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Pseudo-rigidBodyModelingofIPMCforaPartiallyCompliantFour-barMechanismforWorkVolumeGeneration

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StructuresJournal of Intelligent Material Systems and

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 DOI: 10.1177/1045389X08088784

2009 20: 51 originally published online 11 July 2008Journal of Intelligent Material Systems and StructuresDibakar Bandopadhya, Bishakh Bhattacharya and Ashish Dutta

GenerationPseudo-rigid Body Modeling of IPMC for a Partially Compliant Four-bar Mechanism for Work Volume

  

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by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

Pseudo-rigid Body Modeling of IPMC for a Partially CompliantFour-bar Mechanism for Work Volume Generation

DIBAKAR BANDOPADHYA,* BISHAKH BHATTACHARYA AND ASHISH DUTTA

Department of Mechanical Engineering, IIT Kanpur, Kanpur 208 016, India

ABSTRACT: Conventional four-bar crank rocker mechanisms made of rigid links cangenerate only one path, at the rocker tip, for one revolution of the crank. However, if therocker length can be actively changed then its tip can generate a work volume. This studydescribes an application of ionic polymer metal composite (IPMC) as a partially compliantrocker in a four-bar mechanism for work volume generation. First, an experiment isconducted to study the voltage verses bending characteristics of IPMC and based on theexperimental data the IPMC is modeled using a pseudo rigid body model. The model is basedon the fix-pin support type of cantilever mode and its derivation is explained in detail.The maximum and minimum length of the rocker is controlled by changing the voltageapplied to it and this generates a work volume for one revolution of the crank. Simulationresults are compared with the experimentally obtained work volume and the differences arefound. The proposed mechanism has the potential for application in micro positioning,compliant structures, etc.

Key Words: ionic polymer metal composite, four-bar mechanism, work volume, pseudo-rigidbody model.

INTRODUCTION

FOUR-BAR mechanisms have traditionally been madeof rigid links. In the case of a conventional crank

rocker mechanism the rocker tip (or a coupler point)generates a single path for one revolution of the crank.In case the length of the rocker is varied, a work volumecan be generated. Such adjustable four-bar mechanismswith an adjustable rocker have been made earlier using aprismatic joint in the rocker to vary its length (Hong andErdman, 2005). However, such mechanisms are quitelarge in size and require high power due to the prismaticjoint and actuators required to change the rocker length.In this study, the authors propose a small partiallycomplaint four-bar mechanism with ionic polymer metalcomposite (IPMC) as an adjustable rocker. The IPMCbeing small in size, the overall length of the four-barmechanism can be kept small. In addition, the actuationof the IPMC requires very low voltage and its controldoes not involve sophisticated controllers. Hence, it isideal for micro-scale applications.The objectives of this study are:

(a) Experimentally measure the deflection of an IPMCdue to varying voltage inputs and, based on it,model the IPMC using pseudo-rigid body technique.

(b) Develop a four-bar mechanism with a compliantIPMC rocker in which the other links are rigid.

(c) Vary the length of the IPMC for one revolution ofthe crank and obtain a work volume.

(d) Compare the simulation and the experimentalresults.

Several types of compliant mechanism and theiranalysis using pseudo-rigid body model have beenshown in Howell and Midha (1994) and Midha et al.(2000). Flying insect robots, where compliant four-barmechanism has been used successfully for high degree ofmaneuverability (Shimoyama et al., 1993; Garcia andGoldfarb, 1998; Fearing et al., 2000; Yan et al., 2001) andcompliant wing structure for micromechanical flyinginsect (MFI) robot (Sitti, 2001). A few of the varyingpathgenerationof closed loop linkageshavebeen reportedinTao and Sadler (1995), Lin andChen (1996), Tokuz andUyan (1997) and Zhang et al. (1999). Saggere and Kota(2001) have developed a four-bar mechanism for segmen-ted path generation with a flexible coupler which under-goes prescribed shape along with the rigid body motion.

The IPMC is a class of electro-active polymer (EAP)that is gaining importance as an actuator due to its largebending deflection with low actuation voltage. EAPs aredivided into two groups: those driven by an electric fieldand those driven by the diffusion of ions. Details of theworking of an IPMC is given in Shahinpoor et al.(1998), Bar-Cohen et al. (2002), Zeng et al. (2005) and

*Author to whom correspondence should be addressed.E-mail: dibakarb@iitk.ac.inFigures 2–4, 6 and 11–21 appear in color online: http://jim.sagepub.com

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by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

Lee et al. (2007). Several aspects of the manufacture ofIPMC have been described in Shahinpoor and Kim(2000, 2001, 2004, 2005) and Kim and Shahinpoor(2003). Applications of IPMC for vibration attenuationare given in Bandopadhya et al. (2007a,b).The motion of the proposed four-bar mechanism is

not in the gravity plane and hence the IPMC is notaffected by gravitational forces. In addition, the IPMCdoes not transmit any torque as the input to the systemis given to the crank by a DC motor. The rocker followsthe motion of the crank via the coupler. As the crank isoperated at very slow speeds the dynamic forces on therocker are negligible. Hence the low load carryingcapacity of the IPMC does not affect the functioningof the mechanism. The IPMC used is of size40� 5� 1mm3. An experiment is conducted to studythe relation between the deflection and the voltageapplied to the IPMC in cantilever mode. Cantilevermode of one end fixed and other pin joint for the IPMCrocker has been analyzed using pseudo-rigid bodymodeling technique as proposed in Howell (2001).Simulations are performed using the pseudo-rigid bodymodeling technique and it is found that it matcheswith the circular bending model of IPMC as obtainedfrom the experiments. Using these results a four-barmechanism is designed with IPMC based rocker.Experimentally, it is shown that by applying a voltagesource at the rocker end a work volume generation ispossible. In the next section, IPMC has been studiedexperimentally to obtain its bending characteristics toobtain the bending moment–curvature and curvature–voltage relationship. The pseudo-rigid body modeling ofthe IPMC-rocker is then presented. The limit positionsof the partially compliant four-bar mechanism havebeen discussed. The circular bending model of theIPMC-rocker for varying work volume generation isthen discussed. The simulation results and comparativediscussions are then presented while the section thatfollows deals with the experiment results. Finally,conclusions are drawn.

BENDING CHARACTERISTIC OF IPMC

An IPMC is kept in the cantilever mode with one endfixed and a controlled voltage is applied at the fixed end.The voltage is increased in steps and the correspondingdeflections are recorded on a graph paper. The IPMC ofsize 40� 5� 1mm3, purchased from Environment Inc.,Virginia, USA, is used and water is used as the polarsolvent. The experiment is conducted in cantilever modesubjected to an input voltage up to 4.5V to calibrate itsbending characteristic. Copper strips are used at one endand voltage is applied quasi-statically from a DC powersupply (0–60V, 0–10A). IPMC is bent under an inputvoltage up to 4.5V starting from 0.5V with an

increment of 0.5V in each step. For each input voltageafter 30 s tip deflection of the IPMC has been recorded.The radius of curvature R for an input voltage V isobtained as R¼ 1/�, where, l is the length of the IPMCand � is the tip angle of IPMC. From Figure 1, for anarbitrary tip position P(u, v), tan �¼ v/u, further usingEuler–Bernoulli equation of curvature one can get,

d�

ds¼

Mb

EIð1Þ

where, d�/ds is the rate of change in angular deflectionalong the IPMC strip (curvature s), Mb is the bendingmoment generated due to input voltage V, and EI is theflexural rigidity of IPMC. From Equation (1) and onintegration one can obtain the end point coordinates as:

u ¼W sin�, v ¼Wð1� cos�Þ ð2Þ

where, W¼EI/Mb. Combining Equations (1) and (2)one gets,

tan� ¼W 1� cos�ð Þ

W sin�¼

1� cos�

sin�¼ tan

�

2) � ¼ 2�:

ð3Þ

Experimentally � is known to us by locating tipposition of the IPMC. Therefore, curvature–end tipangle relationship is obtained as:

� ¼1

R¼�

l¼

2�

lð4Þ

Figure 2 shows the change in radius of curvature withrespect to the tip angle of the IPMC for each inputvoltage. The tip angle is obtained experimentally forevery bending sequence of IPMC. This clearly demon-strates how the radius of curvature is varying even for

x

U

P

Y

vle l

a

F

R

IPMC

El

O (0,0)

Figure 1. Bending configuration of IPMC under applied voltage.

52 D. BANDOPADHYA ET AL.

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a small variation of tip angle. Further, it is assumed thatIPMC is bending with linearly varying curvature. Inorder to verify this assumption experimentally, tipangles of the IPMC have been obtained and subse-quently curvature of IPMC is plotted for each inputvoltage. After measuring the end point deflection ofIPMC for each input voltage, the relation given inEquation (4) is used to obtain the curvature–voltagerelationship for both increasing and decreasing inputvoltage as plotted in Figure 3. The average curvatureis then used to obtain the bending moment andinput voltage relationship. The experiment clearlydemonstrates the hysteresis losses it undergoes due todehydration.Therefore, from the experiment one can assume,

� ¼1

R¼ k� V 0 � V � 5 ð5Þ

where, k is a path constant, which can be determinedfrom the experimental data and V is the applied voltage.The value of k depends on the backbone materials,counter ions and its amount, and the property of thesolvent present as given in (Shahinpoor et al., 1998).The bending moment generated due to the input voltageto IPMC rocker is given by

Mb ¼ EI� � ð6Þ

where, E is the modulus of elasticity of IPMC and I isthe area moment of inertia. Bending moment generatedcompletely depends on the moisture content andsubsequently movement of ions within IPMC and thebackbone materials. Hence, the electrode properties donot affect the generated moment and also the thicknessof the electrode is very small so it does not affect theelastic modulus of the base polymer material. The IPMCused in the experiment is thick (1mm) and can withstand

a large potential of around 5V for a short time.However, dehydration due to moisture loss throughthe porous metal electrode on the polymer surfaceaffects the repeatability of the IPMC. Each cycle of thefour-bar motion is carried out in about 10 s and hencethe IPMC does not get enough time to back relax in thisshort time.

PSEUDO-RIGID BODY MODELING

OF IPMC-ROCKER

The experimental results obtained in the previoussection are used to model the bending characteristics ofthe IPMC using a pseudo rigid body model (Howell,2001). The maximum bending moment generated foreach input voltage is given as (Mb¼EI� k), where, k isthe average curvature obtained experimentally for eachinput voltage, which includes all the distributed pressureeffects. An analogy has been drawn that the bendingphenomena is equivalent to the same amount of tipdeflection that is caused by an external bending momentMb acting at the tip of IPMC. This concept is employedhere to model the IPMC for each bending momentobtained for each input voltage. The pseudo-rigid bodymodeling technique has been followed considering end-moment at the rocker point, to obtain the end-pointdeflection of IPMC-rocker in cantilever mode ofone end fixed and other pin joint. Voltage is applied atone end and subsequently end point deflection hasbeen obtained. Bending moment and the deflectionangle are related as (Mb¼Ks�) where, � is thecorresponding pseudo-rigid body angle and Ks isthe spring constant. Figure 4 shows the equivalentbending model in cantilever mode for an inputvoltage V. The main objective is to find out the springconstant and pseudo-rigid body angle of the IPMC.

4.53.5

Forward pathBackward path

45

40

35

30

25

20

Cur

vatu

re (

1/m

)

15

10

5

02.5

Voltage (V)

1.50.50 1 2 3 4

Figure 3. Voltage–curvature relationship of IPMC obtained experi-mentally.

500

450

400

350

300

250

200

150

100

50

0.2

Rad

ius

of c

urva

ture

(m

m)

0.4 0.6

Tip angle (rad)

0.8 1 1.2

Figure 2. Change in radius of curvature of bent IPMC with thechange in tip angle.

Pseudo-rigid Body Modeling of IPMC 53

by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

Using the curvature–moment relationship one canobtain:

d�

ds¼

Mb

EIð7Þ

where, � is the tip angle of the rocker. Both the tipangles, � (Equation (1)) and � are equivalent to eachother. For small length of flexural pivot one can assumethat �¼ �, and then spring constant is obtained as:

Ks ¼EI

l3ð8Þ

where, l3 is the length of the IPMC-rocker. Duringmotion of the four-bar mechanism, a tangential forcewill act at the tip of the rocker, this will produce aresultant bending moment at the rocker. Consideringresultant bending moment acting at the rocker is Mt

then:

Mt ¼ Kt�: ð9Þ

This moment can also be expressed in terms of theresultant transverse force Ft multiplied by the momentarm (the length of the pseudo-rigid link), thus one canwrite, Mt¼Ftgl3 where, g is the characteristic radiusfactor. Combining with Equation (9) Ft¼Kt�/gl3,introducing force–deflection relationships in terms ofthe nondimensionalized load term (Howell, 2001).

ð�2t Þ ¼Ftl

23

EI: ð10Þ

The spring constant is directly obtained as:

Kt ¼ �K�EI

l3¼ 1:5164�

EI

l3ð11Þ

where, characteristic radius factor g¼ 0.7346 and stiff-ness coefficient K�¼ 2.07 (Howell, 2001).

Pseudo-rigid Body Modeling of Fixed-pinned

IPMC-rocker

The pseudo-rigid body model of the IPMC-rocker isshown in Figure 5. Here, the rocker is replaced by a link

with a characteristic pivot located at a length gl3 fromthe free end of the rocker. It is assumed that a circularpath can be accurately modeled by two rigid links thatare connected at a pivot along the IPMC strip.A torsional spring at the pivot represents the IPMCstrip’s resistance to deflection. The location of thispseudo-rigid-body characteristic pivot is measuredfrom the IPMC tip end as a fraction of the IPMClength, where the fractional distance is gl and g is thecharacteristic radius factor. The product gl, the char-acteristic radius, is the radius of the circular deflectionpath traversed by the end of the pseudo-rigid body link.It is also the length of the pseudo-rigid body link.

Now, from Figure 5 one can obtain end pointcoordinates of the link as:

v ¼ �l3sin �, u ¼ l3 � �l3ð Þ þ �l3 cos �

¼ l3 � �l3 1� cos �ð Þ ¼ l3 1� � 1� cos �ð Þ� � ð12Þ

where, ED denotes the length of characteristic pivotlocated from the fixed end, DC denotes the pseudo-rigidbody length, and EC is the effective length. Therefore,effective length of the rocker is found out as:

l3e ¼ l3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2� 1� cos �ð Þ � � 1ð Þ

pð13Þ

Figure 6 shows the change in tip position of theIPMC-rocker for different input voltages.

Force using Free Body Diagram

The free body diagram of the mechanism is shown inFigure 7. For the links to be in static equilibrium thefollowing conditions must be satisfied:

XFx ¼ 0

XFy ¼ 0

XM ¼ 0:

The reaction forces on each link are labeled as Fij

i.e., the subscripts indicate the link number. Theequations of equilibrium may be written for link 1 as

Y l3l3–Yl3

u

v

− x

YF

CFn

Ft

Characteristic pivot

Torsionalspring

Pseudo-rigidbody angle

D

E

Figure 5. Pseudo-rigid body modeling of rocker made of IPMC.

Copper strip

Electrode

Equivalent bending model

Cantilever mode

IPMC

V+−

Mb = El × K

Figure 4. Equivalent bending model of IPMC for input voltage V.

54 D. BANDOPADHYA ET AL.

by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

(where considering u, v are the variables along x, ydirection, respectively):

F41u þ F21u ¼ 0,F41v þ F21v ¼ 0:

Taking the moment about point ‘A’ one gets

Ti þ F21vl1 cos �1 � F21ul1 sin �1 ¼ 0 ð14Þ

Similarly for link 2,

F12u þ F32u ¼ 0,F12v þ F32v ¼ 0,

Taking moment about B

T2 þ F32vl2 cos �2 � F32ul2 sin �2 ¼ 0 ð15Þ

And finally for link 3,

F43u þ F23u ¼ 0,F43v þ F23v ¼ 0,

Taking moment about point C one has

Ts þ F43vr3 cos �0

3 � F43ur3 sin �0

3 ¼ 0 ð16Þ

Considering F41u ¼ a, then F21u ¼ �a, F12u ¼ a andF32u ¼ �a ¼ F23u (joint C is fixed), F43u ¼ a similarly

F41v ¼ b, then F21v ¼ �b, F12v ¼ b and F32v ¼ �b ¼ F23v,F43v ¼ b, solving Equations (14) and (16) one gets

a ¼ Ftd ¼Tir3 cos �3 þ Tsl1 cos �1

l1r3 sin �3 þ �1ð Þð17Þ

where, Ftd is a tangential force. Similarly,

b ¼Tir3sin �3 þ Tsl1 sin �1

l1r3 sin �3 þ �1ð Þð18Þ

Therefore, bending moment generated is found out as:

Md ¼ Ftd � �l3sin �3

¼Tir3 cos �3 þ Tsl1 cos �1

l1r3 sin �3 þ �1ð Þ� r3 sin �3

ð19Þ

where, ð�0

3 ¼ �3 þ �Þ, r3 is the length of the pseudo-rigid body link that is given by r3¼ gl3, Ts is thetorque produced by the torsional spring and isgiven by:

Ts ¼ �K�EpI

l3� ð�3 � �3, 0Þ ð20Þ

where, �3,0 is the angle of the undeformed IPMC. Thefour-bar mechanism with compliant IPMC-rocker ismodeled using pseudo rigid body method as shown inFigure 8. The pseudo-rigid body links (CD) are given bygl3 and length (DE) given by r4¼ l3� gl3.

The coordinates of the point C is obtained as:

xc ¼ l4 þ l3e cos �3

yc ¼ l3e sin �3ð21Þ

where, �3 is the IPMC-rocker angle during rotation. Theend point tip position calculations are obtained by usingstandard rigid body mechanism analysis.

−10−10

−5

0

5

10

15

20

30

40

25

45

35

−5 0 5 10

Pseudo-rigid tip position of IPMC-rockerY-

coor

dina

te (

mm

)

15 2520

X-coordinate (mm)

5V

D ′C ′ = g l3

E ′ D ′ = l3 – g l3

4V

3V

2V

1V

0.5VE′

C′

30 35 40 45

D′

Figure 6. Change in tip position of the IPMC-rocker following thepseudo-rigid body modeling, voltage ranges from 0.5 to 5 V.

Y

C

EX

D

Coupler

Crank

Fixed link

B

A

Yl3

l2

q3

q4

q2

q1

l1

r4l4

Pseudo-rigid bodylink

Figure 8. Pseudo-rigid body modeling is applied to four-barmechanism.

Pseudo-rigid body link

Torsional spring F43u

F43v

F23u

F23vF32u

F32v

F21v

F41v

F12v

F12u

F21u

F41u

T2

Ti

Ts

r3

C

B

B

A

C

D

l2

l2

q2

q1

′q3

q3

Figure 7. Free body diagrams for the three moving links of the four-bar mechanism.

Pseudo-rigid Body Modeling of IPMC 55

by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

LIMIT POSITIONS OF THE PARTIALLY

COMPLIANT FOUR-BAR MECHANISM

A four-bar mechanism reaches its limit positions forone complete revolution of the crank i.e., angle �3reaches its maximum and minimum position. Nortonet al. (1994) have introduced the triangle inequalityconcept to study the inequality concept for a non-Grashofian four-bar mechanism based on their mobility.Ting (1989) has modified the triangle inequality conceptand introduced the theorem of assemblability andrevolvability pertaining to the four-bar mechanism.The triangular properties was used to find out thelimit condition of the four-bar mechanism. The limitposition of the mechanism depends on the range of theangle � as shown in Figure 9.

Limit Positions of the Rocker

Limit position of the rocker is obtained when �3reaches its minimum value i.e., �3 ¼ �

min3 and when it

reaches maximum i.e., �max3 ¼ �min

3 þ �. From Figure 9one can obtain

� ¼ cos�1l24 þ l23 � l22

2l4l3

� �ð22Þ

� ¼ cos�1l24 þ l23 � l1 þ l2ð Þ

2

2l4l3

� �� � ð23Þ

Therefore, limit position is obtained as:

�min3 ¼ 1808� �þ �ð Þ, �max

3 ¼ 1808� � ð24Þ

For the partially compliant rocker, limit position isobtained similarly as:

�min3e ¼ 1808� �e þ �eð Þ, �max

3e ¼ 1808� �e ð25Þ

where, �e, �e are the angles corresponding to theeffective length l3e of the rocker. For link lengthsl1¼ 15mm, l2¼ 40mm, l3¼ 40mm, l4¼ 40mm limit

position of the rocker is obtained as �min3 ¼ 93:128 and

�max3 ¼ 1208. Length of the path generated by the rockertip is obtained as s¼ l3 �¼ 18.75mm. A partiallycompliant rocker can have different limit positionsaccording to its effective length.

Limit Angle Positions of the Crank

Limit position of the crank is obtained similarly when�1 reached its minimum value i.e., �1 ¼ �

min1 and when it

reaches maximum i.e., �max1 ¼ 180o þ �min

1 þ �. Thusfrom Figure 9,

�min1 ¼ cos�1

l1 þ l2ð Þ2þl24 � l23

2 l1 þ l2ð Þl4

� �ð26Þ

�min1 þ � ¼ cos�1

l22 þ l24 � l232l2l4

� �ð27Þ

Therefore, limit position of the crank is obtained forthe link lengths specified above as �min

1 ¼ 46:6o and�max1 ¼ 240o. Limit position of the crank is also variedaccording to the effective length of the compliantIPMC-rocker.

CIRCULAR BENDING MODEL OF

THE IPMC-ROCKER

Circular bending model has been used to validate theproposed pseudo-rigid body model. The effective lengthof the IPMC-rocker is obtained through free bodydiagram as shown in Figure 10 and comparison ismade subsequently with the proposed model for tippositioning.

In the figure, OF¼OG¼R is the radius of curvature,FG is the effective length of IPMC of length l3 and isdenoted by l3e, O is the center of the curvature and � isthe angle between the radii of curvature meeting two end

Y

GX

O

F

l3e l3

IPMC

Effe

ctiv

e le

ngth

j

j

Figure 10. Free body diagram of the bent IPMC-rocker.

Y

A

Y

D

a

b

C

B

Rocker

θ1min

θ 3min

Coupler

Fixed link

Crank

Figure 9. Limit positions of a four-bar mechanism.

56 D. BANDOPADHYA ET AL.

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points of IPMC i.e., hFOG ¼ �. The relation can befound as:

l3 ¼ R�) � ¼l3R

ð28Þ

Therefore, � can be calculated for a known bendingconfiguration of IPMC. Using triangular properties,from triangle OFG one can get:

cos� ¼OF 2 þOG 2 � FG 2

2OF:OGð29Þ

Thus, effective length is obtained as:

l3e ¼ Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1� cos�ð Þ

pð30Þ

Figure 11 shows the circular bending pattern of IPMCfor maximum input voltage of 5V starting with initialinput of 0.25V.

SIMULATION RESULTS AND DISCUSSION

This section describes the generation of work volumeby changing the effective length of the IPMC rocker.A program has been developed in MATLAB forpath generation taking into account of the complianceof IPMC and the properties listed in Table 1.

Coordinates of the points P1, P2, P3 and P4 are givenin terms of link length and variable angle. Consideringpoint P1 at the origin, the coordinates of the other pointsare given by:

P4 x4 ¼ l4, y4 ¼ 0ð Þ,

P2 x2 ¼ l1 cos �1, y2 ¼ l1 sin �1ð Þ

P3 x3 ¼ l4 þ l3e cos �3, y3 ¼ l3e sin �3ð Þ:

A motor torque of 10Nmm at the crank is used forthe simulation.

Figure 12 shows the tip position of the IPMC-rocker(AB) after application of input voltages. The resultsshow that pseudo-rigid body modeling matched with thecircular bending model of IPMC. Figure 13 shows thedecrement of effective length for maximum input voltageof 5V. Figure 14 shows the work volume generated foran input voltage of 5V. It is found that increment ofIPMC-rocker length, increases the work volume ofthe four-bar mechanism. Figure 15 shows the workvolume (a�b�c�d�a) generated by the variablelength rocker for an input voltage of 5V followingthe circular bending model of IPMC. It is observedthat the two work volumes generated are the same,although two different modeling techniques for IPMCare used.

IPMC-rocker

5V4.5V

3.5V

2.5V

1.5V

0.5V

4540353025201510500

5

10

15

Y-co

ordi

nate

(m

m)

X-coordinate (mm)

20

25

30

4V

3V

2V

1V

Figure 11. Exact bending pattern of the IPMC- rocker for maximuminput voltage of 5 V with increment of 0.25 V.

Pseudo-rigid pathCircular bending path

50

40

30

20

10

0

−10−10 100 20

X-coordinate (mm)

Y-co

ordi

nate

(m

m)

30 40 50

Figure 12. Path followed by IPMC both for pseudo-rigid bodyanalysis and circular bending model for an input of 5 V.

Table 1. Four bar crank-rocker mechanism properties.

Crank length (l1)¼ 15 mm Width of IPMC (w)¼10 mm Spring constant Kt¼ 37.91 N mmCoupler length (l2)¼ 40 mm Thickness of IPMC (h)¼ 1 mm Total time (t)¼ 10 sRocker length (l3)¼ 40 mm Modulus of elasticity (E)¼1.2 GPa No of steps (n)¼ 500Fixed length (l4)¼ 40 mm increment (�1)¼ 0.0126 rad Maximum voltage input¼ 5 V

Pseudo-rigid Body Modeling of IPMC 57

by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

EXPERIMENT RESULTS AND DISCUSSION

An experimental setup of a partially compliant four-bar mechanism is designed as shown in Figure 16, withIPMC-rocker. The links are made of lightweight plasticand a DC motor (made by KTA Japan) is used toprovide input torque to the crank. Crank and couplerform revolute joints, while one end of the IPMC link isfixed with a coupler and the other end with a revolutejoint with the fixed link. Thus pseudo-rigid bodymodeling of rocker of fixed-pin support has beenevaluated for work volume generation to compare withthe theoretical results. The length of the crank is 15mmwhile the coupler and fixed frame are each of 40mmlength. The size of the IPMC strip is 40� 5� 1mm3 and

a set of two electrodes are placed at the joint end of theIPMC. The experiment is conducted to generate a singlepath for one complete revolution of crank in 10 s.The necessary control input and controller are shown inFigure 17.

In the experiment, the motion of the four-bar isrecorded by a camera and the frames are processed toobtain the coordinates of the rocker tip. Figure 18 showsthe comparison of the theoretical and experimentalresults for generating a single path. Figure 19 shows thevariation of effective length of the rocker obtained fromthe experiment and the model. Figure 19 shows both thetheoretical and experimental variation of effectivelength of the rocker. It is observed that the change intheoretical effective length is more compared tothe experimental, as the change in effective lengthis affected due to dehydration, as time progresses.

Pseudo-rigid body modelCircular bending model

Input voltage (V)

0.5 0.5

5

4

3

Effe

ctiv

e le

ngth

dec

rem

ent (

mm

)

2

1

0.5 0.5 0.5 54321

Figure 13. Decrement of effective rocker length for maximum inputvoltages of 5 V for both pseudo-rigid body modeling and circularbending model.

Work volume generated due to bending of IPMC

1–2–3–4–1 :Work volume

Coupler

−40 −20

50

40

30

20

10

−10

−20

−30

−40

−50

0

0 20

X-coordinate (mm)

Y-co

ordi

nate

(m

m)

40 60 80

Crank

Rocker

2

3

1

4

P1P4

P3

P2

Figure 14. Work volume generated by variable length rocker for aninput voltage of 5 V following pseudo-rigid body modeling.

Work volume generated due to bending of IPMC

a–b–c–d–a : Work volume

Y-co

ordi

nate

(m

m)

X-coordinate (mm)−40 −20

−20

20

30

40

50

−10

10

0 A

B

ad

C

RockerCoupler

Crank

c

b

D

−30

−40

−50200 40 60 80

Figure 15. Work volume generated following circular bendingmodel of the IPMC-rocker for an input of 5 V.

Voltage suppliedto motor

Voltage suppliedto IPMC

IPMCCoupler

Crank

Motor

Figure 16. Rocker is bending due to input voltage during motion ofthe four-bar mechanism.

58 D. BANDOPADHYA ET AL.

by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

Moreover, misalignment and bending resistance at therocker point also affect effective length.After comparing the paths generated by the rocker tip

position, next the lengths of the IPMC is varied frommaximum to minimum for one revolution of the crank.Figure 20 shows the two instantaneous IPMC-rockerend positions during the motion of the four-barmechanism. Figure 21 shows the corresponding workvolume for a maximum input voltage of 3V and aminimum of 0V. The outer curve is obtained bymaintaining the IPMC at 0V and then near the end ofthe curve (on the left side) the voltage is reduced to 3Vso that the rocker length is reduced. This generates thework volume. It is observed that the theoretically

(a)

(b)

Figure 20. (a) and (b) shows the end position of the bending motionof the IPMC-Rocker during work volume generation.

Coupler

Fixed join

Fixed link

IPMC-rocker

Pin join

Crank

Motor

PC DAQ Amplifier

τ

Figure 17. Schematic diagram of the four-bar mechanism with thecontrol strategy.

Experimental

X-coordinate (mm)

Y-co

ordi

nate

(m

m)

24

8

10

12

14

16

18

20

22

24

26

26 28 30 32 34 36 38

Pseudo-rigid

Figure 18. Comparative study on rocker tip position experimentallyand theoretically (pseudo-rigid body) for an input of 4.5 V with anincrement of 0.5 V starting from 1 V.

1 1.5 2.5

Voltage (V)

Effe

ctiv

e le

ngth

(m

m)

Maximum voltage input = 4.5V

40

39

38

37

36

35

38.5

37.5

36.5

35.5

39.5

Voltage increment = 0.5V

TheoreticalExperimental

3.5 4.52 3 4

Figure 19. Variation of effective length obtained experimentally andtheoretically (pseudo-rigid body) for a maximum voltage input of4.5 V with an increment of 0.5 V starting from 1 V.

Pseudo-rigid Body Modeling of IPMC 59

by Bishakh Bhattacharya on September 30, 2010jim.sagepub.comDownloaded from

obtained work volume (shown in Figures 14 and 15) islarger as compared to the experimental one as higherchange in effective length is recorded as shown inFigure 19. It is observed that after each revolution of thefour-bar in 10 s the IPMC undergoes dehydration whichaffects the bending of IPMC for the next revolution.In addition, there may be some degree of misfit andmisalignment at the joint connecting the rocker and thecoupler which may affect the bending of the IPMCduring rotation of the crank. These two parameters willcause some errors in the theoretically and experimentallyobtained results.

CONCLUSIONS

This study presents a partially compliant four-barmechanism for work volume generation in which thetraditional rigid-body rocker has been replaced by anIPMC. Pseudo-rigid body modeling technique has beenfollowed to model the rocker for variable path genera-tion. It is proved theoretically and experimentally thatpseudo-rigid body modeling technique gives preciseresults. This was also verified with the circular bendingmodel of IPMC-rocker. This mechanism has thepotential for application in micro-robotics and also incompliant mechanisms.

APPENDIX

Equation (7), integrating throughout rocker lengthone can obtain,

Z �

0

d� ¼Mb

EI

Z l3

0

ds) � ¼Mbl3EI

,

for small length flexural pivot one can assume, pseudo-rigid body angle � is equal to the tip angle � of IPMC.Therefore, Mb ¼ Ks� ¼

EIl3

�:Equation (13) is obtained as:

l3e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2þ v2

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil23 1�� 1� cos�ð Þ� �2

þ �l3 sin�ð Þ2

q

¼ l3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�� 1� cos�ð Þ� �2

þ�2 sin2 �

q

¼ l3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�2� 1� cos�ð Þþ�2 1�2cos�þ cos2 �ð Þþ�2 sin2 �

q

¼ l3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�2� 1� cos�ð Þþ2�2 1� cos�ð Þ

p

Equation (17) is obtained as

a ¼Tir3 cos �

0

3 � Tsl1 cos �1

l1r3 sin �0

3 þ �1� � ¼

Tir3 cos �3 þ Tsl1 cos �1l1r3 sin �3 þ �1ð Þ

:

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90

70

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60

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x-coordinate (mm)

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