A New Type Of Hydraulic Muscle.pdf

actuators Article

A New Type of Hydraulic Muscle Nitai Drimer 1, *, Jonathan Mendelson 1 and Amitai Peleg 2 Received: 9 August 2015; Accepted: 30 December 2015; Published: 4 January 2016 Academic Editor: Delbert Tesar 1 2

*

Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel; [email protected] Inertia Innovative Design, Haifa 35470, Israel; [email protected] Correspondence: [email protected]; Tel.: +972-4-829-2070

Abstract: This paper presents the invention and development of a new fundamental type of hydraulic actuator, aimed at delivering better actuation efficiency. This actuator is a flexible tube, composed of two different materials, which deflects while applying inner pressure. This concept is simple to produce, and allows adaptation of the deflected shape by the design parameters (radius, wall thickness, geometry, etc.). Among other applications, it is mostly suitable for the activation of fins of nature-like marine robots. Theoretical formulation, production of prototypes and actuation experiments are presented, as well as material hysteresis research and an application example. Keywords: hydraulic muscle; soft actuator; inflated structure

1. Introduction Hydraulic and pneumatic actuators are developed for a wide range of applications, including soft robotics and marine propulsion. Hydraulic actuators provide smooth motion with a simple mechanism composed of a minimal amount of parts. Some hydraulic actuators are inspired by natural muscles that contract and expand, often actuating a kinematic structure which performs a desired motion. For example, [1] presents a pneumatic tubular actuator that shortens when internal pressure is applied, pulling on a flexible rib to create a flapping motion in a ray-inspired robotic swimmer. The traditional fluidic muscle is the McKibben family of artificial muscles which are based on a soft tube that expands and shortens when pressure is applied. They have been studied extensively since the 1950s, with relations between tension, length, contraction velocity and activation pressure well established and tested [2]. They have seen applications in robotics, bio- engineering and artificial limb design. Zhang and Philen [3], as well as Daerden and Lefeber [4], give an expansive review on PAMs (pressurized artificial muscles) including the McKibben muscle. Many variants of PAMs are shown along with analysis of response and actuation force and, along with some applications in prosthetics and robotics. Newer pneumatic actuators such as the Double-Acting sleeve muscle [5] show improvements over traditional muscles such as greater force capacity, lower energy consumption and bi-directional force application. Many methods are used to realize bending motion in fluidic actuators. For instance, [6] presents a pneumatic bellows having an asymmetric section. The bending moment is generated due to the offset between the center of pressure and the centroid of the section. Reference [7] presents a flexible actuator having two axial chambers along the length of the actuator. A pressure difference the two causes the actuator to bend and stretch axially, in effect yielding a 2-DOF actuator. This muscle was used to power a ray-inspired robot. A different way to realize this motion is by limiting the elongation of a flexible shell or tube, in which pressure is applied. An example of this is the FPA [8], which consists of a flexible rubber tube and an internal spiraling steel wire which allows only for axial elongation of the actuator. The FPA can be used as a bending actuator by introducing a “restricting wire”, that is a steel wire running along the Actuators 2016, 5, 3; doi:10.3390/act5010003

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can be used as a bending actuator by introducing a “restricting wire”, that is a steel wire running 2 of 12 along the axis of the FPA. Bending occurs to the side of the actuator in which the wire is embedded. A similar approach is used in [9]: a flexible Silicone tube has both a circumferential fiber wound around it to restrict radial expansion, and aoflongitudinal fiber which theAactuator axis of the FPA. Bending occurs to the side the actuator reinforcing in which the wireabout is embedded. similar deforms when pressure applied.Silicone This allows to an arbitrary shape,around determined approach is used in [9]: ais flexible tube the hastube bothtoa deform circumferential fiber wound it to by the layout of the longitudinal fiber. restrict radial expansion, and a longitudinal reinforcing fiber about which the actuator deforms when The is creation of more complex sometimes achieved byshape, grouping a few flexible actuators pressure applied. This allows themotion tube toisdeform to an arbitrary determined by the layout together. For instance, of the longitudinal fiber. reference [10] describes the usage of three parallel bellows to realize a continuum actuator as ancomplex undersea gripper. Differences in pressure betweena the bend the The creation of more motion is sometimes achieved by grouping few bellows flexible actuators actuator in required direction 3D space. bellows actuators (PBAs) have been used for together. Forthe instance, reference [10]indescribes theParallel usage of three parallel bellows to realize a continuum underwater propulsion; [11] describes a series of PBAs connected with a flexible membrane. This actuator as an undersea gripper. Differences in pressure between the bellows bend the actuator in setup allowsdirection the formation a propulsive in order to provide Suchused implementation can the required in 3Dofspace. Parallel wave bellows actuators (PBAs)thrust. have been for underwater control different patternsa of membrane Grouping a fewmembrane. stages of simple single-motion propulsion; [11] describes series of PBAs motion. connected with a flexible This setup allows the actuators can yield bending, twisting, and translation when controlled properly. Such a is formation of a propulsive wave in order to provide thrust. Such implementation can controlconcept different assessed of in membrane [12]. patterns motion. Grouping a few stages of simple single-motion actuators can yield While many designs exist, actuation efficiencyproperly. and energy in operation areinnot usually bending, twisting, and translation when controlled Suchlosses a concept is assessed [12]. addressed; these designs issues are ofactuation interest when dealing with marine propulsion While many exist, efficiency and energy losses in operation which are not requires usually preformingthese many repetitive or dealing rotational propulsive motions. which requires preforming addressed; issues are of “flapping” interest when with marine propulsion paper “flapping” presents the development of a new fundamental type of hydraulic actuator and manyThis repetitive or rotational propulsive motions. demonstrates itspresents application. name it HELM—Hydraulic Equal-strain Muscle, indicating This paper the We development of a new fundamental type ofLinear hydraulic actuator and its features. We the idea and principles of the HELM, the structural theory, theindicating creation ofitsa demonstrates its describe application. We name it HELM—Hydraulic Equal-strain Linear Muscle, few prototypes and verification of principles the theory.ofOne application is theory, an assembly of few of HELMs features. We describe the idea and thepossible HELM, the structural the creation a few to activate and a fishlike fin, atofa the desired motion to obtainapplication thrust. Theis last section illustrates such an prototypes verification theory. One possible an assembly of few HELMs to assembly. The following section presents the ideas and principles of the HELM and clarifies its activate a fishlike fin, at a desired motion to obtain thrust. The last section illustrates such an assembly. advantages. The following section presents the ideas and principles of the HELM and clarifies its advantages. Actuators 2016, 5, 3

2. Ideaand andPrinciples Principles 2. Idea HELM HELM has has aa tubular tubular shape shape made made of of two two membrane membrane materials: materials: a flexible flexible material material of of low tensional tensional rigidity relatively“stiff” “stiff”material materialwith withgreater greaterrigidity. rigidity.The The flexible material cutouts in rigidity and aa relatively flexible material fillsfills cutouts in the the stiff material, named hinges. Each hinge has a triangular shape in a profile view of the HELM stiff material, named hinges. Each hinge has a triangular shape in a profile view of the HELM (Figure (Figure 1). Internal causes the flexible membrane toand stretch andthe rotate theabout hingethe about 1). Internal pressurepressure causes the flexible membrane to stretch rotate hinge headthe of head of the triangle. Let R denote the of radius of the2α tube, 2α theangle head of angle of the hinge, τ the thickness the triangle. Let R denote the radius the tube, the head the hinge, τ the thickness of the of the hinge membrane and E its modulus of elasticity. By increasing the inner pressure, the is hinge membrane and E its modulus of elasticity. By increasing the inner pressure, the hinge is hinge rotated, rotated, while uniform strain in themembrane whole membrane This the is since theof length of the while uniform strain exists inexists the whole material.material. This is since length the flexible flexible fiber L(y) linearly increases with the distance from the rotation point (theofhead of the triangle). fiber L(y) linearly increases with the distance from the rotation point (the head the triangle). Thus, Thus, the active material is evenly utilized. the active material is evenly utilized. The The design design parameters: parameters: diameter, number and distribution of hinges, head angle of each hinge, hinge hinge material material and and wall wall thickness, thickness, input pressure versus versus time; time; control control the the kinematics kinematics and and power power of of the the HELM. HELM.

Figure 1. Schematic representation of the HELM hinge and relevant parameters. Figure 1. Schematic representation of the HELM hinge and relevant parameters.

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3. Motion Analysis We assume transverse cross sections remain planes (beam theory), and the tube material between hinges exhibits zero strain as it is selected to be significantly stiffer compared with the hinge membrane. A pressure P of the hydraulic fluid rotates the hinge by an angle β p . External loads may apply a moment M about the hinge, which rotates the hinge to an angle β M . The combined rotation of the hinge is therefore: β “ β M ` βP (1) We set the x axis along the centerline of the tube, and y to be aligned with the head of the hinge. The fiber length (along the x axis) of the hinge material at distance y off the head is: Lpyq “ 2pR ` yqtanpαq

(2)

Under a longitudinal strain of magnitude ε the elongation of this fiber is: ∆L “ Lε

(3)

Assuming the bending angle of the hinge is small, the angle of rotation may be approximated as: β“

∆L pR ` yq

(4)

yielding β “ 2tanpαqε, or ε “

β 2tanpαq

(5)

Thus the strain does not depend on y; for a given angle of rotation the entire hinge membrane exhibits uniform strain. This is a key aspect of the HELM. The tension stress in the x direction is composed of a stress σ M by the external moment and a stress σ p by the internal pressure. Assuming linear elastic material: σ “ σM ` σP “ Eε

(6)

The stress due to pressure at zero bending moment can be found by equilibrium of longitudinal force (assuming a thin walled tube): ż2π 2

πR P “

τσp Rdθ, or σp “

RP 2τ

(7)

0

Using Equations (4) and (5), the rotation angle due to internal pressure is: β P “ 2tanpαqε P “ tanpαq

RP Eτ

(8)

The tension force applied to a segment Rdθ of the membrane is related with the rotation angle by: Tpθq “ EετRdθ “

Eτβ Rdθ 2tanpαq

(9)

while the bending moment about the head of the hinge is: ż2π M“ 0

β EτR2 TpθqpR ` yqdθ “ M 2tanpαq

ż2π p1 ` sin θqdθ “ β M 0

πEτR2 tanpαq

(10)

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Consequently, by Equations (1) and (7), under the combination of internal pressure and external moment: Consequently, by Equations (1) and (7), under the combination of internal pressure and external moment:  R 2 E M   R 33P  πR2 τE  (11) M ` πR P “ tan( ) β (11) tanpαq Section Section 5.4 5.4 includes includes aa more more precise precise nonlinear nonlinear computation computation of of the the rotation rotation using using experimental experimental stress-strain stress-strain relations, relations, instead instead of of the the linear linearrelation, relation,assumed assumedby byEquation Equation(6). (6). Elastomers, which are used in different types of hydraulic muscles ([1–5]) as well as the HELM, Elastomers, typically exhibit loss of energy in a cycle of tension and release, in a phenomenon named hysteresis. typically When the the hydraulic hydraulic muscle muscle is designed to do work efficiently, efficiently, itit is is essential essential to to minimize minimize the the energy energy When losses. For a given elastomer, it is is possible possible to to select select an an optimal optimal strain strain range range exhibited exhibited by by the the active active losses. material that that will will minimize minimize energy energy loss loss by by hysteresis. hysteresis. Unlike other types of bellows bellows and and beams, beams, the the material bending of of which which applies applies aa linear linear strain strain distribution, distribution, the HELM is bended bended at uniform uniform strain strain so the the bending selected optimal optimal range range of of strain strain exists exists in in the the entire entire flexible flexiblematerial. material. selected 4. Prototypes 4. Prototypes Three Three prototypes prototypes of of HELM HELM were were produced produced of of three three different different hinge hingematerials: materials: EPDM EPDM (ethylene (ethylene propylene InIn the first two, thethe hinges were glued to propylenediene dienemonomer) monomer)natural naturalrubber rubberand andnitrile nitrilerubber. rubber. the first two, hinges were glued the outer tube made of Hypalon (the stiff material). The third prototype utilizes 0.8 mm thick membrane to the outer tube made of Hypalon (the stiff material). The third prototype utilizes 0.8 mm thick of fabric absorbed with non-vulcanized nitrile rubbernitrile as therubber stiff tube material, and material, a 3.5 mmand thicka membrane of fabric absorbed with non-vulcanized as the stiff tube membrane of non-vulcanized nitrile rubber with fabric with as theno hinge material (see Figure 2). The 3.5 mm thick membrane of non-vulcanized nitrilenorubber fabric as the hinge material (see cutouts were made in the fabric sheets in advance. A thin coating of liquefied nitrile rubber primer was Figure 2). The cutouts were made in the fabric sheets in advance. A thin coating of liquefied nitrile applied between theapplied layers. After setting all the layers, a vulcanization process took place at process 140 ˝ C, rubber primer was between the layers. After setting all the layers, a vulcanization resulting unified vulcanized with embedded fabricwith on the inner and outer took placeinata 140 °C, resulting in connection a unified vulcanized connection embedded fabric ondiameters. the inner Kevlar wire was wrapped around each hinge to minimize hoop strain and prevent circumferential and outer diameters. Kevlar wire was wrapped around each hinge to minimize hoop strain and expansion. This manufacturing process proven to beprocess very durable. prevent circumferential expansion. Thishas manufacturing has proven to be very durable.

Figure 2.2.Clockwise Clockwisefrom from top: Layout of third the third prototype, before andbending, during Figure top: Layout of the HELMHELM prototype, HELM HELM before and during bending, and the vulcanized hinge. and the vulcanized hinge.

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5. Verification 5. Verification Tests Tests 5.1. Tensile Tensile Testing Testingofofthe theHinge HingeMaterials Materials 5.1. Application of thethe mechanical properties of the material. This Application of the thetheory theoryrequires requiresknowing knowing mechanical properties of hinge the hinge material. section presents the tensile tests performed and the acquired mechanical properties. This section presents the tensile tests performed and the acquired mechanical properties. By reducing reducing the the circumferential circumferential expansion expansion of of the the hinge hinge membrane membrane using using the the Kevlar Kevlar wires, wires, plane plane By strain is assumed. The effective elastic modulus for the longitudinal elongation is therefore: strain is assumed. The effective elastic modulus for the longitudinal elongation is therefore:

Eeefff f  E “

E E 2 (1  p1 ´ν2 q)

(12) (12)

In the the tensile tensile tests, tests, the the specimen specimen is is free free to to contract contract perpendicular perpendicular to to the the direction direction of of tension tension so so is is In at a state of plane stress, the stress to strain ratio provides the modulus of elasticity, while Poisson’s at a state of plane stress, the stress to strain ratio provides the modulus of elasticity, while Poisson’s ratio νν is is obtained obtained by by recording recording the the width width and and thickness thickness of of the thespecimen specimenat ateach eachload loadstep. step. ratio Tensiletests testswere were also performed a loading full loading in to order to determine energy loss Tensile also performed for for a full cycle cycle in order determine energy loss through through hysteresis. The material was pre-stretched and went through a few load cycles before hysteresis. The material was pre-stretched and went through a few load cycles before recording stress recording strainand during loading A and relaxation. A few cycles were conducted in a and strain stress duringand loading relaxation. few load cycles wereload conducted in a series of strain series ofThis strain This is necessary since the relaxation curve and resulting ranges. is ranges. necessary since the relaxation curve and resulting energy losses energy dependlosses on thedepend strain on the strain range exhibited by the material. Figure 3 presents a few load-release curves for nitrile range exhibited by the material. Figure 3 presents a few load-release curves for nitrile rubber and for rubber and for natural rubber. natural rubber. Nitrile Rubber

Nitrile Rubber

Natural Rubber

350

400

110

300

350

100 90

300 250

80

150

Tension[N]

Tension[N]

Tension[N]

250 200

200

70 60

150 50 100 100 50

0

40

50

0

50

100 150 Elongation[%]

200

250

0

30

0

50

100 150 Elongation[%]

200

250

20

0

50

100 150 Elongation[%]

200

250

Figure 3. 3. Experimental Experimentalload loadcycle cyclecurves curves natural rubber (right) nitrile rubber. Leftmost is a Figure forfor natural rubber (right) andand nitrile rubber. Leftmost is a 40% 40% elongation and center for 80%. elongation rangerange and center is foris80%.

For a given load cycle, the average value of energy loss per unit volume can be obtained by the For a given load cycle, the average value of energy loss per unit volume can be obtained by the area inside the hysteresis loop. Figure 4 compares relative energy loss versus starting elongation for area inside the hysteresis loop. Figure 4 compares relative energy loss versus starting elongation for the the curves in Figure 3. Averaging all the measured load cycles (including those not presented in curves in Figure 3. Averaging all the measured load cycles (including those not presented in Figure 3) Figure 3) the nitrile rubber exhibited 23.8% relative energy loss, while the natural rubber exhibited the nitrile rubber exhibited 23.8% relative energy loss, while the natural rubber exhibited about 1.9% about 1.9% relative energy loss. relative energy loss. The parabolic fits are 0.0000∆L2 + 0.001L + 1.99 for natural rubber, 0.0007∆L2 ´ 0.2409∆L + 27.5939 for nitrile rubber at 40% elongation range and 0.0010∆L2 ´ 0.3469∆L + 39.5437 for nitrile rubber at 80% elongation range.

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Figure 4. Relative hysteresis versus mean elongation exhibited in a load cycle. Asterisks mark natural rubber load cycles. Red circles present nitrile rubber at elongation range 80%. Blue circles present nitrile rubber at elongation range 40%. Lines are parabolic fits.

The

parabolic

fits

are

0.0000L2  0.001L  1.99

for

natural

rubber,

Figure 4. 4. Relative hysteresis versus mean elongation exhibited in a load cycle. Asterisks mark natural 2 Relative hysteresis versus mean elongation exhibited in a load cycle. Asterisks mark natural Figure nitrile rubber at range 40%80%. elongation range 0.0007 Lload - 0.2409 L  circles 27.5939 rubber cycles.Red presentfor nitrile rubber at at elongation Blue circles circles present present and rubber load cycles. Red circles present nitrile rubber elongation range 80%. Blue 2 forLines nitrile at 80% 0.0010 Lrubber - 0.3469 L  39.5437 nitrile rubber at elongation elongation range 40%. 40%. arerubber parabolic fits. elongation range. nitrile at range Lines are parabolic fits.

5.2. Parameters and Response of HELM Prototype 1 2 The parabolic are Prototype for natural rubber, 0.0000 5.2. Parameters and Responsefits of HELM 1 L  0.001L  1.99 2 nitrile rubber elongation range andis 0.0007 Lfirst - 0.2409 L has 27.5939 The prototype ten hingesfor of head angles radius 2 ˝8at,10 ,12 ˝ 40% ˝ ,...26 ˝. The tube The first 2 prototype has ten hinges of head angles 2α = 8 , 10 , 12 , . . . 26 . The tube radius is forhinge nitrile rubberisatτ 80% elongation range. mm L and Lthickness  39.5437 and of hinge the membrane = 1mm . The hinge is of made of EPDM R= 3535 ,- 0.3469 R0.0010 mm, the the thickness of the membrane =is1τmm. The hinge is made EPDM while while HYPOLON is used for the inflexible tube. HYPOLON is used for the inflexible tube. 5.2. Parameters and Response of HELMtoPrototype 1 and the rotation angle  was measured at each Pressurewas wasslowly slowlyapplied applied HELM, Pressure to thethe HELM, and the rotation angle β was measured at each hinge. hinge. These measurements were compared with theory, anelastic effective elastic The first prototype has ten hinges of theory, head angles tube modulus radius is 2Equation  8assuming ,10(7), ,12assuming ,...26 . The These measurements were compared with Equation (7), an effective modulus based on the tensile tests EPDM. Figure 5 is presents the static of E=35 2.279 on the tensile tests of the EPDM. Figure presents the static pressure response E based 2.279MPa and the thickness of the hinge membrane isofτ the = 51mm . The hinge made of EPDM R mmof,MPa of the prototype 1. pressure responseisofused the prototype 1. while HYPOLON for the inflexible tube. From the analysis isisapplicable for HYPOLON relatively small FromFigure Figure thelinear linear analysis applicable forthe HYPOLON hingeswith relatively small head Pressure was5,5, slowly applied to the HELM, and rotation hinges angle was measured at head each with angles atatlow pressures. This isismost likely due to the small angle nature of our formulation, as well angles low pressures. This most likely due to the small angle nature of our formulation, as well hinge. These measurements were compared with theory, Equation (7), assuming an effective elastic as material nonlinearity and effects of deformations occurring outside of the hinge (which we have as material and effects occurring the hinge (whichthe westatic have modulus of nonlinearity based of ondeformations the tensile tests of the outside EPDM. of Figure 5 presents E  2.279MPa neglected neglectedininour ourformulation). formulation). pressure response of the prototype 1. From Figure 5, the linear analysis is applicable for HYPOLON hinges with relatively small head 14 angles at low pressures. This is most likely due to the small angle nature of our formulation, as well as material nonlinearity and 12 effects of deformations occurring outside of the hinge (which we have neglected in our formulation). 10

 [deg]

14 8

12 6

10

 [deg]

4 8 2 6 0 0

4

1

2

3 Pressure [Pa]

4

5

6 4

x 10

2 response of prototype 1. Each color corresponds to each hinge (of a specified Figure5.5.Static Staticpressure pressure Figure response of prototype 1. Each color corresponds to each hinge (of a specified head angle). Squares mark theexperiment, experiment,while whilelines linesrepresent representthe thetheoretical theoreticallinear linearresponse. response. head angle). Squares mark the 0 0

1

2

3 Pressure [Pa]

4

5

6 4

x 10

Figure 5. Static pressure response of prototype 1. Each color corresponds to each hinge (of a specified head angle). Squares mark the experiment, while lines represent the theoretical linear response.

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5.3. Parameters and Response of HELM Prototype 2 5.3. Parameters and Response of HELM 2 The second prototype of Prototype the HELM has eight hinges of natural rubber with ˝ , τ =assumed and an effective elasticof modulus of E =with 2α=12°, τ=2.3mm and R=25mm 1.098 The second prototype of the HELM has eight hinges natural rubber 2αMPa = 12was 2.3 mm

based tests. Figure 6 presents the of angle at theMPa tip ofwas theassumed HELM, accumulated all 8 and R =on 25the mmtensile and an effective elastic modulus E = 1.098 based on the by tensile 7 of the angle at the tip of the HELM, accumulated by all 8 hinges (dubbed B). 13

Actuators 2016, 5,61presents hinges (dubbed B ). tests. Figure

5.3. Parameters and Response 50 of HELM Prototype 2 45 The second prototype of the HELM has eight hinges of natural rubber with and an effective elastic modulus of E = 1.098 MPa was assumed 2α=12°, τ=2.3mm and R=25mm 40

based on the tensile tests. Figure 6 presents the angle at the tip of the HELM, accumulated by all 8 35 hinges (dubbed B ). B [deg]

30 5025 4520 4015 3510

B [deg]

30 5 25 0 20

0

2

4 6 Pressure [Pa]

8

10 x 10

4

15 response of prototype 2. Black squares mark the experiment, while red line Figure 6. Static pressure Figure 6. Static pressure response of prototype 2. Black squares mark the experiment, while red line represents the linear 10 theory. represents the linear theory. 5

5.4. Parameters and Response of HELM Prototype 3 5.4. Parameters and Response 0 of HELM Prototype 3 0 2 10 The third prototype has six4 hinges6 made8 of nitrile rubber with [Pa] The third prototype has six hinges made of Pressure nitrile rubber with 2α = 12˝x, 10 τ 4= 3.5 mm and R = 25 mm. 2α = 12°, τ = 3.5 mm and R = 25 mm. In order to test our theoretical formulation for external loading; In order to test our theoretical formulation for external loading; a test was conducted where the HELM Figure 6. Static pressure of prototype 2. Black squares mark theat experiment, while red line a test was conducted whereresponse the HELM was loaded by an external force its tip, while simultaneously was loaded by an external force at its tip, while simultaneously increasing internal pressure. represents the linear theory. increasing internal pressure. Let us define the relative angle of each joint to the x-axis as: 5.4. Parameters and Response of HELM Prototype 3 ÿ n Bn “ (13) m “1 β m The third prototype has six hinges made of nitrile rubber with at the HELM B6 (accumulated by the formulation six equal hinges) was recorded In order to test our theoretical for external loading; 2α = The 12°, τangle = 3.5 mm andtip R =of25the mm. with the external load its angle application γ F andforce internal Figure 7 for aalong test was conducted where theF,HELM wasof loaded by an external at itspressure tip, while(See simultaneously a schematic representation). increasing internal pressure.

Figure 7. Schematic representation of the kinematics of the external loading experiment.

Let us define the relative angle of each joint to the x-axis as:

Bn  m1  m n

(13)

Figure Figure 7. 7. Schematic Schematic representation representation of of the the kinematics kinematics of of the the external external loading loading experiment. experiment.

Let us define the relative angle of each joint to the x-axis as:

Bn  m1  m n

(13)

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The angle at the tip of the HELM with the external Actuators 2016, 5, 3

B6 (accumulated by the six equal hinges) was recorded along

F , its angle of application  F and internal pressure (See Figure 78 for of 12a

load

schematic representation). Applying a pressure and moment, we can express the total stress in each hinge (which is uniform Applying a pressure and moment, we can express the total stress in each hinge (which is uniform in the whole membrane of the hinge), from Equations (5), (7), and (10): in the whole membrane of the hinge), from Equations (5), (7), and (10):

RP

M

 σpp

M σM

  RP  M σ “ 2 `2 R 22 2πτR loo2τ moon loomoon

(14) (14)

Then, we can find the strain by the experimental stress-strain curve, and, from Equation (5), Then, we can find the strain by the experimental stress-strain curve, and, from Equation (5), obtain obtain the angle of bending of each hinge along the HELM. the angle of bending of each hinge along the HELM. In order to retroactively compute the moment applied about each hinge by the external load, we In order to retroactively compute the moment applied about each hinge by the external load, we must furnish furnish the the location location of denotethe thedistance distancebetween betweenadjacent adjacent must of each each joint joint relative relative to to the thetip. tip.Let Let llnn denote joints. Assuming Assuming the the HELM HELM is is rigid rigid between between hinges, hinges, the the location location(x, (x,y) y)of ofeach eachhinge hingeis: is: joints.





ÿ n 1 ÿ n 1 cosBBmm ` eeyy nmm´“111llmm sin sin BBmm RRnn “ eexx nmm´“111llmm cos

(15) (15)

Experimental angles angles were were compared compared with with nonlinear nonlinear theory, theory,using using the the experimental experimental stress-strain stress-strain Experimental curve of of the the hinge hinge material. material. Knowing Knowing the the external external moment moment about about each each joint, joint, the the stress stress applied applied to to the the curve hinge membrane can be found using the tension–moment relation in Equation (10) and pressure hinge membrane can be found using the tension–moment relation in Equation (10) and pressure stress stress at Equation The corresponding strain was interpolated from the experimental stress-stain at Equation (7). The(7). corresponding strain was interpolated from the experimental stress-stain curve curve (Figure 3) and finally the rotation angle was calculated by Equation (5). Figure 8 presents the (Figure 3) and finally the rotation angle was calculated by Equation (5). Figure 8 presents the static static response for HELM prototype 3, with and without external load. For the case with no external response for HELM prototype 3, with and without external load. For the case with no external load, load, comparison with linear by Equation also presented. comparison with linear theorytheory by Equation (11) is (11) alsoispresented. 90 1.51 Kg 80 70 0 Kg

B [deg]

60 50 40 30 20 10 0

0

0.5

1

1.5

2 2.5 Pressure [Pa]

3

3.5

4

4.5 x 10

5

Figure 8.8. Static Static response response for for HELM HELM prototype prototype 3, 3, with with and and without without external external loading. loading. The The red red line line Figure presents linear linear theory theory with with EE== 1.709 1.709 MPa. MPa. The The black black curve curve presents presents nonlinear nonlinear analysis, analysis, while while blue blue presents squares mark the experiment. squares mark the experiment.

For more complete characterization of this prototype, a series of blocked moment experiments For more complete characterization of this prototype, a series of blocked moment experiments were conducted. In each experiment, the angle of the HELM was held constant using a cable with a were conducted. In each experiment, the angle of the HELM was held constant using a cable with force gage connected to the tip of the actuator. The cable was perpendicular to the last long section a force gage connected to the tip of the actuator. The cable was perpendicular to the last long section of the HELM. For each constant angle, the pressure was applied in steps and the force and arm where of the HELM. For each constant angle, the pressure was applied in steps and the force and arm where measured to yield the moment about the root of the HELM (see Figure 9). This process was repeated for a few angles.

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measured to5,yield the moment about the root of the HELM (see Figure 9). This process was repeated Actuators 2016, 3 9 of 12 measured to yield the moment about the root of the HELM (see Figure 9). This process was repeated for a few angles. for a few angles.

Figure 9. Blocked moment Experiment. Figure 9. Blocked moment Experiment. Figure 9. Blocked moment Experiment.

Figure 10 presents the measurements compared with theory. Figure 10 Figure 10 presents presents the the measurements measurements compared compared with with theory. theory.

Figure 10. Blocked moment versus pressure, for prototype 3 of the HELM at different angles. Circles Figure 10. Blocked moment versus pressure, for prototype 3 of the HELM present experimental measurements, while lines present theoretical results.at different angles. Circles Figure 10. Blocked moment versus pressure, for prototype 3 of the HELM at different angles. Circles present experimental measurements, while lines present theoretical results. present experimental measurements, while lines present theoretical results.

6. Optimization of Use 6. Optimization of Use This section of considers a simple test case to study the effects of the design parameters on the 6. Optimization Use This section considers case to study theateffects of lifts the design parameters w locatedonatthe efficiency. Suppose a HELMaofsimple lengthtest l with a single hinge the base a weight its This section considers a simple test case to study the effects of the design parameters on the w efficiency. Suppose a HELM of length l with a single hinge at the base lifts a weight located at its wlbase coslifts tip. The moment by gravity about thea single hinge hinge is Matthe Equation (11) can  B a. weight efficiency. Supposeload a HELM of length l with w located at be its tip. The moment load by gravity about the hinge is M  wl cos  B  . Equation (11) can be numerically solved toby yield the deflection angleisfor supplied and can by calculating the tip. The moment load gravity about the hinge M the = ´wl cos(B). pressure, Equation (11) be numerically numerically solved to yield the deflection angle for the supplied pressure, and by calculating strains in yield the active membrane, thefor energy loss is interpolated from the approximation given in solved to the deflection angle the supplied pressure, and by calculating the strains in the strains inYong’s the active energy is1.709 interpolated from the approximation in Figure 4. modulus is taken approx.loss for nitrile rubber; Other parameres of E =from MPa active membrane, themembrane, energy lossthe isasinterpolated the approximation given in Figure 4.given Yong’s Figure 4. Yong’s modulus is taken as approx. for nitrile rubber; Other parameres of E = 1.709 MPa o modulus taken E =and 1.709RMPa for nitrile rubber; Other parameres of the HELM are α =as 15approx. , τ = 2mm = 30mm. the HELMisare ˝ o α = HELM 15 , τ =are 2 mm mm.and R = 30mm. α =and 15 R , τ= =302mm the

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Figure Figure 11 11 presents presents the the results results of of this this test test case. case. The efficiency efficiency was was computed computed for for two two cases: cases: when pressure is raised continously from 0 to 3 bar, and when it is raised from 1.5 to 3 bar. For nitrile pressure is raised continously from 0 to 3 bar, and when it is raised from 1.5 to 3 bar. Forrubber, nitrile in the former thecase, bestthe efficiency is at theishighest load (10load N),(10 and, inand, the latter, occures at the rubber, in the case, former best efficiency at the highest N), in the it latter, it occures lowest load (1load N) due to due changes in mean and elongation range. range. Little differences occur at the lowest (1 N) to changes in elongation mean elongation and elongation Little differences with natural rubber as the lines coincide. This demonstrates the importance of the selection of proper occur with natural rubber as the lines coincide. This demonstrates the importance of the selection of parameters to optimize a specific oparation of the HELM. proper parameters to optimize a specific oparation of the HELM.

Figure 11. (Left)—Response Angle of the HELM under different loading weights, W = 1, 2...10 N Figure 11. (Left)—Response Angle of the HELM under different loading weights, W = 1, 2 . . . 10 N marked by red to violet lines, respectively. Solid lines are for natural rubber and dashed lines for marked by red to violet lines, respectively. Solid lines are for natural rubber and dashed lines for nitrile nitrile rubber. (Center)—Bending moment at base, for each corresponding case. (Right)—actuation rubber. (Center)—Bending moment at base, for each corresponding case. (Right)—actuation efficiency efficiency vs. weight load, for nitrile (dashed line) and natural (solid line) rubber. Red marks a vs. weight load, for nitrile (dashed line) and natural (solid line) rubber. Red marks a pressure range of pressure range of 1.5–3 Bar and blue 0–3 Bar. 1.5–3 Bar and blue 0–3 Bar.

The above efficiency assessment does not include losses in the pressure supply unit. We have The efficiency assessment include lossesitself, in thewhich pressure supply We have aimed atabove computing energy losses indoes the not HELM actuator stem from unit. hysteresis. In aimed at computing energy losses in the HELM actuator itself, which stem from hysteresis. In principle, principle, the energy supplied to the actuator (the integration of pressure byş volume PdV ) may the energy supplied to the actuator (the integration of pressure by volume PdV) may be recycled be the recycled as releases the HELM releasesThe pressure. overall is efficiency is the mechanical of the as HELM pressure. overallThe efficiency the mechanical efficiencyefficiency of the actuator actuator multiplied by the hydraulic efficiency of the activation pressure device. multiplied by the hydraulic efficiency of the activation pressure device.

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7. Conclusions Conclusionsand andFuture FutureWork Work HELM is a new new type type of of hydraulic hydraulic actuator actuator that may be used for for soft soft robotics robotics and and marine marine propulsion. propulsion. Through the range of design parameters (head angle, radius, material, material, etc.), its its motion motion characteristics controlled to yield various bending shapesshapes at different sizes andsizes inputand pressures. characteristics can canbebe controlled to yield various bending at different input The linear theoretical formulation of the HELM hinge was found to be reasonably applicable pressures. for small rotation angles. formulation Nonlinear formulation, experimental curves is found The linear theoretical of the HELMbased hingeon was found to be material reasonably applicable for to be more suitable larger deflections andbased nonlinear materials such as nitrile rubber. Further small rotation angles.for Nonlinear formulation, on experimental material curves is found to be experimentation with different radii andand head angles, exhibiting strain ranges, is necessary. more suitable for larger deflections nonlinear materials various such as nitrile rubber. Further The next step would dynamicradii analysis and experimentation, minimizing losses experimentation withbedifferent and head angles, exhibitingaiming variousatstrain ranges,energy is necessary. during cyclic bending and relaxation. The next step would be dynamic analysis and experimentation, aiming at minimizing energy losses Ascyclic previously stated, one of the key benefits of the HELM design is the ability to choose the during bending and relaxation. strainAs range exhibited by the active material in order hysteresis, to better previously stated, one of the key benefits of to theminimize HELM design is thecontributing ability to choose the actuation efficiency. strain range exhibited by the active material in order to minimize hysteresis, contributing to better For example, based on the material characterization preformed, a HELM made of nitrile rubber actuation efficiency. with aFor required motion that an elongation range of 40% awould use a starting elongation example, based oncorresponds the materialtocharacterization preformed, HELM made of nitrile rubber of 160%. Similarly,motion at an elongation range of 80%, starting elongation 120%would woulduse be selected to with a required that corresponds to ana elongation range ofof40% a starting minimize (seeSimilarly, Figure 4). at an elongation range of 80%, a starting elongation of 120% would be elongationlosses of 160%. selected to minimize losses (see Figure 4). Preferably, the HELM would have been made of natural rubber and integrated by a vulcanization process. The natural rubber presents small hysteresis (about 1%) and better elastic linearity, while vulcanization provides stronger connection of the hinges.

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Preferably, the HELM would have been made of natural rubber and integrated by a vulcanization process. The5,natural rubber presents small hysteresis (about 1%) and better elastic linearity, 11 while Actuators 2016, 1 of 13 vulcanization provides stronger connection of the hinges. One One possible possible application is an assembly of few HELMs to activate a fishlike fin at aa desired desired kinematics to obtain thrust. Figure 12 illustrates three HELMs embedded into a silicone fin construction. kinematics to obtain thrust. Figure 12 illustrates three HELMs embedded into a silicone fin This prototypeThis has been fabricated and evaluated towing tank experiments. Currently, we are construction. prototype has been fabricatedbyand evaluated by towing tank experiments. developing a hydro-elastic mathematical model to predict the performance of the propulsion fin. Currently, we are developing a hydro-elastic mathematical model to predict performance ofThe the aim of this paper is aim to present HELM, while a the future publication will present the mathematical propulsion fin. The of this the paper is to present HELM, while a future publication will present model and experiments theexperiments propulsion on fin.the propulsion fin. the mathematical modelon and

Figure 12. 12. Prototype Prototype for for underwater underwater propulsion propulsion mechanism mechanism powered powered by by three three HELMs HELMs embedded embedded in in Figure silicon fin. fin. aa silicon Acknowledgments: This research was supported by the Technion Autonomous Systems Program (TASP) Land Acknowledgments: This research was supported by the Technion Autonomous Systems Program (TASP) Land & Sea Sea Research Research Centers. Centers. & Author Contributions: Contributions: N. Drimer conceived (MSc. Thesis) and A. Author conceived the theHELM HELMconcept conceptand andguided guidedJ. J.Mendelson Mendelson (MSc. Thesis) and A. Peleg (System Engineer); Mendelsonperformed performedmathematical mathematicalformulation formulationand and modeling, modeling, experiments, data Peleg (System Engineer); J. J.Mendelson data processing testing of of the the prototypes. prototypes. processingand and documentation; documentation;A. A. Peleg Peleg conducted conducted the the detailed detailed design, design, fabrication fabrication and and testing Conflicts of Interest: The authors declare no conflict of interest.

References 1.

Cai, Y.; Bi, S.; Zheng, L. Design and Experiments of a Robotic Fish Imitating Cow-Nosed Ray. J. Bionic Eng. 2010, 7, 120–126.

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Conflicts of Interest: The authors declare no conflict of interest.

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