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Sensors and Actuators A 167 (2011) 484–494

Contents lists available at ScienceDirect

Sensors and Actuators A: Physical

journa l homepage: www.e lsev ier .com/ locate /sna

i-bellows: Pneumatic bending actuator

oel Shapiroa,∗, Alon Wolfa, Kosa Gaborb

Technion, Faculty Mechanical Engineering, Haifa, 32000, IsraelTel Aviv University, School of Mechanical Engineering, Faculty of Engineering, Tel Aviv 69978, Israel

r t i c l e i n f o

rticle history:eceived 20 September 2010

a b s t r a c t

We present a compliant single degree-of-freedom pneumatic actuator with large bending capabilities.Several actuator designs are compared and validated against the suggested actuation model. Repeatabil-

eceived in revised form 2 March 2011ccepted 3 March 2011vailable online 10 March 2011

eywords:yper-redundant robotontinuum robot

ity, some dynamic properties and the affect of external loads are examined as well.© 2011 Elsevier B.V. All rights reserved.

neumatic actuator

. Introduction

Fluidic actuators (hydraulic and pneumatic) hold typical advan-ages over the more common electric actuators; they consist ofewer parts and have a lower weight and lower cost. These charac-eristics have drawn the attention of many researchers interested in

iniaturization [1], several of whom reported that, at micro scale,uidic actuators hold an advantage over electric actuators thankso a higher force/volume ratio. Fluidic actuators are attractive for

edical devices [2], not only because of miniaturization capabilitiesut also due to safety considerations, including high compliancend low working-temperatures.

Power supply is still a limiting factor for fluidic actuators. Com-act and portable fluidic power sources are still exceptional, suchs the unique idea to use a hydrogen storage alloy [3]. Most fluidicctuators include compressors and bulky supply lines, which makehem cumbersome. Despite these difficulties, several inflatable bel-ows actuators have been developed in the past. One of the earliest

orks was that of McKibben [4] on artificial muscles in the 1950s,hich opened the way to various applications [5] based on differ-

nt types of artificial muscles, e.g., Yarllot muscle [6], Morin muscle7] and ROMAC micro actuator [8]. While a single artificial-musclectuator only generates linear motion, other bellows actuators haveeen designed to generate rotation or bending. Balloon rotary joints

Abbreviations: CAD, computer assisted design; COP, center of pressure; CSM,ontinuum snakelike manipulators; UV, ultra violet.∗ Corresponding author. Tel.: +972 528695589; fax: +972 48295711.

E-mail addresses: yoelsh@tx.technion.ac.il (Y. Shapiro), alonw@tx.technion.ac.ilA. Wolf), kosa@vision.ee.ethz.ch (K. Gabor).

924-4247/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2011.03.008

usually have a limited working range but they have been used suc-cessfully for finger-like apparatus, such as a hand-prosthesis [9] ora micro-gripper [10].

Bending bellows actuators are usually long beam-like elementswhere one side stretches in the longitudinal direction more thanthe other, thus bending the actuator. Often hoops or spoke-likerestraining-beams are used to resist swelling of the bellows. Oneconcept used to achieve differential stretching is partitioning thebellows into several compartments and applying different pres-sures to each chamber. This approach introduces an additionaldegree of freedom – the pressure difference is attributed to bendingand the total pressure is attributed to elongation or stiffness. A bel-lows with three compartments [11] has three degrees of freedom(DOF) and a bellows with two compartments has two DOF – the lat-ter type was used to propel a manta-like swimming robot [12]. Itshould be noted that, in comparison to most non-partitioned singleDOF bellows, multiple DOF bellows also have an increased rangeof motion. Multiple DOF necessitate multiple pressure supplies,valves and sensors as well as complicated manufacturing.

Single DOF bellows can generate bending if they are anchoredat each end to a flexible non-extending backbone while beingconstrained to the backbone’s proximity. A multi-segment hydro-dynamic active catheter [13] utilized this concept where the supplyline served as a flexible backbone. This active catheter is especiallynoteworthy for its band-pass valves, which allow actuation of mul-tiple segments with a single supply line. Other single DOF bellows

rely on a different principle – asymmetrical profiles are employedto make one side more susceptible to longitudinal strain than theother side, so that each side stretches to a different extent and thebellows bends when the internal pressure changes. Most bellowsare actuated by increasing the internal pressure so that the bending

Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494 485

ide vi

iambiotoestboj

taeZakBewie

f

Fig. 1. Typical Bi-bellows cut section; (a) frontal view (b) s

s towards the stiffer side. A few examples use negative pressure forctuation, such as a hydraulic catheter [14] consisting of an asym-etrical silicon tube and an asymmetrical elastic support which

ends towards its weaker side. A hydraulic forceps [15] consist-ng of a bending bellows was designed by making incisions on theuter side of the curved actuator but a flexible outer sheath hado be added to retain the driving fluid. Instead of weakening theuter side, the inner side could be more resilient. This idea wasxploited in a gripper for picking up wafers [16]. The fingers con-isted of a bellows with one wall thicker than the other walls, sohat the bellows bends towards the thick side when inflated. Theending capability was increased with the assistance of groovesn the outside of the thicker wall, acting somewhat like fingeroints.

Several research groups incorporated a kinematic constrainto generate bending motion. Hirai et al. [17] explored differentrrangements of fibers embedded within the bellows-wall. Brettt al. [18] used a spring-metal cantilever to enforce bending whilehang et al. [19] used a thin cantilever and a chain to dictate thectuator’s motion. Brett’s and Zhang’s groups employed a naïveinematic model i.e. the bellows always bends along a perfect arc.oth groups focused on static aspects; Zhang et al. [19] were inter-sted in the force exerted by the actuator (for gripping applications)

hile Brett et al. [18] deployed strain gauges along the cantilever

n order to determine external forces acting on the actuator (gen-rating a sensitized end effector).

In this report we will review our initial work on a new conceptor a bending inflatable bellows actuator. Our concept is inspired by

Fig. 2. Notation for beam deflection along arc.

ew with pressure-driven loads. (COP – center of pressure).

bi-metallic strips; an actuator made of two materials with slightlydifferent shear modules. By applying pressure inside the bellows,each of the materials would tend to deform according to its stiff-ness while maintaining boundary constraints imposed by the otherhalf. The bimetallic problem was solved by Timoshenko [20]. In thisreport we will use a simpler model comprising the same mechanicalprinciples, with an asymmetrical actuator replacing the compositebellows.

We will present a theoretical model used to describe the actu-ator deflection. We will also present a validation of the modelthrough experiments that provided both the mechanical proper-ties of the device and its kinematics. Without loss of generality,this work used rapid-prototype polymer tubes, but the same con-cept and methodologies can be applied to actuators manufacturedby different means from other materials.

2. Theory

The suggested Bi-bellows is an elastic tube with one half thickerthan the other, giving it a plane-symmetrical cross section insteadof an axis-symmetrical cut section; see Fig. 1. Increasing the internalpressure P applies a tensile force T = P·Ain, where Ain is the internalcavity’s area as opposed to Aout which will be used later to denotethe area of the cross-section wall. Because the cut-section is not

axis-symmetrical, its centroid is slightly removed from the centerof pressure (COP) by a small distance Cy; this results in a bendingmoment M = P·Ain·Cy (see Fig. 1(b)).

A simple Euler–Bernoulli beam model is suggested for model-ing the bellows’ bending under quasi-static loading. The following

Fig. 3. Shape coefficient for a bent beam moment of inertia.

486 Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494

Fig. 4. A silicone rubber actuator in initial (a) and pressurized (b) positions accompanied by swelling.

F hoopss

a

i

i(tmtitwbt

ig. 5. Actuator cut-section and isometric view. (a) Square (b) Stamp, body yellow,entence, the reader is referred to the web version of the article.)

ssumptions were used for the bending model:

i. The cross-section does not change along the actuator.ii. Clamping effects are neglected.ii. The Young modulus of the material, E, remains constant during

actuation.

A moment applied to the free end of a beam induces a constantnternal bending moment, deflecting the beam along a circular arcconstant curvature). The bending curvature is defined accordingo the bending radius, �C = 1/RC. According to the Euler–Bernoulli

odel �C = M/E·Ics where M is the external bending moment, E ishe young modulus, and ICS is the cross section’s inertia moment

n the appropriate direction. The subscript C is used to point outhat the curvature being considered is that of the neutral axishich passes through the cross sections’ centroids. The angle

etween the beam’s ends and the floor is equal to the cen-ral angle the beam lies on, as shown in Fig. 2. The bending

Fig. 6. Pressure control loop implemented in Labview. (a) front panel and (b)

green, markers red, inlet blue. (For interpretation of the references to color in this

angle is equal to the beam length divided by the bending radius;ˇ = L/Rc.

Hook’s law dictates a linear relation between P and the beam’schange of length �LC, while Euler–Bernoulli’s law dictates a linearrelation between P and �C. We define two spring coefficients in (1)and (2):

k�L = P

�LC= Aout

Ain · L0· E (1)

kk = P

kC= ICS

Ain · Cy· E (2)

LC denotes the length of the neutral axis and L0 its initial length.Since ˇ = L/R , we can write a second order polynomial for the bend-

c

ing angle in (3):

ˇ(P) = P

k�

(P

k�L+ L0

)(3)

block diagram; the PI controller sends ‘digital’ commands to the valves.

Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494 487

Fig. 7. Fitting a circle through the markers.

Ffipi

istfiaadf

ig. 8. Maximal bending angle. ˇ depicted in red, marker centroids in green, Taubin-tted circle in blue. On the top-right we show the actuation pressure, bendingarametes and circle-fitting factor (Q). (For interpretation of the references to color

n this sentence, the reader is referred to the web version of the article.)

External forces may also be taken into consideration in bend-ng angle predictions. Gravity acts as a distributed load and causeself-loading proportional to the actuator’s weight. If the actuatorip moves across a surface instead of being suspended in the air,riction forces arise. Friction forces are practically more challeng-ng to consider, without having direct sensing of their magnitude

nd direction. If the beam depicted in Fig. 2 bends upwards (againstgravity field), then integrating over the beam can yield the beameflection as follows; If s is the arc length variable (s ∈ [0,1]), then aorce element of magnitude q0 at point s2 applies a bending moment

Fig. 9. Actuation parameters under quasi-static loading.

Fig. 10. Bending angle repeatability.

at s1 (s2 > s1) equal to (4):

dmS1 (S2) = −q0 · RC · [sin (S2ˇ) − sin (S1ˇ)]ds2 (4)

Replacing s1 with s and integrating s2 over [s,1] yields (5) theinternal bending moment along the arc:

m(s, ˇ) = q0 · LC

ˇ

[(s − 1) sin(s · ˇ) + cos(ˇ) − cos(s · ˇ)

ˇ

](5)

where RC is replaced by LC/ˇ. The (negative) contribution to thebending angle is given in (6):

� = LC

E · ICS

∫ 1

s=0

m(s, ˇ)ds = q0L2C

E · ICS· 1

ˇ2

[1 + cos(ˇ) − 2 sin(ˇ)

ˇ

](6)

If we wish to consider an upright beam bending in agreementwith the gravitational force, then we can return to (4) and replacethe sine functions with cosine functions and repeat the two integra-tion steps. Notice that for the simple straight beam (ˇ ≡ 0, RC → ∞ )(4) does not hold and (5) and (6) suffer from singularity.

The dynamic behavior of the Square model was examined undersinusoidal actuation at different frequencies. We compare these tothe SquareIn model, to demonstrate the significant difference. Mod-eling the beam’s deflection as a mass-spring system allows makinga crude estimation1 of its peak response frequency, f. Denoting thebeams inertia moment around an axis going through its base as Ib,f can be derived from (7):

Ib · ¨̌ + Ics · E

L0ˇ = Cy · Ain · P (7)

So a rough estimation of the peak response frequency f can begiven in (8):

f =√

Ics · E

Ib · L0(8)

It should be noted that Ib was taken to be constant but, in fact, itdiminishes as the actuator bends. Consider a 1D beam; the inertiamoment taken from its clamped edge is L2/3. If the beam is curved,

then its moment of inertia is give by (9):

Ib(ˇ) = 2(

L

ˇ

)2(

1 − sin (ˇ)ˇ

)(9)

1 A high enough damping coefficient can shift the peak-response-frequency sig-nificantly downwards.

488 Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494

Table 1Model properties.

L0 [mm] Cy [mm] My [mm] Aout [mm2] Ics [mm4] W [g] Ib [kg mm2] k�L [N/mm3] kk [N/mm3] f [Hz]

Square 124 1.2 2.8 55.85 607.55 10 55.71 5.27 × 10−3 5.729 0.2Stamp 124 2.2 6.5 61.66 933.59 11 80.15 5.81 × 10−3 4.985 0.2

%Stra

n

bejˇ

3

utstelepF1PtS3u

siBas

in LabVIEW. The actuators were tested under two modes; cyclicloading (sine wave) or quasi-static loading (pyramid step function).Only positive pressure was used for actuation, i.e., the sine wavewas biased instead of having a zero mean. An Intersema MS54XX

Fig. 11. Segment lengths (a) and

The expression for Ib(ˇ) can be broken up into L2 times Cs, aon-dimensional shape coefficient, displayed in Fig. 3:

The formula for Cs agrees with the simple cases of a straighteam (ˇ = 0, Cs = 1/3) and a circle (ˇ = 2�, Cs = 1/2�2). If we re-stimate the peak response frequency f with a mean Ib instead ofust using a constant value, then f would increase. For example, if

is in the range [0,�], f would increase by ∼15%.

. Methods

Several models of bi-bellows actuators were designed and man-factured using a rapid prototype machine. The 3D printer usesiny droplets of resin to create a slice of the CAD model, cures thislice with UV light and then prints the next slice on top. Severalypes of resin can be used; structural materials with different prop-rties and also support material which is washed away later-on,eaving cavities. Each model consisted of a flexible body and sev-ral rigid components; inlet, hoops, markers. The flexible body islane-symmetrical and resembles a Square or Stamp, as shown inig. 5. The internal cavity of each model is ∅7 mm in diameter and20 mm long, while the top half has a ∅10 mm external diameter.reliminary experiments were conducted with a silicone rubberube, where one side was reinforced by gluing half a tube to it.ignificant swelling was noticed with the silicone actuator (up to2% change in diameter) as shown in Fig. 4. The swelling was notniform, displaying kinks.

Rigid hoops (green elements in Fig. 5) were used to resistwelling of the rapid prototype models. The significant difference

n Young modules2 (Tango Black ∼0.45 MPa for the bellows, Tangolack + ∼3.5 MPa for the hoops) is sufficient for the hoops to serves rigid constraints. Each hoop’s diameter is ∅1 mm, and they arepaced along the bellows with a 4 mm distance between centers,

2 Results of tensile test courtesy of OBJETTM.

in (b) as a function of pressure.

as demonstrated in Fig. 5. Other rigid elements include an inlet andmarkers; 15 small markers,∅1.5 mm and 0.5 mm high, are equallyspaced along each side of the actuator. The rigid inlets were usedfor anchoring the bellows.

Two other models, SquareIn and StampIn, were also used; theywere similar to the Square and Stamp models but with smallerhoops buried inside the body’s wall (imitating the techniquedescribed in [11,12]). Both had a shorter effective length (75%) andrecurrent end effects so they were soon abandoned, but we will usesome results to demonstrate qualitative phenomena in the resultssection. Features of the Square and Stamp models are summarizedin Table 1.

The symbols in Table 1 stand for; L0 initial length, Cy centroid-COP distance (Fig. 1) My centroid-marker distance, Aout cut-sectionwall area, Ics cut-section inertia moment round centroid in y direc-tion (see Fig. 1), W actuator’s weight (excluding the inlet’s weight,2 g), Ib actuator inertia moment round clamped edge at ˇ = 0, k�Land k� effective spring coefficients ((1) and (2)), f approximatedpeak response frequency (8).

The pressure control loop and image storing were implemented

Fig. 12. Markers against the end cap (a) and inlet (b) (hoops not shown).

Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494 489

Table 2Repeatability: errors of observations vs. fitted curves.

ˇ [◦] L0� %Strain

Mean 1.43 0.028 0.45

pb

soeFla

pTciap

Ti

Q

PcmdEm

εC = εM + My − Cy

2�L0· ˇ (11)

Std 1.51 0.031 0.54Max 5.88 0.178 2.07

ressure sensor was used to read the actual pressure inside theellows, relative to atmospheric pressure.

A PI controller calculates the pressure error between the sen-or reading and the set point. A single supply/release valve ispened when the error exceeded 0.01 MPa (0.1 bar); when the errorxceeds 0.02 MPa (0.2 bar), two valves are opened, as shown inig. 6. A simple USB camera is used to take photographs of the bel-ows and all the relevant information (iteration, pressure reading,nd pressure command) is stored in the file name.

The bending radius, bending angle and strain are calculated byost processing each image. Once the markers are identified, theaubin algorithm [21] is used to fit a circle passing through theirentroids, as shown in Fig. 7. The bending angle is computed accord-ng to the two extreme markers. Strain is derived by comparing therc length to its minimal value; arc length is calculated using theroduct of the circle radius and the bending angle.

A slight adaptation was made to Chernov’s version [22] of theaubin algorithm in order to include Q, a quality estimator definedn (10):

= max{||P̄i − P̄C ||} − RM

RM(10)

¯ i is a vector to the centroid of marker i, P̄C is a vector to the cir-le center and RM is the radius of the circle passing through thearkers. The non-dimensional quality estimator Q normalizes the

istance of a marker from the circle’s circumference by its radius.

xceedingly high Q values allow automatic detection of markerisidentification.

Fig. 14. Friction vs. gravity: (a) �

Fig. 13. Non-uniform curvature, due to gravity.

4. Results

The actuator is able to achieve bending angles greater than 180◦

at moderate pressures, as demonstrated in Fig. 8 for a Stamp model.Strain was measured by tracking the markers’ centers so it rep-

resents the strain on that fiber; other fibers experienced greaterstrain. For example, the strain on the neutral axis can be calculatedaccording to (11):

ˇ – pressure, (b) �ˇ − �S.

Strain values, even on the outer most fiber, remained under 20%strain.

490 Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494

F gravid

divt

ig. 15. Bending angles of two different actuator models actuated at 0.25 Hz againstotted line shows the gravity-corrected prediction.

No hysteresis was detected under quasi-static loading, asemonstrated in Fig. 9 for a StampIn model actuated against grav-

ty (hence the negative initial bending angle). However, even underery slow cyclic loading of 0.25 Hz, hysteresis was detected for allhe observed parameters; ˇ, RM and %Strain.

Fig. 16. Bending with (upright starting position) or

ty: (a) Square model, (b) Stamp model. The full line shows the naïve prediction, the

In

Actuator repeatability was examined more closely on a Squaremodel under quasi static loading, actuated against a frictional force.Fourteen repetitions of a step function were used and a total of 870frames were analyzed. Results for the bending angle are shownin Fig. 10. Errors (distances along the “y” axis) between the mea-

against (horizontal starting position) gravity.

Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494 491

p fide

sb

atFsi

iq+bF

antbpattTfiit

d

Fig. 17. Pressure control loo

urements and the fitted-curve pair are summarized in Table 2 forending angle, bending radius and strain.

Examining the strain of each segment along the actuator isnother way to assess its shape. In order to give a clear visualiza-ion of segment length, second order regression curves are used inig. 11(a). The regression curve is used to calculate segment strain,hown in Fig. 11(b), where the intercept of each curve in Fig. 11(a)s taken as the initial length of each segment.

The results in Fig. 11 were taken from a SquareIn actuator, bend-ng against gravity from a horizontal beginning position, underuasi-static actuation. The bending angle was in the range [−10◦,60◦]. The reduced strain of segments 1 and 14 can be explainedy the actuators’ design near the inlet and end-cap, as shown inig. 12.

A slight decay in strain can be noticed in Fig. 11(b) when movingcross segments 2-13 from base to tip. Recall that a fiber under-eath the neutral axis undergoes compression under bending, sohe lower strain of the upper half could be attributed to greaterending. The distributed gravitational load is responsible for thishenomenon; it induces a bending moment which opposes thectuating moment and its magnitude grows towards the actua-or’s base. An example from a Square model is used to validatehis claim; Fig. 13 shows an actuator with a blue circle fitted by theaubin method [21] through all markers (Q = 0.026), and two circles

tted manually: yellow to the top half and red to the lower half;

t is evident that the curvature increases towards the actuator’sip.

Results from a SquareIn model under quasi-static actuationemonstrate how the strain-pressure curves alternate under dif-

lity at different frequencies.

ferent external loads. Fig. 14 shows �ˇ against pressure (a) and�S (b); �ˇ is the angle change and �S is the strain change fromambient pressure (P = 0).

One of the ideas for the future is to incorporate PVDF as a strainsensor in order to measure the bending angle. The results displayedin Fig. 14 imply that measuring the overall actuator strain in orderto estimate ˇ cannot produce an injective relation, not even approx-imately. This might be overcome by separate strain measurementsof several segments along the actuator or by placing additionalforce-sensors along the actuator.

Quantitative comparison of the Square and Stamp models isgiven in Fig. 15. Both models were examined under a 0.25 Hz sinu-soidal actuation, bending against gravity from a horizontal startingposition. The solid lines depict the prediction of ˇ according to (3)and the dotted lines depict the gravity correction of ˇ according to(6). Notice that correcting ˇ according to gravity requires previousknowledge of the actuator’s alignment. The Stamp model displaysmore, probably because of faster ˇ changes. As predicted, the Stampmodel is superior to the Square model. The gravity compensationcorresponds to observation.

In Fig. 14 we appended two predictions; the ‘traditional’ naïvepressure-bending prediction and a prediction that considers grav-ity. Apparently in both cases the theory does not describe theSquare model’s behavior very well. In Fig. 16 we add a comparison

of the Square model bending with or against gravity to demon-strate that this discrepancy is consistent. Results are shown inFig. 16 as a blue or red scatter plot, against the naïve prediction(black dotted line) and the gravity corrected predictions (red orblue, respectively).

492 Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494

e loop

stooasto

tw

Fig. 18. State domain pressur

The dynamic behavior of the Square model was examined underinusoidal actuation at different frequencies. We compare these tohe SquareIn model, to demonstrate the significant difference. Inrder to estimate the ability of the pressure-control loop to carryut sinusoidal commands at different frequencies we replaced thectuator with a small dead volume (cap) and compared the mea-ured pressure (P) against the pressure set-point (SP). Examininghe results displayed in Fig. 17 we decided to examine the actuator

nly up to 4 Hz, because of the control loop’s limitations.

Fig. 18 displays state domain loops of measured pressure (P) vs.he set point (SP); blue dots were taken from the cap and red dotsere taken from a Square model with an upright starting posi-

Fig. 19. Bending angle at different actuation frequencies.

s at six different frequencies.

tion. Evidently, the actuator does not affect the pressure profilesignificantly.

Next we show the bending angle of the SquareIn actuator isagainst the measured pressure in Fig. 19, and the strain in Fig. 20.The regression curves are used to capture the loops’ main axis.Recall that the sine wave is biased: the regression curves rotate(increasing phase) approximately around the middle of ˇ’s rangeand slightly off center of the strain-percentage range, approxi-

mately around the first quarter.

Using regression curves helps in depicting the actuators’response and comparing it at different actuation frequencies. Theslopes of the regression curves were used to generate amplificationfunctions and to compare the models against each other; amplifi-

Fig. 20. Strain at different actuation frequencies.

Y. Shapiro et al. / Sensors and Actuators A 167 (2011) 484–494 493

(a) (b)

Am

plifi

ca�

on [d

eg/b

ar]

Am

plifi

ca�

on [%

str

ain/

bar]Square

Square

Square In

Square In

ion tr

cQta

1t

rtra

5

(Thb

bwwt

i

ii

iv

v

A

s

R

[

[

[

[

[

[

[[

[

[

[[

[

in biomedical engineering from Technion-I.I.T. in 2006(summa cum laude). He completed a M.Sc. in 2009 onhaptic control for an orthopedic-surgery robot in theMechanical Engineering faculty, Technion-I.I.T. under thesupervision of Dr. Alon Wolf. He is currently a Ph.D. stu-

Actua�on frequency [Hz]

Fig. 21. Bending angle amplificat

ation functions of slope-values are depicted in Fig. 21(a) and (b).ualitatively, one can say that the actuators are over-damped, as

hey do not vibrate at all when released from a large initial bendingngle.

The decay of Square and SquareIn in Fig. 21(a) is 4.1 and2.8 dB/decade and in Fig. 21(b) it is 3.3 and 5.3 dB/decade, respec-ively.

Further investigation is required to distinguish whether theeduced strain of the markers’ fiber (which is below the neu-ral axis) corresponds to increased bending or to over-all actuatoreduced strain, e.g., by measuring strain on a fiber above the neutralxis.

. Discussion

The bi-Bellows actuator can achieve large bending angles+180◦) at moderate pressures (<1.25 atm above ambient pressure).he beam deflection model is adequate for models with externaloops. Inadequacy for models with internal hoops might be causedy multiple end conditions recurring within the actuator.

Measuring strain can give an indication of the bending angleut must be done carefully, i.e., in a combination of force sensors orith sub-actuator resolution. Generally the predictions present aeaker non-linearity than that observed. Further study is necessary

o reconcile these discrepancies but candidate factors may be:

i. out of plane bending caused by external forces (gravity, friction)which are not traceable by the current measuring system andwere not accounted for in the current model

i. swelling increases Cy, Ain, Ics; reducing bending and also increas-ing the weight of the second order term

i. volume conservation under tension reduces Aout, increasing theweight of the second order term

. stiffening of the polymer (strain induced increase of E) wouldintroduce variability into the weight of the second order term;decreasing with strain

. cross-sectional shape factors increasing stiffness and damping.

cknowledgement

We would like to thank Mr. Zviki Kinstler for his contribution toetting up the experimental system.

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Biographies

Yoel Shapiro was born in 1980 and received a B.Sc.

dent under Dr. Wolf.

4 Actu

ir

94 Y. Shapiro et al. / Sensors and

Alon Wolf earned all his academic degrees fromthe Faculty of Mechanical Engineering at Technion-I.I.T. In 2002 he joined the Robotics Institute ofCarnegie Mellon University and the Institute for Com-puter Assisted Orthopedic Surgery as a member ofthe research faculty. He was also an adjunct AssistantProfessor in the School of Medicine of the Univer-sity of Pittsburgh. In 2006 Dr. Wolf joined the Facultyof Mechanical Engineering at Technion, where he

founded the Biorobotics and Biomechanics Lab (BRML).The scope of work done in the BRML provides theframework for fundamental theories in kinematics,biomechanics and mechanism design, with applications

n medical robotics, rehabilitation robotics, and biorobotics, such as snakeobots.

ators A 167 (2011) 484–494

Gábor Kósa received his B.Sc. in mechanical engineeringfrom Technion-I.I.T. in 1995. He served in the IDF from1995–1998 as a research engineer. In 2001 received aM.Sc. on non-linear dynamics and control from Technion-I.I.T. He was employed at RAFAEL as a MEMS R&D engineerfrom 2000–2001. He received a Ph.D. on medical micro-robots from Technion-I.I.T. in 2006. Since 2007, he hasbeen working as a post-doctoral fellow and biomedicalmicrosystems group leader in the Computer Vision Lab-

oratory in ETH Zurich, investigating novel force sensorsand swimming micro robots. Research interests includemodeling, designing, fabricating and testing of micro sys-tems, micro fluidics and medical robotics. He was recently

appointed as a lecturer in the School of Mechanical Engineering at Tel Aviv Univer-sity.