Robotics and Autonomous Systems 30 (2000) 103–113

A hydrostatic robot for marine applications

Ravi Vaidyanathana, Hillel J. Chielb,c,∗, Roger D. Quinnaa Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, OH 44106, USA

b Department of Biology, Case Western Reserve University, 10900 Euclid Ave., Cleveland, OH 44106, USAc Department of Neuroscience, Case Western Reserve University, Cleveland, OH 44106, USA

Abstract

Invertebrates that use a fluid-filled cavity surrounded by contractile tissue (a hydrostatic skeleton) propel themselvesefficiently underwater and penetrate regions inaccessible to legged or wheeled devices. A hydrostatic robot would thereforebe invaluable for exploring marine environments. We have constructed a three-segment hydrostatic robot that locomotesunderwater. Each segment consists of two solid circular disks connected by four equidistant shape memory alloy springs.A fluid-filled bladder in the center of each segment provides hydrostatic skeletal support. The 15.5 cm long robot moves atspeeds up to 0.6 cm/s, can turn up to 21◦, and can change its length by up to 16%. A model of the robot’s kinematics has alsobeen developed. ©2000 Elsevier Science B.V. All rights reserved.

Keywords:Muscular hydrostats; Worm-like robots; Underwater locomotion

1. Introduction

1.1. Background

Over the last few years, there has been consider-able interest in developing autonomous underwater ve-hicles that can function in marine environments forextended periods of time [1,2]. Many of these vehi-cles are torpedo-shaped, are propelled by turbines, andhave rigid external shells [3].

Engineers inspired by the design of biological or-ganisms have pointed out that other body plans maybe useful for robotic applications [4]. Indeed, aquaticorganisms show a wide variety of different body plansthat are successful in many different marine environ-ments. A variety of investigators have begun to de-

∗ Corresponding author. Tel.: +216-368-3846;fax: +216-368-4672.E-mail address:hjc@po.cwru.edu (H.J. Chiel).

velop biomimetic underwater robots. Triantafyllou etal. [5,6] have developed two different robotic fish, onemodeled after the bluefin tuna, the other after the chainpickerel. They have used these devices to explore theways in which the fins of fish can control and reducedrag. In addition, they have studied the effectivenessof the whole body movements made by fish that al-low them to rapidly accelerate as they initiate swim-ming [6]. Other investigators have also begun to ex-plore the properties of fish-like bodies as the basis forautonomous robots [7]. Ayres et al. [8] have begun todevelop lobster-like and lamprey-like robots that uti-lize finite state machine controllers based on biolog-ical central pattern generators. The prototypes of thelegs of the lobster-like robot and a prototype of sev-eral segments of the central body of the lamprey-likerobot have been implemented using shape memoryalloy (Nitinol) wire [9].

A marine robot capable of locomotion over irregu-lar terrain, complex maneuvering through tortuous or

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104 R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113

sinuous crevices such as those found in coral reefs, andwith the ability to swim short distances would be aninvaluable tool for performing a wide variety of tasks.Furthermore, if it could utilize the hyper-redundantkinematics [10,11] typical of vertebrates such assnakes, or invertebrates such as caterpillars or leeches,it could adapt to a wide variety of surfaces, includ-ing irregular or shifting terrains [12] far better thanstructures with limbs or wheels. Finally, because thisbody design is streamlined, it would also require lesspower to propel itself through water than other bodytypes.

Worms, jellyfish, snails and slugs have hydrostaticskeletons, i.e., fluid-filled spaces surrounded by mus-cle, that allow them to locomote through extremelytortuous crevices, and to insinuate themselves into ex-tremely irregular spaces. The flexible body wall ofthese organisms provides them with fewer constraintson their degrees of freedom and allows them to equal-ize pressure over a wide range of depths. Some ofthese organisms are nearly neutrally buoyant, so that itis only necessary for them to overcome drag in orderto move forward.

A robot with a hydrostatic or hydraulic skeleton,i.e., a fluid-filled cavity surrounded by contractile tis-sue [13,14], as is observed in segmented worms, wouldhave further advantages for marine applications thanwould a snake-like robot, in which muscular elementsare attached to a hardened skeleton. Since the centralcavity would be filled with water surrounded by flexi-ble walls, such a robot would have the ability to oper-ate over a wide range of pressures, thus giving it accessto very deep regions beneath the ocean’s surface. Be-cause its fluid-filled cavity would be neutrally buoyant,it would only need to overcome drag in order to propelitself through water or across irregular terrains. Thus,we have focused our attention on constructing a robotinspired by the body plans of segmented, worm-like(annelid) invertebrates. To begin to explore the utilityof these properties for autonomous underwater vehi-cles, we have successfully modeled and constructed ahydrostatic robot that can locomote underwater.

1.2. Kinematic model of hydrostatic structures

The robot mimics the body plan of segmentedworms (e.g. leeches or marine worms). Each seg-

ment of a worm consists of a muscular body wall,surrounding a fluid-filled body cavity. Muscles mayact circumferentially or longitudinally. Segments areseparated from one another by cross-walls, known assepta.

Although more complex models have been pro-posed [15], it is possible to model each segment as acircular cylinder. Assuming that the volume of eachsegment remains constant, that its cross-section re-mains circular as it changes length, and treating thecircumferential muscles (which are quite thin) as hav-ing essentially zero thickness, one can express the vol-ume of the cylinder asV= p(rL + rc)2LL, whererL isthe radius of the longitudinal muscle,LL is its length,and rc is the radius of the central fluid filled cavity.The length of the circumferential muscle (Lc) is equalto the circumference of the longitudinal muscle, thusLc = 2p(rL + rc).

These equations lead to a simple expression for therelationship between the length of the circumferentialand longitudinal muscles [16]:

LC =√

4πV

LL. (1)

This inverse square relationship between the longi-tudinal and circumferential muscles (length and cir-cumference of the cylinder) implies that when a seg-ment is long, small decreases in diameter can leadto large changes in length. The elongation/contractionbehavior and corresponding length changes undergoneby one hydrostatic segment are shown in Fig. 1 (mod-ified from [16]). A pressure vessel model was used toderive the relationship between the longitudinal force

Fig. 1. Model of a simple muscular hydrostat. Since muscle isessentially incompressible, any change in a linear dimension mustbe compensated by changes in other linear dimensions to main-tain constant volume. Thus, when the central longitudinal musclecontracts, the structure becomes short and fat, stretching the cir-cumferential muscle. In contrast, when the circumferential mus-cle contracts, it causes the structure to elongate and becomethinner.

R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113 105

Fig. 2. Schematic representation of the kinematic model for the hydrostatic robot. (A) Schematic representation of a three-segment robotas it takes an undulatory step (compare Fig. 5 (A)). (B) Geometric basis of Eqs. (4)–(6).Lchord: distance between outermost segments;L1:distance between midpoints of lower ends of a single segment. See text for further details.

FL and the circumferential forceFC [16]:

FL = FC

√πV

L3L

. (2)

This equation predicts that when the segment islong, the ability of the circumferential muscles to exertforce in the longitudinal direction will rapidly decline.

Segmented worms may locomote by sequentiallycontracting and elongating entire segments, or by dif-ferentially contracting longitudinal muscle bands orappendages. A general kinematic model of locomotionfor structures with very high numbers of degrees offreedom (“hyper-redundant” structures) has recentlybeen described [10,11], which can characterize thekinematics of either extensible or inextensible travel-ing or stationary wave gaits. For the purposes of thiswork, a simpler kinematic model has been developedfor two common forms of locomotion: alternating con-traction and elongation of segments, and undulatorymovements induced by alternate contraction of groupsof longitudinal muscle bands.

The step size for a single segment that undergoes auniform contraction (see Fig. 1) is simply the differ-ence between its relaxed length,Lrel, and its contractedlength,Lcont. Thus, the total step size forn segments,Sunif , would be this difference timesn, or:

Sunif = n(Lrel − Lcont). (3)

An undulatory step occurs when the bottom con-tractile elements of a hydrostatic segment shorten in-dependently of those in the top of the same segment.This differential contraction causes the entire structureto form an arc that bends in the direction of the con-

tracting elements (Fig. 2 (A) illustrates this movementfor a three-segment hydrostatic robot). Relaxing thecontracting elements while simultaneously flexing thetop elements will subsequently cause the entire hydro-static segment to move forward (provided there is noslip in the reverse direction). Assuming each segmentbends over an arcθ , the forward step sizeSund of nhydrostatic segments in series may be described as therelaxed length of each of the segments,Lrel, timesn,minus the length of the chord connecting the ends ofthe bending segments at maximum contraction,Lchord,or:

Sund = nLrel − Lchord. (4)

This length may be expressed as a function of thearc over which each segment bends (θ ), the numberof hydrostatic segments in series (n), and the length ofeach chord connecting the ends of each of then hy-drostatic segments when their bottom contractile ele-ments are fully shortened (L1).

As illustrated in Fig. 2 (B) for a three-segment hy-drostatic robot, the bases of the bending segments formthe tops of a series of isosceles triangles. For each ofthe small triangles whose sides have lengthsL1 andL2, by the law of sines,L2/sinα = L1/sinθ . Becauseeach triangle is isosceles,α = 1

2 (180− θ). Thus,L2 = (L1/sinθ)cos1

2θ . Furthermore, by again apply-ing the law of sines to the isosceles triangle formedby Lchord and the outermost sides of lengthL2, andrecognizing that the angleα′ = 1

2 (180− nθ), it isclear thatL2/sinα′ = Lchord/sinnθ , which in turnimplies thatLchord = sinnθ (L2/cos1

2nθ). Substitut-ing the formula that describesL2 in terms ofL1, we

106 R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113

obtain

Lchord = L1sinnθ cos1

2θ

cos12nθ sinθ

. (5)

Since sinθ = 2cos12θ sin1

2θ , this simplifies to

Lchord = L1sin1

2nθ

sin12nθ

. (6)

Finally, if each segment is capable of turning at anangleθ , the total angular change in the direction ofthe front segment, and hence the direction in whichthe robot will step, will be

β =⌊n

2

⌋θ, (7)

where the fraction in brackets indicates thatn shouldbe divided by 2 with any remainder truncated. Thetruncation is necessary since the front half of the robotmust be lifted in order for it to turn, and thus a robotwith an odd number of segments will have the sameturning angle as a robot with one less segment, sincethe center segment will make almost no contributionto the turning angle.

2. Hydrostatic actuator requirements

A major obstacle to construct hydrostatic robots hasbeen the difficulty of obtaining actuators with the ap-propriate properties. After describing these properties,in roughly descending order of importance, we willbriefly review current technologies and describe therationale for the actuators that we finally chose to use,shape memory alloy wires:1. The actuator should be able to function underwater.2. The actuators should be compliant in three dimen-

sions.3. They should have constant volume to allow them

to exert usable force in more than one direction atonce (see Eq. (2)).

4. The actuators should be able to conform to dynam-ically changing curved surfaces along their lengthwithout constraining the deformation of the hydro-stat.

5. The actuators should be capable of large excursionsrelative to their length when activated.

6. The actuators should generate a force that is dis-tributable over a large surface area, as point forceshave the potential to produce excessive distortion.

7. The speed of response (rate of contraction) mustbe suitable for useful applications.

8. Delivery of power and dissipation of heat must beeasily accomplished.

9. The actuators should have externally controllablestiffness in order to create a moveable equilibriumpoint; they should have sufficient viscosity to pre-vent oscillations.

10. The actuators should be easily controllable througha computer interface.

3. Actuators

We carefully reviewed the literature on the prop-erties of a variety of actuators with these criteria inmind. Piezoelectric actuators are unlikely to be usefuldue to their small operating strains. Hydraulic ac-tuators, despite being one of the best macro-motionactuators in classical robotics, show little promisefor this application because of their inability to ap-ply a force over a large area or along a dynamicallychanging surface. Although polymers, given theirflexibility and mechanical properties (electrome-chanical or chemomechanical materials), hold greatpromise [17–21] electromechanical polymers are notat a state of development where they can be readilyused as hydrostatic robotic actuators. Furthermore,we found that without some kind of motion enhance-ment, chemomechanical polymers may not be usefulfor macro-robotic applications [22]. Electric motorsare also unlikely to achieve hydrostatic motion inisolation. Although snake-like creeping devices andmanipulators have been successfully constructed us-ing electric motors [11,12], such mechanisms consistof either several hardened skeletal elements linkedtogether, or require a large drive and control unit thatmust be rigidly attached to the manipulator.

Inflatable pneumatic bladders meet most of the cri-teria for hydrostatic actuators. The chief shortcom-ings of these actuators is the difficulty of controllingthem due to their inherent compliance. Furthermore,although pneumatic bladders themselves are flexible,the pumps supplying their pressure, and often the tubesconnected to them, are not, and this could inhibit

R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113 107

hydrostatic motion. The device might also have to berigidly attached to its drive unit, further limiting itsflexibility.

Shape memory alloy (SMA) springs have severaladvantages for the construction of a hydrostatic robot:(1) They can function underwater. (2) By combininggroups of such springs and actuating them differen-tially they can move in three dimensions. (3) Althoughtheir linear dimensions may change, their volume isessentially fixed, allowing them to exert usable forcein more than one direction. (4) Since their “remem-bered” shape is arbitrary (i.e., the configuration atwhich the material is annealed), SMAs can conformto dynamically changing curved surfaces, exert forcesalong curved paths, thus functioning as desired whiletheir shape changes. (5) When SMA wires are woundinto springs, large deformations relative to the springlength are possible. (6) In well designed configura-tions, the force they generate may be distributed overa large surface area. (7 and 8) When used underwater,they can be rapidly cooled by the surrounding water,greatly decreasing their cycle time. (9) They have acontrollable stiffness and sufficient viscosity to damposcillations. (10) They only require a thin, flexible wireto connect them to an electrical power supply, allow-ing a greater degree of autonomy, and relatively con-venient control. Although SMAs have low efficiency,power optimization was not a primary concern for afirst prototype. These considerations determined ourchoice of SMAs for actuating our hydrostatic robot.

4. Materials and methods

4.1. Introduction

Each segment of the robot consisted of circularwooden disks connected to one another using fourtension SMA springs placed at 90◦ intervals aroundthe disks. Circular disks were used to anchor the ten-sion springs between segments so as not to hinder theradial motions of the segment. A water-filled latexbladder was placed within the four springs, providingeach segment with a hydrostatic skeleton. This designclearly parallels that of a segmented invertebrate, inwhich the springs act as muscles (force producing el-ements) contracting against a hydrostatic (hydraulic)

element providing skeletal support (the bladder).When the SMA springs were activated by passingan electric current through them, the segment con-tracted; after current was turned off, they cooled, andthe bladder caused the segment to expand. Since theSMA springs surrounding the disk may be actuatedindividually, several types of bending motion are pos-sible. For example, actuating any single one of thesprings, or any two in combination, will cause anangular motion against the bladder and the springsthat are not actuated. Initially, a single hydrostaticsegment was fabricated and tested both in air and inwater. Two and three segments were then coupledtogether. Elongation/contractile and undulatory loco-motion as well as turning movements were observed.The accuracy of the kinematic model for describingthe motion of the robot was also assessed.

4.2. Materials

The hydrostatic segment was assembled in the fol-lowing manner (Fig. 3):1. Four SMA tension springs (nickel–titanium, 48◦C

activation temperature, No. 3-096, Mondo-TronicsInc., San Leandro, CA 94577) were attached to theoutside of circular wooden disks at 90◦ intervals.The disks were made of pine wood, were 3 cm indiameter, and 1.7 cm in thickness.

2. A fluid-filled bladder was inserted into the structureformed by the springs and the disks. The bladderwas large enough to cause the springs to expandslightly to accommodate it, but not so large that itprevented movement when the springs were acti-vated. Each bladder was made of a latex balloonthat was filled with tap water.

Fig. 3. Schematic illustration of the assembly of one segment ofthe robot.

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3. In order to resist reverse slipping, two “feet” wereattached to the bottom of each disk in such amanner that the segment could slide forward, butresisted backwards motion. The feet consistedof nails, with heads removed, that were inserted1.5 cm apart in each disk at an angle of roughly20◦ from vertical.The SMA springs used were modeled as springs

with two different constants. When heated (activated),the spring constantKh was 200 N/m, and when leftpassive at room temperature, the spring constantKcwas 105 N/m. Heating changed the spring’s constantfrom Kc to Kh, causing it to contract against the restor-ing force of the bladder.

4.3. Single segment operating parameters

Due to the variability in both the materials usedand the methods of fabrication, several individual seg-ments were built and tested in order to determinean average value for each of the parameters of themodel. The following averages were obtained frommeasurements of single segments and used in Eqs. (3),(4), (6) and (7) to predict the kinematics of a hydro-static robot of any number of segments:Lcont= 2.5 cm,Lrel = 2.9 cm,L1 = 2.3 cm,θ = 23◦.

4.4. Power requirements and control

To actuate the SMA springs, a simple resistive cir-cuit was implemented so that a four switch binarycontroller could control the behavior of the three seg-mented robot. Each switch actuated three springs inthe same longitudinal direction, which were connectedin series. Each switch thus controlled a line of threesprings, each located at 90◦ intervals across the robot.Once the voltage of the power supply was adjusted toallow each of the groups of springs at least 6 A of cur-rent, the cycle times of the strokes were the same asfor a single segment, although the step size was sub-stantially larger. The SMA springs had a resistance of1.1�. A highly flexible wire tether that did not hinderlocomotion was used to transmit current to the robot.Underwater testing was done on a styrofoam surfacein a 25 gallon fish tank. For the studies reported in thispaper, switches were controlled manually.

Fig. 4. A view of the robot underwater. Note the styrofoam surfaceand the ruler, marked in cm, beneath the robot.

5. Results

5.1. Prototype performance

Several different tasks were attempted with the threesegmented hydrostat to judge its overall performanceas an aquatic locomotive device: speed, ability to turn,and ability to adapt to terrain. All trials were video-taped. The robot in its relaxed position is illustratedin Fig. 4, and a summary of its physical properties areprovided in Table 1.

The three segment robot was capable of some for-ward progression when all springs were simultane-ously activated, causing it to contract, although muchof the return stroke was lost due to slip. Based on mea-sured operating parameters of a single segment, Eq.(3) gives a predicted step size of 1.2 cm for a threesegment hydrostatic robot. Because the robot slippedon the styrofoam surface during uniform steps, thegreatest values for the actual step size of the robotwere around 0.5–0.7 cm; average step sizes were evensmaller.

Undulatory steps of the robot were far less ham-pered by backward slipping. When all the bottomsprings were activated, the front feet were forced downinto the substrate, while the back feet were pulled to-wards the front of the robot (Fig. 5 (A). When thedevice was allowed to relax from this position, the op-posite movement was observed (Figs. 5 (B) and 5(C)).This behavior minimized slip, allowing the robot toachieve its full range of motion. Based upon parame-

R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113 109

Table 1Physical specifications for hydrostatic robot

Total robot length, all 3 segments relaxed 15.5 cmTotal robot length,

all segments uniformly contracted14.3 cm

Total robot length,bottom springs of all 3 segments contracted

13 cm

Diameter of circular disk 3 cmWidth of disk 1.7 cmBladder diameter (relaxed) 3.5 cmV, bladder volume 22 cm3

Lrel (distance between disks,single segment relaxed)

2.9 cm

Lcont (distance between disks,single segment contracted)

2.5 cm

L1 (distance between disks,bottom springs contracted, single segment)

2.3 cm

θ (single segment maximum bend,bottom springs contracted)

23◦

Length of feet 2 cmDistance between feet 1.5 cmPlacement of 4 SMA springs

between each pair of disks90◦ intervals

Wire thickness for springs 750mmSpring diameter 6 mmTotal spring length 29 mmLength of spring coil 16 mmKc, spring constant, passive (room temperature) 105 N/mKh, spring constant, activated (heated) 200 N/mTime to activation upon heating in water (thw) 1.3 sTime to relaxation upon cooling in water (tcw) 1.8 sCycle time for activation and relaxation 3.1 sPredicted turning angle (3 segments) 23◦Actual turning angle (3 segments) 18◦–21◦Predicted uniform step size (3 segments) 1.2 cmActual uniform step size (3 segments) 6 0.7 cmPredicted undulatory step size (3 segments) 2.2 cmActual undulatory step size (3 segments) 2.1 cmPredicted maximal speed for undulatory stepping 0.71 cm/sActual speed for undulatory stepping 0.6 cm/s

ters of a single segment, Eqs. (4) and (6) predicted astep size of 2.2 cm for a three-segmented hydrostat lo-comoting by means of undulatory motions. The actualrobot typically moved at step sizes within a millimeterof this predicted value.

5.2. Turning ability

The degrees of freedom provided by the multipleSMA springs can be exploited to allow the robot toturn, or to free its legs from small obstacles. Turn-ing behavior was executed as follows: (1) All the top

Fig. 5. An undulatory step. (A) The bottom actuators in eachsegment are activated, causing the robot to contract, press its frontfeet into the substrate, and pull forward. (B) The top actuators ineach segment are activated, causing the robot to lift up and movethe feet in the middle segment forward. (C) The robot returns toits relaxed position before initiating the next step, having movednearly a centimeter forward.

springs were contracted in order to lift the front halfof the robot (Fig. 5 (B)). (2) The bottom springs onthe side towards which the robot was to turn were con-tracted while simultaneously relaxing the top springson the side away from the turning direction, whichmoved the front segment in the direction of the turn(Fig. 6 (A)). (3) Once the front segment was facing inthe direction of the turn, an undulatory step was takenin that direction (Fig. 6 (B) and 6 (C)). This schemeallowed the robot to turn successfully, although therewas some slipping in the direction away from the turnduring the stepping phase.

The ability to lift the front of the robot was alsovery useful whenever the feet became lodged in thesubstrate, or when a part of the robot was hindered byuneven terrain. By lifting its front half (Fig. 5 (B)),the robot could release its feet or climb over objects.

Eq. (7) predicts a turning angle of 23◦ for thethree-segment hydrostat fabricated. The actual deviceturned at angles between 18◦ and 21◦ because of

110 R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113

Fig. 6. A 90◦ turn around an obstacle.

some losses due to slip. To test its overall ability tomove through complex terrain, the robot was con-trolled so as to make a 90◦ turn around an obstacle(Fig. 6 shows views of this motion from above) anda “bell” shaped turn which took the robot around twoobstacles and into a narrow gateway (Fig. 7 showsviews of this motion from above).

5.3. Speed of movement

The speed of the contraction–relaxation cycle un-dergone by the tension SMA springs was much greaterin water than in air, though a great deal more currentwas necessary underwater. We compared the perfor-

Fig. 7. A “bell”-shaped turn around an obstacle.

mance of the robot on a solid surface in air as well asunderwater. Heating times were measured as the timefrom when current was applied until visible contrac-tion was observed; cooling time was measured as thetime from current removal to visible relaxation. In wa-ter of roughly room temperature, the springs showedthe following heating (thw) and cooling (tcw) timeswhen heated with 6 A:thw = 1.3 s, tcw = 1.8 s. Sincecurrent was removed as soon as contraction was visibleto minimize cooling times, these measurements pre-dict a total cycle time of just over 3 seconds, a speedincrease by a nearly a factor of 17 over cycle timesin air at room temperature. Larger currents did not in-crease speed due to overheating of the springs. Thesecycle times may be applied to a hydrostatic robot ofany number of segments actuated with these particularsprings. Thus, a predicted speed of locomotion maybe derived by dividing the step size by the period ofthe contraction–relaxation cycle. For a three-segmenthydrostat working in water this gave a predicted speedof locomotion of 0.71 cm/s, which correlates well with

R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113 111

the actual measured value of just over 0.6 cm/s. Minorslipping accounts for this small discrepancy.

The amount of time that a hydrostatic robot of anynumber of segments will take to turn may also bepredicted using these cycle times. Two full cycle timesplus one additional relaxation phase will be neededin order to execute a turn in any direction. This givesa predicted total time of 8 seconds to finish a turn.The three-segmented hydrostatic robot took nearly 9seconds to complete a turn.

5.4. Longitudinal and circumferential forces

During the contractile stroke, the compres-sive longitudinal force exerted by the four SMAsprings against the bladder (Flcomp) will be equal to4Kh(Lrel − Lcont), i.e., the spring constant of theSMA spring when heated times the distance of thestroke, or 3.2 N.

The force exerted by the bladder against the springsin the return stroke may be estimated from the distancethe springs are stretched back when they are not actu-ated (i.e., as they cool). This is the force necessary toexpand the equivalent spring in its lower temperature,deformable phase by the same distance. This longitu-dinal restorative force (Flres) will therefore be equalto 4Kc(Lrel − Lcont) = 1.7N.

If the robot is modeled as a constant volume cylin-der, both the equivalent lengths and forces in thecircumferential direction may be derived from Eqs.(1) and (2) respectively. Since the volume of eachbladder was around 22 cm3, Eq. (1) predicts that thelengths of the equivalent circumferential muscle var-ied from 9.8 cm at its shortest length (LCshort, whichcorresponds toLLrel, i.e., when each hydrostatic seg-ment is at its longest, relaxed length) to 10.5 cm at itslongest length (LClong, which corresponds toLLcont,i.e., when the segments are maximally compressed).Note that this change is not linear, but proportional tothe inverse root of the longitudinal length.

Eq. (2) predicts that the antagonistic circumferen-tial compressive force in the longitudinal direction(FCcomp) when the hydrostatic segment is maxi-mally compressed, and thus at lengthLcont, will beFLcomp/2.1 = 3.2 N/2.1 = 1.5 N. In contrast, whenthe hydrostatic segment is maximally elongated,and thus at lengthLrel, the antagonistic circumfer-

ential restorative force in the longitudinal direction(FCres) will be FLres/1.7 = 1.7 N/1.7 = 1 N. Thus,as predicted by Eq. (2), the equivalent circumferen-tial muscle exerts greater forces when the hydrostatis compressed, i.e.,FCcomp> FCres. Note that thesecalculations are based on the assumption that the be-havior of the robot is analogous to that of a cylinderwith constant pressure.

6. Conclusions

To our knowledge, this is the first successful ex-ample of an underwater hydrostatic robot. We havedemonstrated that such a robot is capable of straightline locomotion, turning around obstacles, and loco-moting through a complex terrain. Moreover, a simplekinematic model has proven to be accurate in predict-ing the movements of this device.

Future work could lead to significant improvementsin the performance of this robot. Power requirementscould be reduced by covering SMA springs with poly-mers that were electrically insulating but heat con-ducting. The robot could be made fully autonomousif it was attached to a self-contained power supply,such as a battery. The weight of the power supplycould be counterbalanced by reducing the density ofthe fluid in the bladders (e.g., using oil rather thanwater) as well as reducing the density of the materialfrom which the septa are made. It would also be valu-able for the robot to have the ability to sense its ownmotions, as well as perturbations due to obstacles ormoving objects. Adding strain gages to the circum-ference of the bladders might provide sensory feed-back similar to that provided by pressure and touchsensors in the leech. Moreover, it would be relativelyeasy to replace the current binary switch control witha computer controller using neural networks based onthose that control locomotion in invertebrates such asthe leech. In particular, the recent work of Kristan andLewis [23] has provided a circuit architecture that,given touch sensors distributed around the perimeter ofeach hydrostatic segment, could generate a populationvector that would lead to appropriate bending move-ments away from obstacles. More flexibility in actu-ation could be obtained by individually wiring eachshape memory alloy, but this would require a largenumber of additional wires that might be difficult to

112 R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113

maintain. It might be possible for the robot to moveboth backwards and forwards if the fixed feet were re-placed with other, biologically-inspired actuators. Forexample, leeches attach their posterior suckers, con-strict and elongate their bodies, attach their anteriorsuckers to a new location, release the posterior suck-ers and shorten in order to progress [24]. Other an-nelids use stiff bristles (setae) that can repeatedly at-tach to and detach from the substrate, and some evenhave small extensible and retractable “feet” (parapo-dia) on each segment that allow them to flexibly turnand to swim [24]. Furthermore, a larger number ofsegments coupled with individual control of segmentscould allow for gliding serpentine locomotion, whichis very efficient for swimming [12], and thus allow the“feet” to be eliminated entirely. Finally, miniaturizingthe robot and using MEMS technology may extend itsutility into other areas, such as visualizing and clear-ing blood vessels.

Despite the current limitations of the robot, we be-lieve that this work may serve as a foundation for thecreation of hydrostatic robots capable of complex mo-tions in an aquatic environment.

Acknowledgements

This work was supported by NIH grant HL-25830,and ONR grant N00014-90-J-1545. We thank WilliamCondit for help with the robot power supply.

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R. Vaidyanathan et al. / Robotics and Autonomous Systems 30 (2000) 103–113 113

Ravi Vaidyanathan received his M.S. de-gree in Mechanical Engineering in 1996from Case Western Reserve University,where he is pursuing his Ph.D. de-gree through the BioRobotics Laboratorywithin the Mechanical and Aerospace En-gineering Department. Presently, he is alsoworking as a Mechanical and Controls En-gineer at Orbital Research, Inc., in Cleve-land, Ohio. His current research involves

biologically inspired control of autonomous vehicles, swarm in-telligence for multiple agent coordination, and flexible automationfor industrial robotics. He is member of the ASME and also holdsthe license of Engineer in Training in the state of Ohio.

Hillel J. Chiel is Professor on the Facultyof the Biology Department of Case West-ern Reserve University, with joint appoint-ments in the Departments of Neuroscienceand Biomedical Engineering at CWRU. Hereceived his B.A. degree (1974) from YaleUniversity, and M.S. (1976) and Ph.D.(1980) degrees from MIT. After postdoc-toral work in the Center for Neurobiologyand Behavior at the College of Physicians

and Surgeons of Columbia University and in the Molecular Bio-physics Research Department at AT&T Bell Labs, he joined thefaculty of CWRU in 1987. His research focuses on experimental

and theoretical analyses of the mechanisms of adaptive behavior.His research group has pioneered the development of kinematicand kinetic models of muscular hydrostatic (tongue-like) structures,which served as the basis for the development of the worm-likerobot described in their article. These models are currently beingused to understand the biomechanics and neural control of feedingin the marine molluskAplysia californica.

Roger D. Quinn is Professor on the fac-ulty of the Mechanical and Aerospace En-gineering Department of Case Western Re-serve University, having joined the depart-ment in 1986 as the General Motors Assis-tant Professor. He received the B.S. (1980)and M.S. (1984) degrees in MechanicalEngineering from the University of Akronand the Ph.D. degree in Engineering Sci-ence and Mechanics from Virginia Poly-

technic and State University, Blacksburg, in 1985. He has directedthe Biorobotics Laboratory at CWRU since its inception in 1991:three insect inspired robots and one worm inspired robot havebeen developed. The second legged robot received an award at the1995 IEEE International Conference on Robotics and Automation.The third legged robot was a finalist for the 1998 Discover Maga-zine’s Technology Awards. He was also team leader for the AgileManufacturing Project in the Center for Automation and Intelli-gent Systems Research at CWRU from 1994–98. His research isdevoted to biorobotics, robotics for manufacturing and multibodydynamics.