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Design considerations for an underwater soft-robot inspired from marine invertebrates

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2015 Bioinspir. Biomim. 10 065004

(http://iopscience.iop.org/1748-3190/10/6/065004)

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Bioinspir. Biomim. 10 (2015) 065004 doi:10.1088/1748-3190/10/6/065004

Design considerations for an underwater soft-robot inspired frommarine invertebrates

Michael Krieg1,3, Isaac Sledge2,3 andKamranMohseni1,2,3

1 Department ofMechanical andAerospace Engineering, University of Florida, USA2 Department of Electrical andComputer Engineering, University of Florida, USA3 Institute forNetworked Autonomous Systems, University of Florida, USA

E-mail:mohseni@ufl.edu

Keywords: optimization, jets, cephalopods

AbstractThis article serves as an overview of the unique challenges and opportunitiesmade possible by a soft,jellyfish inspired, underwater robot.We include a description of internal pressuremodeling as itrelates to propulsive performance, leading to a desired energy-minimizing volume flux program.Strategies for determining optimal actuator placement derived frombiological bodymotions arepresented. In addition a feedbackmechanism inspired by the epidermal line sensory systemofcephalopods is presented, whereby internal pressure distribution can be used to determine pertinentdeformation parameters.

1. Introduction

The field of soft robotics has received much attentionlately. One reason is that rigid structures are prone tocritical failure when disturbances or perturbationsresult in high enough stress; whereas, soft structuresare capable of bending to accommodate the distur-bance and return to an operational state [53]. Inaddition, there are several situations, such as tactilesensing or grasping where flexibility in an actuator canprove advantageous to a desired action. This is oftenthe case in biological underwater locomotion [32], butgenerally not reflected in propeller-dominated under-water robotics.

Flexibility is a defining property of propulsion inmarine animals [31]. Fine musculature and compliantmembranes permit both active and passive control ofswimming appendages. The degree of flexibility exhib-ited in propulsive mechanisms varies widely acrossmarine animals. The flexibility ranges from thunni-form swimmers, which can be generalized as con-joined rigid elements, to invertebrates like jellyfishcontaining no rigid elements whatsoever. One com-mon element seems to be that flexibility in flappingfoils enhances propulsive output, propulsive effi-ciency, or both [32, 59–61].

Marine mammals and fish utilize an inner rigidskeletal structure with muscles, tendons, and flexiblemembranes layered on top. Thunniform swimmers,

which are mostly rigid with oscillatory bending in thecaudal fin, can be replicated by a series of joints andrigid tail sections creating the biological kinematicswithout specifically incorporating flexible materials[3, 5, 58]. During testing on these platforms, it isobserved that the body deformations significantlyreduce the drag of the vehicle compared to a com-pletely rigid hull [5]. This represents a form of flex-ibility which is actively controlled with no freedom ofmotion. Pectoral fin paddling in labriform swimming,on the other hand, employs passive flexibility via flex-ible rays running down the fins [16, 28]. This flexibilityactually allows the fin to generate maneuvering forcesin the desired direction, whether the fin is flapping intowards the body or out away from it, a feature whichis present in passively flexible biomimetic fins [51], butnot rigid ones [18]. Many experimental studies havelooked into the advantages of flexible flapping foils interms of thrust production [1, 15, 52] and power input[13]. Quinn et al [42] investigated the effects of multi-ple, self-propelled swimming bodies, showing anincrease in propulsive economy with higher flexibilityand slower swimming speeds.

Other marine animal swimming modes like angu-illiform and rajiform require large scale undulations.Even though the deformation of the overall bodyshape is actively controlled, external flexible mem-branes become necessary to maintain the body shape,and thus play some role in the propulsivemechanisms.

RECEIVED

27March 2015

REVISED

27August 2015

ACCEPTED FOR PUBLICATION

14 September 2015

PUBLISHED

29October 2015

© 2015 IOPPublishing Ltd

Clark and Smits fabricated an actively oscillating,deformable fin, derived frommanta rays, and analyzedits hydrodynamic properties [11]. Additional studieswere carried out by Moored et al [36] to assess itsswimming potential and efficiency. Further progress ishighlighted in the survey article ofMoored et al [37].

So far, all the examples we have highlighted can beadapted to standard robotic platforms relatively easily,in that the flexibility exists in the propulsor, and therobot itself, or internal structure, is still a rigid body towhichmotors and standard actuators can be affixed. Itis in this regard that marine invertebrates, specificallythose belonging to phylum Medusozoa and classCephalopoda, are unique. These marine animals haveno rigid internal structures and maintain desiredshape utilizing what is referred to as a ‘muscularhydrostat’. In a ‘muscular hydrostat’ arrangement,muscles not only drive bodymotions, but also provideskeletal support for those motions. As such, these ani-mals provide a biological model to help guide designsof a completely soft robot, absent of any rigid struc-tures. One common feature to the locomotion of thesemarine invertebrates is the periodic generation offinite propulsive jets. Much like synthetic jet actuatorsused for flow control, these animals alternate betweeningesting and expelling fluid so that there is no netmass flux over an entire cycle, but there is a positivemomentum flux providing propulsion. As an exam-ple, consider jellyfish body deformations used to gen-erate the propulsive jets, which are shown for distincttime segments of Sarsia tubulosa swimming later infigure 6. This propulsion mechanism begins when thejellyfish body contracts which reduces the sub-umbrellar cavity volume and forces out a propulsivejet. Immediately after that the body expands back to itsformer volume refilling the cavity. After a period ofcoasting, the pulsation cycle is complete and the jelly-fish starts a new cycle.

The vortex ring formation associated with pulsedjets is known to enhance the impulse transfer com-pared to continuous jets with equivalent volume flux[27] due to a converging radial velocity induced by theforming vortex ring [23]. This type of propulsion has

been utilized to provide accurate low speedmaneuver-ing to slender, low drag, underwater robots [19–21, 35]. Since the thruster can be placed just inside thevehicle hull, the streamlined hull shape is unaffectedby adding thrusters for maneuvering, as can be seen infigure 1. Pulsed-jets have also been used as the primarymeans of propulsion, demonstrating a higher propul-sive efficiency than previously assumed for highmomentum jet propulsion [38, 39], much like squidthemselves [6, 7]. Although the thrusters draw inspira-tion from the propulsion of squid and jellyfish, theyhave generally been used on traditional rigid under-water vehicles. Clearly the morphology of these ani-mals indicates that this type of propulsion could easilybe applied to underwater soft robotics. Some softrobots inspired by this type of propulsion include a jel-lyfish shaped robot employing shape memory alloycomposite actuators to drive bell contraction [55, 56],and a soft robot was created by casting amold from themantle of Octopus vulgaris that expelled jets bymimicking the radial contraction of the mantle [48].In addition the fiber support structure in squid man-tles can also be used as inspiration for soft robot bodydesign. The interwoven tunic fiber structures helpmaintain the shape of the squid and distribute theaction of the muscles [17, 34, 57]. Furthermore, theiralignment maximizes both the propulsive output for agiven contraction and the energy storage in fibersreleased during mantle re-inflation [22]. A similarsupport structure could greatly reduce the number ofrequired actuators in a soft robot, reducing itscomplexity.

In this paper we examine the functional require-ments of a soft robot with no rigid parts, utilizing notonly the propulsion mechanisms of cephalopods andjellyfish, but also aspects of their musculature, mor-phology, and sensory mechanisms for inspiration inthe design and control strategy. This discussion beginswith a brief overview of the thrust and pressuredynamics inside flexible jetting cavities, which identi-fies energy optimal volume flux programs in section 2.In section 3, methods for determining the best loca-tions of body deformation actuators from biological

Figure 1.Traditional rigid underwater robot utilizing cephalopod inspired thrusters. Formore information on this vehicle refer to[19].

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Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

models are presented. Finally, we make it possible toclose the control loop, in section 4, by introducing abioinspired sensory feedback mechanism capable ofdetermining body deformation parameters from pres-sure distributions.

2. Pressure dynamics andminimalwork

When squid and jellyfish eject jets to propel themselvesthrough the water, they create an imbalance betweenthe pressure of the internal cavity and the pressure ofthe external fluid, which drives locomotion. There-fore, the internal pressure dynamics are of vitalimportance when designing a soft underwater robotinspired by this type of propulsion. The externalpressure distribution also strongly affects the swim-ming of these animals; however, the dynamics asso-ciated with the external pressure distribution arefundamentally different than those for the internalpressure. The external pressure forces on movingunderwater bodies are generally categorized as drag oradded mass forces and there are multiple studiesfocusing on reducing these forces. The analysis of thissection does not consider the effect of body deforma-tion on external pressure, but rather focuses on howbody deformations can minimize total work requiredfor propulsive jetting. However, these deformationsare likely still optimal when taking into accountexternal pressure forces as well, given that small scaledeformations required for jetting will have very littleeffect on the overall drag profile. Also, during jettingthe jetting forces themselves are much stronger thanthe drag forces, as evidenced by the massive accelera-tions of these animals.

A model for the internal pressure distribution ofany axisymmetric deformable cavity body is providedin [24]. Themodel allows us to relate the internal pres-sure to the deformation of the cavity, which is whatwill be directly controlled by the proposed soft robot.In that study, it was shown that there is a direct linkbetween the pressure inside of a jetting cavity and thetotal circulation dynamics of the system:

Pt t

ud

d

d

d

1

2. 1b b

Jet Cav 2˜

( )=G

+G

-

In this equation, Pb is a reference pressure at theintersection of the inner cavity surface and the axis ofsymmetry, ub is the velocity of the surface at that point,

JetG is the circulation of the entire region downstreamof the cavity opening (which we refer to as the jetregion for simplicity), and CavG is the circulationwithin the cavity not including the circulation termsspecifically due to stretching boundaries.

The usefulness of such modeling lies in the factthat circulation is an invariant of motion for inviscidflows, and thus the rate of change of circulation ineither cavity or jet region can be calculated as a sum ofvorticity flux and source terms. The simplest case todescribe is when the cavity body is ejecting a

propulsive jet starting from rest. Subsequent jettingcycles involve interactions between incoming fluidvortex structures and the internal boundaries, asdescribed by the impingement circulation terms in[24]. As a first step, this paper will discuss control algo-rithms for a soft jetting robot during this first jettingcycle to illuminate the control strategy before movingon to themore complex cases.

For this case, there are only two sources of circula-tion. The emanating fluid carries with it a shear layercreated on the surface of the nozzle opening. This canbe considered a flux of vorticity into the jet region. Therate at which vorticity is expelled into the jet regionscales with the jet velocity squared. Inside the cavitythere is a circulation due to the fluid which convergestowards the thruster opening. The flow on the insideof the opening can be modeled as a half-sink of finitecircular area [24]. The circulation of the internal flowscales with the volume flux (strength of the sink), sothat the rate of change of internal circulation scaleswith jet acceleration. Therefore, for this simplifiedcase, the reference pressure on the inside of the cavitycan be calculated as

P C C u¨ 1

2, 2b b1

22

2˙ ( )= W + W +

where W is the volume flux crossing the opening, W isthe rate of change of the volume flux, and C1 and C2

are constants which mostly depend on nozzle geome-try and can be determined from [23] and [24],respectively. For most cases, the cavity surface velocityub is very small compared to jet velocity and this termcan be neglected making the reference pressure afunction only of volume flux and its time derivative.This relationship between pressure and volume fluxparameters (through generation of circulation) wasverified in [24] using a prototype bioinspired jetthruster very similar to ones placed on rigid vehiclesdepicted in figure 2. The prototype jet thrusters expelfluid by moving a semi-flexible mechanism, which isreinforced with a helical spring so that it maintains aconsistent diameter and has a linear relationshipbetween plunger deflection and volume flux. As issummarized in figure 3, the pressure predicted by (2)compared to the pressure measured on the plungersurface shows high accuracy, and is discussed ingreater detail in [24] for different volume fluxprograms.

This form of jet propulsion is inherently unsteady,as the jet flow must be terminated in between pulsa-tion cycles to allow for refilling; which contrasts withcontinuous jet propulsion commonly used in recrea-tional watercraft (jetskis, bow thrusters, etc). There-fore, it is important to know if certain volume fluxprograms are beneficial to propulsion, or more speci-fically to propulsive efficiency, and design the robotcontroller accordingly. The total power whichmust bedelivered to drive fluid motion for any general cavityshape and deformation was defined in [24] as the pro-duct of local pressure and cavity surface boundary

3

Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

velocity integrated over the entire surface

W

tr Pu n s

d

d2 d , 3· ˆ ( )òp=

ss s

whereW is the total work required for jetting, σ is thecavity surface boundary, P is the pressure at anylocation s along the boundary surface, rs is the radialcoordinate at s, n is the normal unit vector at s, and us

is the velocity of the surface at s due to deformation.The geometry of the jetting cavity and controlleddeformations dictate the values for σ, u ,s

and n;ˆ

whereas the local pressure distribution P(s) must becalculated. One method to calculate the pressuredistribution by integrating the momentum equationalong a strategic viscosity free path is described in [24].The integral of (3) cannot be reduced for any generalcavity shape, but often the specific cavity geometry willallow pressure andwork terms to be greatly simplified.For the specific prototype actuator geometry in [24],the pressure is uniform over the internal plunger plate,and the total work is just the product of the reference

pressure, Pb, the area of the plunger, and the velocity ofthe plunger. It was determined in that study that animpulsive volume flux program requires less energythan a smoother sinusoidal program with equivalentmomentum transfer. The power required to drivefluid motion is shown over the entire program for twosuch cases in figure 4, including the total workrequired for each program. The energetic advantage ofthe impulsive deflection program exists because thepeaks in boundary deformation velocity and pressureforce are out of phase for the impulsive program andcoincident for the sinusoidal program. Since thepower required to drive the submerged surface is theproduct of the boundary velocity and the pressure atits surface, which it must act against, it is advantageousto have a low boundary velocity when the pressure ishigh, and similarly a high boundary velocity when thepressure is lower.

As can be inferred from the form of (3), the totalwork required for propulsion will change slightly with

different flexible cavity geometries. However, the

Figure 2.Diagram illustrating the semi-flexible cavity side wall used in thrusters designed to be placed on rigid vehicles. Also shown isa picture of one of the semi-flexible cavitiesmounted on a transparent housing to allow internal viewing.

Figure 3.Pressure on the surface of an internal plunger driving fluidmotionwithin a prototype jet thruster as calculated from thecirculation based pressuremodel summarized by equation (2) comparedwith the actual pressuremeasured on the plunger in [24], for(a) an impulsive plunger deflection (volumeflux) program and (b) a sinusoidal plunger deflection program.

4

Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

swimming behavior of the squid Loligo pealei suggestthat the impulsive volume flux program is actuallybeneficial for amuch larger range of cavity geometries.The squid mantle volume program reported in [2] isreproduced here in figure 5 to show the nearly con-stant volume flux during jetting and refilling, and sig-nificant accelerations when switching back and forth.It should be noted that, in contrast to the prototypeactuator in [24]which generated volume changes withan axially deflecting internal plunger, the squid gen-erates volume change by deforming themantle surfaceradially (circumferential muscle contraction). There-fore, the energetic benefit of offsetting peaks in inter-nal pressure and deformation velocity by utilizing animpulsive flux program are apparently independent ofthe specific deformation geometry.

A volume flux program which reduces the totalenergy required for propulsion is likewise desirable inthe kinematics of soft jetting robots. The discussion ofthis section indicated that an impulsive velocity pro-gram will decrease total required energy for propul-sion, but the optimal program will likely have somevariations with body geometry. In the followingsections, we will discuss deformation of a soft robotwith a specific jellyfish-like geometry, including a

general algorithm for determining actuator placementon bioinspired robots, and introduce a feedbackmechanism inspired by cephalopod epidermal linesensing tomaintain a given desired jetting program. Inaddition, though not discussed in this paper specifi-cally, epidermal-line-inspired sensors have also beenused as a mechanism for determining exact externalhydrodynamic forces. These forces can be used as afeed-forward mechanism to improve vehicle con-troller stability [62].

3. Placement of soft-body actuators

The previous section discussed pressure generationinside jetting cavities, and presented volume fluxpatterns which minimize the total work required forpropulsion. That analysis was based on circulationgenerating mechanisms, without specific considera-tion for the geometry of the cavity or how the cavity isdeformed to generate the desired volume flux. In thissection, we discuss possible cavity geometries withvarying degrees of flexibility and actuation techniques;recognizing that biological morphology has evolved tosimultaneously optimizemultiple objectives, we intro-duce a method for automatically selecting actuatorlocations from recorded body deformations.

Biologically inspired pulsed jet thrusters require atleast some degree of flexibility in order to deform thethruster cavity and drive the volume flux. The thrus-ters placed on rigid vehicles have a flexible cavity sidewall [19, 20], as illustrated in figure 2. Since the mostlyrigid thrusters are reinforced in such a way that thedeformation is limited to a single parameter, there is aunique deformation program for any desired volumeflux. This means that in order to create a desiredimpulsive velocity program, the thruster must per-form an impulsive cavity top plate deflection.

If this propulsive technology is to be applied to asoft robotic platform, it is more difficult to restrict the

Figure 4. Instantaneous power required to drive plungermotion for (a) an impulsive plunger deflection and (b) a sinusoidal plungerdeflection. Also the total work over the entire cycle is listed for each case. Data previously presented in [24].

Figure 5.Mantle volume of Loligo pealei versus time duringthe jetting process. Figure adaptedwith permission fromfigure 5 of [2], including straight lines to show the nearlylinear volumeflux program employed by squid.

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Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

cavity deformation to a single parameter. Further-more, a specific optimal body shape and body shapedeformation for a soft jetting robot are ambiguous,making the restrictions on body shape and deforma-tion similarly nebulous. However, marine inverte-brates have had eons to develop desirable flexible bodyshapes (jellyfish especially, being one of the oldest liv-ing animals on the planet). In this section, we use themorphology of jellyfish, specifically Sarsia tubulosa, asa hypothetical basis for body shape, and to determine aminimum number and location of actuators requiredfor deformation of this body. Although this process isdone for a specific species of jellyfish (and its exactbody shape), because of available data on bodymotions and pressure dynamics of that species, itcould easily be applied to the body motions of anyothermarine invertebrate.

We can find reasonable configurations of actua-tors by leveraging a dynamical-systems-based modelthat we introduced in [50]. This process was originallydesigned to uncover motion primitives, which aresubsets of kinematics sequences from some entity thatcontains variable-duration actions. However, sincethe algorithm also characterizes both the kinematicsand change in kinematics, insight as to where promi-nent changes occur in an entity’s body during amotion sequence can also be drawn. From theseinsights we conclude the necessary number and loca-tions of actuators needed to drive the motion. We willfirst introduce the algorithm developed to segmentcomplex body motions into motion primitives from[50], and brief descriptions of the motion primitivesdetermined for a Sarsia tubulosa swimming cycle,before deriving analysis on how the expectation of sto-chastic systems can be leveraged to define suitableactuator locations for this study.

The motion primitive analysis given in [50] char-acterizes body deformations as an aggregation of sto-chastic differential equations, which capture theaverage spatial and temporal evolution of body kine-matics. The stochastic ODEs take the form: yd

km

t p a; km( ) = - y t p b;

km

km( ( ) )- td

i i km,å k+ t 1 2( )

B td ,km ( ) where ti k

m,

2( )k are draws from a Lévy-drivenGauss–Markov process at time t,Bk

m is a realization of aBrownian motion, the variables k and m are indicesrepresenting different datasets and data modalities,respectively, each differential equation is denoted by p,the average spatial behavior is dictated by b ,k

m and theaverage temporal behavior is governed by a .k

m Thebody kinematics vector x tk

m ( ) is defined at each time tas: x t w P y t p t; ,k

mkm

km

km

km

p( ) ( ) ( )å s= + where Pk

m is

a projectionmatrix weighted bywkm and tk

m ( )s is a ran-dom noise term. We included a nonparametric super-position of Gauss–Markov processes [8] in the modelto handle deviations from the average dynamics beha-vior. The Gauss–Markov processes also help to char-acterize rapid jumps in dynamics due to quickmaneuvers. Given a kinematics sequence, we uncoverthe most likely parameters that could have generatedthe sequence using batch statistical inference [44].Once the motion has been quantified it is segmentedinto primitives by temporal breakpoints associatedwith gradual or abrupt changes in body kinematics(formore details refer to [50]).

Body motions of Sarsia tubulosa were captureddigitally and imported into a Navier–Stokes solverwith an arbitrary Lagrangian–Eularian method[45, 46]. Lipinski andMohseni [33] examined the flowfield in and around the jellyfish with respect to mate-rial transport barriers and jellyfish feeding. When alocomotion sequence of Sarsia tubulosa gathered inthose studies was analyzed using the segmentationalgorithm discussed here, we uncovered five motionprimitives in each pulsation cycle. Detailed analyses ofthe body dynamics and resulting flow field for eachmotion primitive are provided in [50].

A summary of the geometric, physical, and flowcharacteristics associated with each motion primitiveis given in table 1, and the body motions associatedwith each primitive are depicted in figure 6. Despitethe fact that the algorithm receives no information

Table 1.A summary of the geometry, physical, andflow characteristics formotion primitives of Sarsia tubulosa, as determined in [50].

Phase Geometry/Physical change Effects onflow

Pressurize There is an outward rotation of the velar flap and a

pressurization of the subumbrellar cavity. Half-sink

terms dominate.

The outwardflow across the velum increases sharply.

Jetting (Acceleration) The bell compresses and the velar flap continues to

rotate outward. An increase in pressure is seen due to

vorticity flux. The power ismaximized.

An ejecting jet is formed. Part of thefluid slug is ejected

in this phase.

Jetting (Deceleration) Theminimumvelar diameter and bell volume are

achieved. Themaximumbell pressure and upstream

thrust are achieved.

Propulsive vortex ring formation, separation, and

translation occurs. The remainder of the slug ismoved

into the ring before separation.

Refill The cavity volume and velar diameter increase. An

upstream thrust is generated due to the relaxation vor-

tex striking the cavity.

There is an inward flowoffluid, which refills the sub-

umbrellar cavity. A relaxation vortex is also formed

inside the cavity.

Coasting There are slight velar oscillations due to small pressure

changes.

There are slight inward and outwardflow changes

across the velum. The relaxation vortex diffuses.

6

Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

about the surrounding flow field, the kinematic seg-mentation still corresponds to stages of motion withcontrasting circulation generating mechanisms. Thefirst primitive corresponds to an outward swinging ofthe velar flap, due to cavity pressurization, which cau-ses it to protrude in the downstream direction (seefigure 6(a)). With a low volume flux, but substantialrate of change in volume flux (increasing) the cavitycirculation is dominated by half-sink components,which drives pressurization.

The second primitive, figure 6(b), and third primi-tive, figure 6(c), constitute the jetting phase of theswimming cycle. During the second primitive, sinceboth W and W are positive, the vorticity flux and half-sink components work in conjunction to increase theinternal pressure of the cavity generating a high pro-pulsive force. The third primitive marks when thevolume flux begins to decrease and the contribution tototal circulation from the half-sink components oppo-ses the contributions from vorticity flux (W is positive,while W is negative). As such, the internal pressurereaches a maximum at the transition between the sec-ond and third primitives. Toward the end of the thirdprimitive, the reduced volume flux greatly reducescontributions from these components. The half-sinkcomponents eventually become dominant, resultingin the negative pressure force at the end of this primi-tive, despite the continued jetflow.

The remaining two primitives correspond to cav-ity fluid refill, figure 6(d), and coasting, figure 6(e).During the fourth primitive, elastic strain energy isreleased, causing the bell to expand. The rapid

expansion of the bell quickly draws fluid into the sub-umbrellar region, generating a relaxation vortexwithin the cavity. Due to the interaction of the incom-ing fluid with the cavity boundaries, the circulationdynamics become quite complex. For the last primi-tive, the only observed motions are small oscillationsin the velarflap.

In addition to segmenting body motion into pri-mitives, the set of parameters in the stochastic differ-ential equations, which correctly quantify the motion,can also be used to establish candidate positions foractuators. This is done by first determining wherethere are large-magnitude deformations in the jelly-fish’s body, and then determining where the deforma-tions occur for each instance.

For the first part of this procedure, we identifyrapidly fluctuating stochastic differential equationsassociated with fast-paced, large deformations. Thevarious differential equations are ranked according thenumber of Gauss–Markov processes i k

m,k associated

with each differential equation solution y t p; ,km ( )

which indicates a sudden change or divergence of bodymotion from the mean behavior. We then select theequations with the most Gauss–Markov processes torepresent large scale changes. Since each differentialequation is associated with a particular time span, wetherefore know when the body is undergoing sub-stantial changes.

The regions in which those changes occur can bediscerned by analyzing the expectation, y t pd ; ,

km ( ) of

selected stochastic differential equations. The expecta-tion of this dynamics derivative provides a spatially

Figure 6.Body deformations during differentmotion primitives for a swimming Sarsia tubulosa. Themotion primitives in (a)–(e)correspond, respectively, to cavity pressurization, propulsive jet formation, jetting and propulsive vortex formation, cavityfluid refilland relaxation vortex formation, and coasting.

7

Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

averaged view of where and by how much the jelly-fish’s body changes shape. The number of modes ofthe dynamics derivative offers an estimate for thenumber of regions that change throughout the body,and hence a possible number of actuators. The rela-tionship between the kinematics vector, x t ,k

m ( ) andthe stochastic differential equation solution, y t p; ,

km ( )

is used to map the expected values of y t pd ;km ( ) to the

surface of the body. From our observations, the loca-tions of the derivative modes correspond with joint-like areas.

The expectation of the three highest ranking sto-

chastic differential equations describing the motion of

Sarsia tubulosa are shown in figure 7. In this figure, the

expected value is plotted as a color contour on the sur-

face of the jellyfish body with yellow (lighter shade)corresponding to high amounts of deformation while

red (darker) corresponds to little shape change. It can

be observed that the largest shape change occurs in

three regions labeled by the markers in figure 7. The

volume flux is the result of a compression of the bellwhich is driven by a hinge-like flapping at the locationmarked by ③ in figure 7(c) and an expansion at thelocation marked by ②. The velar flap also swings out-ward changing the nozzle geometry and opening dia-meter during jetting, using a hinge-like mechanism atthe cornermarked by①.

Animal body deformations are driven by musclecontractions which will, due to conservation ofvolume, result in either thinning or thickening of thebody. The contraction of the jellyfish bell, which drivesjet expulsion, is accomplished by contraction of cir-cumferential muscles resulting in thickening of thebody at locations marked by ② and ③ in figure 7. Thiscontraction can be accomplished with a single actua-tor on the soft-robot platform stretching from the axisof symmetry to the marker ②, which will be para-meterized by a thickness strain in this region .1 Themotion of the velar flap is correlated to oscillations inthickness of a region at the end of the bell just beforethe velar flap itself, marked by ① in figure 7. Even

Figure 7.The expectation of the dynamics derivative frommotion primitive analysis plotted as a color contour over the surface of thejellyfish body for time periods (a) 0.1–0.25 T (b) 0.25–0.45 T and (c) 0.41–0.75 T, which are the time spans of the three highest rankingstochastic ODEs. This derivative describes which regions of the jellyfish body are deforming andmoving during some time span. Thered (dark) contour corresponds to less deformationwhile the yellow (lighter) contours correspond tomore deformation.

Figure 8. (a)The Sarsia tubulosa strains, 1 and ,2 at two characteristic body locations, tagged bymarkers③ through② and① infigure 7, respectively, are shown for a single jetting cycle. (b)The volumeflux resulting from these body strains, alongwith the volumeflux approximated by (4) trimmed about the initial body shape.

8

Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

though neuromuscular data [47] suggests that themotion of the velar flap is a passive action, we will addan additional actuator in this region to give the soft-robot full control authority, denoting the strain of thisregion by .2 Electro-active polymer actuators wouldbe a good choice for the actuators on the soft robot,given the compression/expansion style of actuation aswell as the required flexibility of the body. The strainsin these body regions over an entire swimming cyclefor Sarsia tubulosa are shown in figure 8(a). Next wediscuss how these actuators should be operated toenact the desired fluid volume flux.

If it is assumed that these actuators deform theseregions uniformly, that the volume of the jellyfishbody remains constant, and that the velar flap itselfdoes not deform in length then we can determine therate of change of the cavity volume in terms of thestrain rates ,1 applied at markers ② and ③ in figure 7,and ,2 applied atmarker① infigure 7

rz

s

r

ss

r s

ll

d

¨

2

2.

4

s

f

f

10

2

22 1 2

22

2

22

2

1

( )( )

˙ ˙

˙ ˙

˙˙

( )

⎜ ⎟⎛⎝

⎞⎠

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

ò

p

W = -¶¶

+¶¶

--

-+ -

ss s

s

In this equation rs and zs are the radial and axialpositions of a point on the inner cavity surface which isat a distance, s along the surface, s1 is the distance alongthe cavity surface to the velar flap joint and lf is thelength of the velar flap. It should be noted here that thevolume flux is a function of both the strain rates at thetwo actuator locations as well as the current shape ofthe cavity surface. This should be expected as a bodywith a larger radial extent will expel more volume dueto the body contraction associated with a given strainin body thickness, but it also means that the controllermust have some knowledge of the current body shapein order to drive actuator strain required for thedesired volume flux. For many simple robots an exactknowledge of the current shape may not be known. Insuch cases we can trim the flux-strain dynamics aboutthe initial shape of the soft body, so that the integralterm, r s ,1( )s and lf in (4) all become constants that canbe calculated from the initial shape, and volume flux isgiven purely as a function of actuator strains. Theaccuracy of such an approximation will decrease withlarge scale deformations, but should be reasonablegiven small deformations in shape.

The volume flux of Sarsia tubulosa during the jet-ting cycle is approximated from the two characteristicstrains, 1 and ,2 equating the shape of the body to itsinitial shape at all times. Figure 8(b) shows the volumeflux computed by this approximation compared withthe actual volume flux during the jetting cycle. It canbe seen that the approximation is very close to theactual flux at the beginning of the cycle, as would beexpected, but loses some accuracy during peak volumeflux. However, throughout the entire cycle the error in

the approximation never gets larger than 15% of theactual volume flux. From a feedback controller stand-point this should be considered an acceptable approx-imation, since small errors in the predicted actuationdynamics can be corrected for in the feedbackmechanism.

4. Epidermal-line inspired jetting feedbackcontrolmechanism

Previous sections examined volume flux programsthat are beneficial to propulsive efficiency, indicatingthat impulsive volume flux programs minimizerequired energy for multiple cavity geometries.Though this optimal program may change slightly fordifferent cavity geometries, we assume that there issome known, optimal, desired volume flux program

,W which can be related to required body deformations(controls) and actuator strains. It is also imperativethat the robot knows what deformations are actuallybeing generated in real time, so that corrective actionscan be taken when the jet volume flux strays from thedesired programs. Here, we define a control law for asoft robot with a geometry and actuator configurationas presented in the previous section and introduce analgorithm for calculating jet flux parameters frominternal pressure sensors inspired by the epidermalline of cephalopods.

The simple single parameter thruster illustrated infigure 2 has been outfitted with a linear potentiometermeasuring real-time plunger deflection in experi-mental studies [20, 21, 23, 24]. If h(t) is the internalheight of this thruster’s cavity, then a proportionalderivative algorithm can be easily implemented toprovide the control signal, u, for the linear actuatordrivingmotion

u t k R h k R h¨ 5p d p2

d d p2( ) ( )( ) ˙ ˙ ( )p p= W - + W -

Here kp and kd are the proportional and derivativecontrol gains, respectively, dW and ¨

dW are the desiredvolume flux and desired rate of change of volume flux,respectively, and Rp is the radius of the flat plungerplate. This is in fact the control algorithm implemen-ted in [24] to generate desired volume flux programs.This implementation works well in this case because hcan be easily determined from a potentiometer whichis rigidly attached to the back of the plunger.

For the case of a jellyfish-shaped soft robot, a simi-lar proportional derivative control law can be createdleveraging the relationship between actuator strainand volume flux (4), linearized about the initial bodyshape. Just like the previous control law, the errorbetween desired volume flux and actual volume flux isused to generate the control signals, hence the con-troller must have some way of determining the actualvolume flow rate at any given time. The flexible robotbody results in complications in this regard, sincewithout rigid joints (and motor encoders or

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Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

potentiometers) it is difficult to determine the exactmagnitude of the deformation. Here again we drawinspiration from the sensory systems of marine inver-tebrates to generate an appropriate feedbackmechanism.

Cephalopods have a sensory system known as theepidermal-line, which is functionally very similar tothe lateral-line sensory system in fish [9, 10]. Muchmore research has been done on the fish lateral line,showing that the system is largely responsible for fishschooling [40], rheotaxis [4], and aids in obstacledetection when other senses are limited [41]. Super-ficial neuromasts on the body surface respond to theshear stress or the net velocity of the flow, whereassubdermal canal neuromasts respond to the pressuregradient or the net acceleration [12, 25, 26]. Someresearchers have attempted to mimic the sensing cap-abilities of lateral line canal neuromasts with arrays ofpressure sensors on robotic platforms [14, 43, 54, 62].Although research on the epidermal line of cephalo-pods has been more limited, it has been observed thatthere are two lines of the system which extend alongthe funnel of cephalopods [30]. Furthermore, thebroad band of ciliated cells on squid funnel as opposedto thin lines of octopus funnel [29, 30, 49] is likely dueto the increased role jetting plays in squid locomotion,suggesting that the pressure distribution during jettingmay serve as a good source of feedback for jet control.In fact, there is strong evidence that pressure variesgreatly with the different mechanisms of thrust gen-eration (i.e. different sources of vorticity generation).

Since there are no internal structures associatedwith the vorticity flux terms, the internal pressure dueto this mechanism of circulation growth can be con-sidered mostly uniform throughout the cavity. Cer-tainly the half-sink flow exists within the cavitywhenever the jet flow is present, but the circulationand pressure dynamics due to this internal flow struc-ture are added separately. As such, a large degree ofnon-uniformity in the internal cavity pressure is indi-cative of a relatively high contribution from the half-sink circulation terms to the internal pressure. As anexample, figure 9 shows the internal cavity pressuredistribution at three characteristic moments through-out the jellyfish jetting cycle. In the first instance,figure 9(a), the jetting cycle has just started, so there is alarge pressure associated with the fluid acceleration/half-sink circulation, and a limited pressure due tovorticity flux. At this time the pressure distribution ishighly non-uniformwith a large peak towards the out-ward radial edge of the cavity. In the second instance,figure 9(b), there is negligible jet acceleration and theinternal pressure is dominated by the vorticity fluxterms, resulting in the nearly uniform pressure dis-tribution. In the third instance, figure 9(c), the fluidhas begun to decelerate towards the end of the jettingcycle. Again the half-sink flow structure makes a largevariation in internal pressure, but in this case the fluiddeceleration acts in opposition to the flux of vorticity

and the pressure profile dips negative before reachinga peak closer towards the velarflap.

If several soft pressure sensors are placed internalto the cavity, similar to epidermal lines in cephalopodfunnels, then the measured pressure distributioncould be used as a feedback mechanism to drive cor-rective body deformations. The reference pressure isthe sum of contributions from terms that scale withvolume flux and terms that scale with the rate ofvolume flux, whereas the non-uniformity in pressuredistribution is only due to the terms which scale withthe rate of change of volume flux. We, therefore, sur-mise that the ratio of area under the non-uniform partof the pressure distribution Au(t) to reference pressurearea Ar(t) (see figure 10(a)) is proportional to the ratioof acceleration pressure terms to reference pressure

A

AC

C

C C

¨

¨. 6u

r3

2

12

2˙ ( )- µ

W

W + W

The term C3 is to account for the fact that when theeffect of the half-sink (acceleration) pressure termsbecome negligible there remains a degree of non-uniformity in the pressure distribution (i.e. thesimplification that pressure distribution due to vorti-city flux alone is uniform is not completely accurate).The coefficient C3 must be calibrated in the laboratoryprior to operation by measuring the ratio of Au to Ar

when ¨ 0,W = andC1 andC2 are constants described insection 2.

The proportionality described in (6) is actuallyquite valid for the majority of the jetting phase, as canbe seen in figure 10(b), where both ratios are shownwith respect to time. The direction (sign) of W can alsobe determined from the pressure distribution since thepressure distribution will only drop below the refer-ence pressure at any point if ¨ 0.W < We denote thissign by s ,W which is −1 when the jet flow is accelerat-ing and+1 otherwise. Combining (2) and (6) allows usto determine the volume flux and rate of volume fluxfrom the characteristic areas of the pressure distribu-tion

C

PA

AC s C

A

AC

Cs C

1

1 11

, 7b

u

r

u

r

4 ¨ 5

42

¨ 5

1 2

˙ ( )

⎡

⎣

⎢⎢⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦

⎥⎥⎥⎥W =

- - +

- - -

W

W

where C4 is the constant of proportionality of (6),given a body-shape similar to Sarsia tubulosaC 0.05.4 = The accuracy of this method at predictingvolume flux during jetting is shown in figure 11. Here,it can be seen that the instantaneous volume flux canbe estimated fairly well by determining the non-uniformity of the internal pressure profile.

With this information the control loop for the softrobot can be closed. The desired volume flux, ,dW is agiven impulsive velocity program which along with theactual volume flux determined from the pressure sen-sor data (7) can be placed into the soft robot control law

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Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

to generate appropriate control signals for the soft-actuators. It should be noted here that the coefficientsC4 and C5 are specific to the Sarsia tubulosa body shapeand must be determined by fitting the model to theactual pressure dynamics; however, the general under-lying principle that non-uniformity in internal pressureduring jetting corresponds to pressure from half-sink(accelerating flow) circulation dynamics is valid for any

flexible cavity geometry. Though the coefficients willneed to be reevaluated for any alternative body shape,themethodology remains the same.

5. Conclusion

This paper provides analysis of several aspects of designspecific to marine-invertebrate-inspired, underwater,

Figure 9.Pressure distribution over the inner surface of the jellyfish Sarsia tubulosa at 0.05 s (a) 0.18 s (b) and 0.28 s (c) into the jettingcycle. The full jetting cycle is 1 s for Sarsia tubulosa.

Figure 10. (a)Diagram illustrating characteristic areas of the internal pressure distribution. (b)The ratio of unsteady pressure terms tothe total reference pressure shown alongwith the ratio of characteristic reference pressure areas to illustrate the nearly proportionalrelationship between the two.

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Bioinspir. Biomim. 10 (2015) 065004 MKrieg et al

soft robots. The pressure dynamics inside a flexiblecavity are described as they relate to the volume fluxcoming out of the cavity as well as the rate of change ofthat volume flux. As a result, it is shown that impulsivevelocity programs minimize the energy required todrivefluidmotion for certain cavity geometries, and it isassumed that any given cavity geometry has an optimaldesired volume flux program (often impulsive) thatshould be achieved to increase efficiency. Next, thedeformation of a hypothetical soft robot with the samebody shape as the jellyfish Sarsia tubulosa is definedusing a finite number of actuators. Motivation foractuator placement on the soft robot is driven here byactual deformations observed in that specific species ofjellyfish, and a general technique is described to identifypossible actuator locations from biological locomotionutilizingmotion primitive analysis. Finally, a theoreticalsensory feedback mechanism, inspired by the epider-mal-line of cephalopods, is derived that allows thevolume flux and flux rate to be determined from thepressure distribution inside the cavity. It is shown thatshape of the pressure distribution inside the swimmingjellyfish is highly indicative of the extent to whichunsteady (rate of volume flux) terms affect the propul-sive force. Furthermore, the exact quantification of thispressure distribution non-uniformity allows for highfidelity determining of volume flux and volume fluxrate of the swimming jellyfish cavity, used as amodel forthehypothetical soft robot body.

Acknowledgments

This work was funded by a grant from the Office ofNaval Research. The second author was additionallyfunded by a University of Florida Graduate ResearchFellowship and a Robert C. Pittman Researchfellowship.

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