Stiffness Modeling for an Orthogonal 3-PUU CompliantParallel Micromanipulator

Qingsong Xu and Yangmin LiDepartment of Electromechanical Engineering, Faculty of Science and Technology

University of MacauAv. Padre Tomas Pereira S.J., Taipa, Macao SAR, P. R. China

{ya47401 & ymli}@umac.mo

Abstract— The stiffness modeling for a compliant parallelmanipulator (CPM) is very important since it provides a basis forthe characterization of static, modal, and dynamic behavior ofthe CPM. This paper presents the stiffness modeling of a three-prismatic-universal-universal (3-PUU) CPM with orthogonallymounted actuators, that is designed to provide three spatialtranslational DOF for nano-scale manipulation. Considering thecompliance of each compliant element, the analytical stiffnessmodel for a spatial CPM is established by a straightforwardapproach, which is then applied to stiffness modeling of the 3-PUU CPM. In addition, the finite element analysis is carried outto validate the developed model, and as a further application,the influence of architectural parameters on stiffness factors arederived based on the stiffness model, which is valuable for a cost-effective design of the CPM.

Index Terms— Compliant mechanisms; Parallel manipulators;Stiffness.

I. INTRODUCTION

As is well-known, parallel manipulators possess inherentadvantages over their serial counterparts in terms of highrigidity, high load carrying capacity, high velocity, and highprecision, etc [1]; on the other hand, compliant mechanisms,i.e., flexure hinge-based mechanisms, possess excellent charac-teristics including vacuum compatibility, no backlash property,no nonlinear friction, and ease of manufacture, etc [2]. Con-sequently, as a combination of these two separate concepts,compliant parallel manipulators (CPMs) which are parallelmanipulators employing compliant hinges at the joints, canprovide the merits of both parallel and compliant mechanisms,and have a potential in various applications where an ultra-precision manipulation is first and foremost required [3]–[12].

The stiffness modeling for a CPM is very important sinceit is directly related to the characterization of static, modal,and dynamic behavior of the CPM. In the literature, mostwork on the stiffness analysis is focused on separated flex-ure hinges including elliptical flexure hinges, corner-filletedhinges, and right circular hinges, etc [13]–[15], and planarCPMs possessing two or three DOF in a plane where only thestiffness of flexure hinges on their working directions is takeninto account [7]–[9] by using the method of finite elementanalysis or analytical modeling approach. Since a spatial CPMwith high stiffness can provide a more dexterous and precisemanipulation with high natural frequency, it is necessary to

P joint

U joint

PZT

Mobile platform

Fig. 1. An orthogonal 3-PUU CPM.

perform the stiffness modeling of it. Unfortunately, only a fewof literatures dealing with this point are available up to now[10], [11].

The primary goal of this paper is to perform the stiffnessmodeling of a 3-PUU CPM with orthogonally arranged linearactuators which is designed for nano-scale manipulation inour previous work [12]. An analytical approach for stiffnessanalysis of a spatial CPM is presented and applied to the 3-PUU CPM. The proposed modeling method employs only onekind of transformation matrix, which is more straightforwardand intuitive than that in reference [10] where too manytypes of transforming matrices concerning the translation androtation of displacements and forces are utilized to derivethe stiffness matrix. Moreover, the finite element analysis iscarried out to validate the established stiffness model, andthe influence of kinematic parameters on stiffness has beenassessed, which is useful in architectural design.

II. DESCRIPTION AND DEVELOPMENT OF THE CPM

As shown in Fig. 1, the designed 3-DOF CPM, which adoptsflexure hinges at all joints, consists of a mobile platform, afixed base, and three limbs with identical kinematic structure.Each limb connects the fixed base to the mobile platformby one flexure prismatic (P) hinge followed by two flexureuniversal (U) hinges in sequence, where the P joint is fixed atthe base and actuated by a piezoelectric (PZT) actuator. Thus,each limb is a PUU kinematic linkage.

It has been shown that a 3-PUU kinematical structure withconventional mechanical joints can be arranged to achieve only

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Proceedings of the 2006 IEEEInternational Conference on Mechatronics and Automation

June 25 - 28, 2006, Luoyang, China

xy

z

O

w

h

l

(a) A cantilever.

r

t

w

xy

z

O

(b) A right circular hinge.

Fig. 2. Basic compliant elements.

translational motions with some certain geometric conditionssatisfied [16], [17], i.e., in each kinematic chain, the firstrevolute (R) joint axis is parallel to the last R joint axis,and the two intermediate R joint axes are parallel to eachother. Meeting these conditions, the proposed CPM possessesthree translational DOF. Furthermore, in order to generatean isotropic manipulator with a cuboid shape workspace, thethree actuated P joints are arranged in such an orthogonalway. Mounting a suitable end-effector on the mobile platformor placing the mobile platform under a specified microscopeas a XYZ-stage, the CPM can be used in 3-D nano-scalepositioning manipulation.

Offering the merits in terms of smooth motion, high accu-racy, and fast response, etc., PZT actuators are much suitablefor precision engineering applications. The drawback is theirlimited stroke. To make a compromise between the stroke andresolution, one type of PZT P-239.80 is selected from thePolytec PI, Inc., which has the stoke of 140μm and resolutionof 1.4 nm. The linear actuator of the CPM is implemented witheach PZT embedded in a flexure P hinge as shown in Fig. 1.

The adopted compact flexure U hinge consists of twoorthogonally intersected flexure R hinges as illustrated in thefollowing Fig. 4(b). The flexure R hinge can be designed tohave profile shapes of right angle, corner filleted, or rightcircular, etc. Here we adopt the right circular one because itsdisplacement accuracy is the best, i.e., the center of flexureis displaced much smaller than the other types, which can befabricated by the method of EDM or laser cutting, etc.

Additionally, because the ratio of yield strength (σy) toYoung’s modulus (E) of the material heavily affects therotary limits of flexure hinges, which partially determine themanipulator workspace, it is necessary to choose materialswith higher ratio of σy/E so as to obtain a larger workspace.Thus we select the material of titanium alloy Ti-6Al-4V tobuild the CPM.

III. STIFFNESS ANALYSIS OF A SPATIAL CPM

Pseudo-rigid-body (PRB) model is an effective approach forkinematic design of CPMs, since it allows compliant mecha-nisms to be modeled as equivalent rigid-link mechanisms andenables the use of traditional mechanism analysis methodology

ix

iy

iz

iO

x

y

z

Oir

Localframe Reference

frameθ z

θ y

θx

θ zi

θxi

θ yi

Fig. 3. Representation of compliance transformation.

to design compliant mechanisms. However, PRB model can notbe utilized for a complete stiffness analysis of spatial CPMssince it only considers the flexure hinges stiffness in theirworking directions.

A. Compliance and Stiffness Matrix of Compliant Elements

When an external wrench F = [fx fy fz mx my mz]T is

applied on a certain point of a compliant element, it causes aninfinitesimal twist X = [ux uy uz θx θy θz]

T of that point.Here we use twists to represent infinitesimal displacements of acompliant element. With the assumption of linear relationshipbetween a force and the caused deflection, the following twoequations hold:

F = KX, (1)

X = CF, (2)

where K and C are respectively called the stiffness andcompliance matrices, which are inverse matrices of each other,i.e., K = C−1.

In view of Plucker’s original conventions, a twist is ex-pressed in the axis coordinate and a wrench is expressed in theray coordinate. The conversion between the two coordinates isallowed by a matrix Δ with the expression of [18]:

Δ =

[0 II 0

], (3)

where 0 is a 3×3 zero matrix, and I is a 3×3 identity matrix.It should be noticed that K and C are also the transformationsbetween the axis and ray coordinates.

For the basic compliant elements of a cantilever and a rightcircular flexure hinge as shown in Fig. 2, the compliancematrices with respect to local frames are described by Ccant

and Ccirc, respectively, which take on the following form:

C =

⎡⎢⎢⎢⎢⎢⎢⎣

c1 0 0 0 c7 00 c2 0 −c8 0 00 0 c3 0 0 00 −c8 0 c4 0 0c7 0 0 0 c5 00 0 0 0 0 c6

⎤⎥⎥⎥⎥⎥⎥⎦

, (4)

where the expressions of factors c1 to c8 are listed in [3].

125

B. Compliance Transformation between Frames

As illustrated in Fig. 3, for a compliant element with localcompliance matrix Ci and stiffness matrix Ki expressed inlocal frame Oi{xi, yi, zi}, the elastic deformations Xi canbe transformed into Xi with respect to the reference frameO{x, y, z} by a 6×6 transformation matrix Ji, i.e.,

Xi = JiXi. (5)

Since the deformations Xi and Xi are the increments ofthe corresponding displacements in essence, the matrix Ji alsocharacterizes a velocity transformation from frame Oi to frameO, which can be derived as follows [19]:

Ji =

[Ri −RiS(ri)0 Ri

], (6)

where Ri denotes the rotation matrix of the frame Oi withrespect to O, ri =

−−→OiO = [rix riy riz ]

T is a vector expressedin the frame Oi, and S(·) represents the skew-symmetricoperator with the notation of

S(ri) =

⎡⎣ 0 −riz riy

riz 0 −rix

−riy rix 0

⎤⎦ . (7)

As revealed in the remainder of the paper, the uniquetransformation matrix Ji can be employed as a foundation forthe analytical stiffness modeling of a spatial CPM.

As far as the wrench transformation is concerned, thewrench Fi described in the local frame Oi can be transformedinto F with respect to the reference frame O via [20]:

Fi = ΔJiΔFi. (8)

Alternatively, by virtue of the kineto-statics duality, thewrench Fi exerted on the compliant element can be trans-formed into Fi simply by JT

i , i.e.,

Fi = JTi Fi. (9)

It is obvious that (9) is equivalent to (8) in view of the factthat J−1

i = ΔJTi Δ. Using (2), (5), and (9), we can get

Xi = CiFi = JiXi = JiCiFi = JiCiJTi Fi, (10)

which holds for any wrench F, consequently

Ci = JiCiJTi , (11)

where Ci is the compliance matrix of the compliant elementexpressed in the reference frame O.

In view of the inverse relationship between stiffness andcompliance matrices, the stiffness matrix with respect to frameO can be derived as follows.

Ki = J−Ti C−1

i J−1

i (12)

C. Compliance Matrix of a Compliant Serial Chain

For a serial chain including m compliant elements, theelastic deformation X at a reference point C is the summationof deformations of each elements, i.e.,

X =

m∑i=1

Xi =

m∑i=1

JiXi. (13)

Expressing the deformation into the form of (2) and in viewof (10), allows the derivation of:

C F =

m∑i=1

JiXi =

m∑i=1

JiCiJTi F. (14)

Then, with the consideration of (11), we can get the compli-ance matrix for a serial chain expressed in the reference frameC:

C =

m∑i=1

Ci = JCdJT , (15)

where J = [J1 J2 . . . Jm], and the diagonal matrix Cd =diag(C1, C2, . . . , Cm).

D. Stiffness Matrix of a Compliant Parallel Manipulator

Considering a parallel manipulator consisting of n compliantserial chains which connect the mobile platform to the fixedbase, the displacements X of the platform at the reference pointP equal to deformations of each serial chain with respect tothe reference frame P , i.e.,

X = JciXci, i = 1, 2, . . . , n, (16)

where Jci is the transformation matrix of the ith chain fromits reference frame Ci to the frame P .

Moreover, the wrench F applied to the reference point P isdistributed to each chain’s reference points Ci, i.e.,

F =

n∑i=1

J−Tci Fci. (17)

Assign that K is the stiffness matrix of the compliant parallelmanipulator expressed in the reference frame P , and Kci is thestiffness matrix of the ith serial chain expressed in frame Ci.Combining (1) and (16), the above equation can be writteninto:

KX =

n∑i=1

J−Tci KciXci =

n∑i=1

J−Tci KciJ−1

ci X, (18)

which holds for any displacement X, thus we have

K =

n∑i=1

J−Tci KciJ

−1

ci = J∗

cKdcJc, (19)

where J∗

c = [J−Tc1 J−T

c2 . . . J−Tcn ], Kd

c =diag(Kc1, Kc2, . . . , Kcn), and Jc = [J−1

c1 J−1

c2 . . . J−1

cn ]T

126

h wl 11y 11z

11x

1y1z

1x

1O

2d

12x12y

12z

11O12O

(a) A prismatic joint.

r

t

w

w

(b) A universal joint.

Fig. 4. Flexure joints.

IV. STIFFNESS MODELING OF THE 3-PUU CPM

A. Stiffness of Flexure Prismatic joint

Referring to Fig. 4(a), we can observe that the matricestransforming the compliance of the cantilevers from their localframes O11 and O12 to frame O1 respectively are:

J11 =

[I −S(r11)0 I

], J12 =

[Rx(π) −Rx(π)S(r12)

0 Rx(π)

],

(20)where I is a 3×3 identity matrix, r11 =

−−−−→O11O1 = [0 0 d]T ,

Rx(π) denotes the rotation of π around the x-axis, and r12 =−−−−→O12O1 = [0 0 d]T .

According to (11), the compliance matrices of the can-tilevers expressed in frame O1 can be respectively derived as:

C11 = J11Ccant1JT11, C12 = J12Ccant1JT

12. (21)

The flexure prismatic joint consists of two cantilevers con-nected in parallel, hence the stiffness matrix can be derived inview of (19) by noticing that Jci (i = 1, 2) becomes identitymatrices:

Kprsm1 = K11 + K12 = C−1

11+ C

−1

12. (22)

B. Stiffness of One Limb

With the assumption that the rigidities of the mobile plat-form and cuboid moving link of the flexure prismatic hingeare infinite, then one limb consists of six compliant elementsincluding one prismatic hinge, one cantilever, and four revolutehinges. Selecting the center point P of the mobile platform asthe reference point, then the compliance matrix of one limbcan be derived in accordance with (15).

Taking limb 3 as an example, the local frames of thesix elements are shown in Fig. 5, referring to which, thecompliance transformation matrices can be identified as:

J1 =

[Rz(

π2)Rx(π

2) −Rz(

π2)Rx(π

2)S(r1)

0 Rz(π2)Rx(π

2)

],

J2 =

[I −S(r2)0 I

], J3 =

[Rz(−

π2) −Rz(−

π2)S(r3)

0 Rz(−π2)

],

J4 =

[I −S(r4)0 I

], J5 =

[Rz(−

π2) −Rz(−

π2)S(r5)

0 Rz(−π2)

],

J6 =

[I −S(r6)0 I

], (23)

a

1l

2l

1y 1z

1x

1O

32 ( )O O

54 6( , )OO O

2x 2y

2z 3x3y3z

5x5y5z6x 6y

6z

xy

z

P

4x 4y

4z

b

Fig. 5. Geometry of Limb 3.

where Rx(π2) and Rz(

π2) denote the rotation of π

2with respect

to the x and z axes, respectively, and the position vectors are

r1 = [l1 + a + l2 0 0], r2 = r3 = [0 0 a + l2],

r4 = r5 = r6 = [0 0 l2], (24)

with l2 = (b + w)/2.Making use of (15) allows the generation of the compliance

matrix for limb 3:Cc3 = JCdJT , (25)

where J = [J1 J2 . . . J6] is a 6×36 ma-trix, and the 36×36 diagonal matrix Cd =diag(Cprsm1, Ccirc2, Ccirc3, Ccant4, Ccirc5, Ccirc6).

Therefore, the stiffness matrix of limb 3 with respect toframe P is

Kc3 = C−1

c3 . (26)

Likewise, the stiffness matrices Kc1 and Kc2 of limbs 1 and2 can also be determined, respectively.

C. Stiffness of the 3-PUU CPM

Taking into account the fact that Jci, for i = 1, 2, and 3,become identity matrices since the local frame of each limbis the same as the reference frame P , the stiffness matrix forthe 3-PUU CPM can be obtained via (19) as follows:

KP =

3∑i=1

Kci. (27)

Further more, the stiffness matrix with respect to other ref-erence frames can also be derived through (12). For instance,the stiffness matrix expressed in frame E{x, y, z}, which islocated at the point above the mobile platform center P , is:

KE = J−TP KP J−1

P , (28)

where

JP =

[I −S(e)0 I

], (29)

127

Fig. 6. FEM model of the 3-PUU CPM.

with the position vector e =−−→PE.

V. STIFFNESS MODELING TESTS WITH FINITE ELEMENT

ANALYSIS

To verify the stiffness modeling processed above, finiteelement analysis (FEA) is undertaken in ANSYS. The meshedthree-dimensional finite element model (FEM) of the CPM isshown in Fig. 6, which is developed utilizing the 10-node solidelement SOLID 92. The bottoms of three flexure P hinges arefixed and a force/moment is exerted on the mobile platform.In the simulation, the material is assigned to be Ti alloy, andthe architectural parameters of the CPM with flexure elementsare described in Table I.

The FEA is performed two times to determine the stiffnessvalues in three directions. Firstly, a force fz = −1 N is appliedat the center E of the upper plane of the mobile platform. Inaddition, point E1 is located on the y-axis and EE1 = dy , asdescribed in Fig. 6. The displacements zE and zE1 of pointsE and E1 are generated after solving the FEM and used tocalculate the stiffness factor Kfz−uz

along the z-axis directionand Kfz−θx

around the x-axis via the following equations:

Kfz−uz=

fz

zE

, (30)

Kfz−θx=

dyfz

zE1 − zE

. (31)

Secondly, a force fz = −1 N is applied at the point E1

which causes a moment mx around the x-axis. The displace-ment zE of point E and zE1 of E1 are used to determine thestiffness factor Kmx−θx

through:

Kmx−θx=

d2yfz

zE1 − zE

. (32)

The three stiffness factors are calculated by the FEA ap-proach and analytical method, respectively, with the resultsshown in Table II. It is observed that the deviation is less than10% with respect to the FEA result. Moreover, the analytical

TABLE I

PARAMETERS OF THE CPM AND TI ALLOY

CPM architectural parameters (mm)a b l1 r t w h l d

60 60 15 2.0 0.5 16 2 16 8

Ti-6Al-4V alloy parametersYoung’s Modulus E Yield strength σy Poisson’s ratio113.8 GPa 880 MPa 0.342

TABLE II

STIFFNESS OF THE CPM

Kfz−uzKfz−θx

Kmx−θxStiffness(N/mm)×103 (N/rad)×105 (N mm/rad)×106

Analytical 3.57 1.76 2.30FEA 3.95 1.69 2.11Deviation (%) 9.6 4.1 9.0

method is cost-effective since the FEA approach requiresconsiderable more calculation time.

VI. INFLUENCE OF ARCHITECTURAL PARAMETERS ON

STIFFNESS FACTORS

The cost-effective analytical stiffness model has been furtherapplied to evaluate the influence of every architectural para-meter of the CPM on each stiffness factor. As an example, theplots of the stiffness factor Kfz−uz

versus all of the parametersof are illustrated in Fig. 7. When two parameters are varied,the other parameters are assigned to be constants as shownin Table I. It can be observed that stiffness factors decreasenonlinearly as the increasing of a, b, l1, r, and l. On the otherhand, the stiffness factors increase when the parameters t, w,h, and d of flexure hinges increases, as expected.

Further more, the analytical stiffness model can be used foroptimal design, or static, modal, and dynamic analyses of theCPM as well.

VII. CONCLUSIONS

The stiffness model of a 3-PUU CPM is established ana-lytically in this paper by an intuitive procedure incorporatingthe compliance of each compliant element into the stiffnessmatrix which relates every architectural parameter of the CPM.The finite element analysis is also performed in ANSYS,and the simulation results reveal that the deviation of theanalytical approach is less than 10% with respect to the FEAmethod. Since the FEA requires considerable much calculationtime, the analytical stiffness model is a cost-effective approachwhich is further applied to generate the variation tendency ofstiffness factors as the varying of architectural parameters, thatis useful in designing the CPM.

The main contribution of this paper is the derivation ofthe analytical stiffness model of a 3-PUU CPM via a simplerapproach, and the generation of the influence of each kinematicparameter on stiffness factors. Moreover, the methodologypresented in this paper can be extended for stiffness modelingof other types of CPMs.

128

1020

3040

5020

4060

80100

0

5000

10000

15000

b (mm)a (mm)

Kfz

−uz

(N

/mm

)

(a)

0

50

10010

20

30

400

5000

10000

15000

a (mm)l1 (mm)

Kfz

−uz

(N

/mm

)

(b)

23

45

0.5

1

1.50

2000

4000

6000

8000

10000

r (mm)t (mm)

Kfz

−uz

(N

/mm

)

(c)

010

2030

4050

0

2

40

2

4

6

8

10x 10

4

w (mm)h (mm)

Kfz

−uz

(N

/mm

)

(d)

010

2030

10

20

30

40

0

2000

4000

6000

d (mm)l (mm)

Kfz

−uz

(N

/mm

)

(e)

Fig. 7. Stiffness factor Kfz−uzversus architectural parameters of a 3-PUU CPM.

ACKNOWLEDGMENT

The authors appreciate the fund support from the researchcommittee of University of Macau under grant no.: RG083/04-05S/LYM/FST and Macao Science and Technology Develop-ment Fund under grant no.: 069/2005/A.

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