et al., 2010). Of theseself-propelled aquatic vehicles, onlythe mechanical pickerel has anythingresembling a vertebral column:a piece of spring steel designed torelease mechanical work to poweraccelerations.
The BVCj presented here was in-vented to propel a surface swimming,0.3-m long tadpole robot (Longet al., 2006; Doorly et al., 2009),known as Tadro4 (Figure 10). BVCj
were attached to a servo motor thatcreated a sinusoidally varying pitchingmotion of a tail. That pitch bent theBVCj, creating a bending momentthat propagated down the length ofthe BVCj in a traveling wave that, inturn, oscillated the terminal caudalfin. In this configuration, withoutdistributed muscles, the BVCj actsas both a transmission system, transfer-ring momentum from the servo motorto the caudal fin, and as a propeller,directly transferring momentum tothe surrounding fluid.
Since Tadro4 was built to behavereactively, with sensorimotor feedbacksystems creating foraging and predatoravoidance, we needed a version thatcould be programmed to swim straightusing a constant flapping frequency ofthe tail, f, and lateral amplitude of thecaudal fin. That modified version ofTadro4 was called MARMT (MobileAutonomous Robot for MechanicalTesting), and it had a hull length of17 cm and a tail length of 10 cm(Long et al., 2011).
Outfitted with a given BVC j ,MARMT’s steady swimming perfor-mance was measured as swimming
FIGURE 8
The mechanical properties of the BVCs (BVCΞ) with variable cone angles, Ξ, and constant jointlength, j. Horizontal bars indicate the median, the lower and upper limits of the box indicate the25th and 75th percentiles, respectively, and the whiskers indicate the range.
IGURE 9
he mechanical properties of the BVCs (BVCj) with variable joint length, j, and constant conengle, Ξ. Top row: points represent the means of E ′ and E ″ pooled across f and κ; error barsre one standard error of the mean. Bottom row: points are not pooled.
speed, U, and stride length, the slopeof the line of U regressed onto f,which measures the distance therobot travels over one period of theflapping tail (Figure 11). As f increased
for any BVCj, so, too, did the U. TheBVCj with greater values of E ′ pro-duced a more rapid increase in U,over the same range of change in f,compared to BVCj, with smaller values
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of E′ (Figure 11, top panel). The stridelength of MARMT increased initially,doubling as E ′ doubled, before taper-ing off.
When the BVCj operates in theflapping tail, MARMT’s swimmingperformance is clearly linked to themechanical properties of the BVCj , E′in the case shown here. Those me-chanical properties are, in turn, underthe control of the structure of theBVCj . Thus these experiments, takentogether, demonstrate the functionalrelationship between the structure ofthe BVCj and the performance of aself-propelled aquatic robot.
SummaryUsing the morphology and me-
chanical properties of the vertebralcolumns of sharks as our biologicaltarget, we built and tested a series ofBVC. The mechanical behavior ofthe BVCs, measured by the storageand loss moduli over a range of bend-ing frequencies and curvatures, can bealtered by changing (1) the materialproperties of the hydrogel that makesup the intervertebral joint, (2) thelength of the intervertebral joint, or(3) the shape of the vertebrae. BVCsare sufficient to function as propulsiveelements in swimming aquatic ro-bots: in Tadro4 and MARMT theBVC converts a simple pitch oscilla-tion from a servo motor into a waveof bending that drives the caudal finlaterally.
Having identified variables that in-fluence the mechanical behavior ofBVCs, we offer a few observations forthose wishing to build jointed, flexiblebiomimetic skeletons for use in flexi-ble, flapping propulsive systems:
(1) Design of biomimetic systems:Engineered systems that are much
FIGURE 10
The aquatic robot, Tadro4, is propelled by a BVC (BVCj). Tadro4 is a fully-autonomous surface-swimmer with a flattened circular body and propulsive undulatory tail. It is modeled after fish likethe extinct Drepanaspis and the living electric ray, Narcine. Using sensory input from photoresis-tors and IR proximity detectors, Tadro4 searches for and swims up light gradients while avoidingcollisions. Tadro4 is propelled by its submerged BVCj, which is wrapped in a thin membrane, at-tached to a caudal fin, and actuated by an oscillating servomotor. Tadro4 was developed by Doorlyet al. (2009). Photo of Drepanaspis specimen 8462, American Museum of Natural History. Photoof adult Narcine is courtesy of Dr. Steve Kajiura.
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simpler than the targeted biolog-ical system can match and extendthe targeted range of mechanicalbehaviors.(2) Control of mechanical proper-ties: The spacing of rigid elementsin a flexible matrix is more impor-tant than the shape of the rigid ele-ments or the material properties ofthe flexible material.(3) Control of reconfiguration: Be-cause of the strain- and strain-rate-dependence of viscoelastic materials,
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no passive, flexible propulsive sys-tem, if its E ′ and E″ matches thatof the vertebral column of sharks,will produce constant motions overa wide range of motor inputs.
This work is a straight-forward exam-ple of one method of biomimeticdesign (Fish, 2006; Long, 2007):describe, test, build, and test. Startby identifying a specific operationalcontext—aquatic undulatory pro-pulsion in this case. Then describe,quantitatively, the biological system’s
al
functional morphology. Next, testthe morphology’s mechanical behav-ior under physiologically relevant test-ing conditions. Finally, build and testsimple biomimetic models of the sys-tem that change just a single structuralvariable over a wide range. Repeat thisprocess with different variables, ceterisparibus, until the designer knowswhich variables permit the natural sys-tem and its operational range to bemimicked or extended in biomimeticform.
AcknowledgmentsWe thank Carl Bertsche, Nicole
Doorly, Carina Frias, AndresGutierrez,Jonathan Hirokawa, Kira Irving, DougPringle, Foster Ranney, HannahRosenblum, Hassan Sahktah, SoniaRoberts, Elise Stickles, Josh Sturm,and Janese Trimaldi for their help indesigning, building, and testing ver-tebral columns, BVCs, and the aquaticrobots. This work was supported bythe National Science Foundationof the USA (DBI-0442269 andIOS-0922605).
Lead Author:John H. Long, Jr.Department of Biology,Vassar College124 Raymond Avenue,Poughkeepsie, NY 12604-0513Email: [email protected]
Swimming performance of a surface-swimming robot, MARMT, propelled by a tail with the BVCj asthe primary skeleton. MARMT is a version of Tadro4 (Figure 10) modified for mechanical testingover a range of flapping frequencies of the tail, f (Hz). For all types of BVCj tested, swimming speedof MARMT, U, increased linearly with increases in f (all R2 values > 0.92). The rate of change ofU with respect to f is the stride length (distance traveled per period of the flapping cycle); it wasgreatest with BVCj having larger storage moduli, E ′. Three replicates of each kind of BVCj weretested (N = 36). Means (N = 12) are shown here. Data reanalyzed from Long et al., 2011.
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