Kinematic Precision Scaling With Physical Errors in

R. J. Chang Professor National Cheng Kung University, Tainan 70101, Taiwan, Republic of China

Y. L. Wang Assistant Professor National Kaohsiung Marine University, Kaohsiung 811, Taiwan, Republic of China

Kinematic Precision Scaling With Physical Errors in Miniaturization of Four-Bar Polymer Compliant Machines A precision-scaling kinematic model with the effects of physical error is investigated in the miniaturization of four-bar polymer machines with compliant joints. A pseudolinkages model (PLM) for the multiple-links compliant machine is formulated. A scaling formulation of the multiple-links compliant machine and its associated PLM is developed. A method for scaled-up test and scaled-down analysis of the compliant mechanism with the considerations of physical errors of fabrication processes, material properties, and experimental tests is proposed. By defining an index of signal-to-noise ratio, the performance of the miniature realization under physical errors is evaluated. The applications of the scaling PLM for the miniature realization of a compliant machine are illustrated by performing both numerical analysis and experimental testing on four-bar compliant polyethylene machines. 关DOI: 10.1115/1.2738512兴 Keywords: miniaturization, S/N ratio, scaling analysis, PLM model, four-bar compliant machine

1

Introduction

The research of compliant machine 关1–4兴 and microelectromechanical system 共MEMS兲 关5–8兴 has stimulated recent interest on realizing miniature compliant polymer machines 关9,10兴. Miniature compliant machines ranging from meso- to microscale can be realized to provide their small and accurate output motions through the deflection of flexible members 关9兴. The development of microcompliant machines is very important in the miniaturization of compliant machines for engineering applications 关5兴. Micromachines, in general, can be classified into two categories 关5,7兴. The first category is a machine with micron-scale size and fabricated by MEMS technologies. The second category is a machine for microoperations and fabricated by precision machining technologies. The classification of micromachines according to the geometrical size or operational range can be extended to a mesoscale in the miniature machine 关10兴. As the geometrical size is reduced from meso- to microscale, a compliant machine with a one-piece mechanism is particularly suited for realization by employing extrusive or moldable material. Actually, the advantages of carbonbased polymer material with a variety of material properties, high manufacturability, and cost effectiveness have been realized in miniature systems 关5,6,9,10兴. Planar four-bar polymer machines with compliant joints have been realized in cost-effective optical read/write heads for fine servooperation. The complex behavior of polymer causes the difficulties of design and realization in engineering applications. However, the engineering realization with polymer, as compared to other materials, in fabricating compliant joints provides a high operational reliability with minimum constraints in stress concentration and fatigue life. With more and more polymer materials employed for microdevices, various lithography and nonlithography methods have been developed for the miniature realization of the compliant machines 关5兴. Miniaturization of machinery has become an ongoing effort by engineers. According to Webster’s Third New International DicContributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received December 7, 2006; final manuscript received December 8, 2006. Review conducted by Kornel F. Ehmann.

tionary, 1986, the word “miniature” means a “copy” on a muchreduced scale. The physical realization of size reduction of a machine to produce a reduced copy, which retains the original functions, is a challenging work for modern mechanical engineers 关5兴. Various issues in the realization of miniature machine have been pointed out by researchers. In Hayashi’s opinion on micromechanisms, the miniaturization of dimensional factor would cause scale effects of acting forces, strength of materials, surface properties, manufacturing accuracy, and traveling speed 关7兴. Hattori suggested that an efficient improvement of power transmission of a micromachine was of great importance during size miniaturization 关6兴. Nicoud pointed out that precision and energy problems were to be considered in microsystems 关11兴. As for micromechatronic system and/or MEMS, the miniature issues in design and fabrication of actuator, controller, and mechanism for micro applications were identified 关8兴. Miniaturization issue of employing different materials in the fabrication was proposed since material-dependent rules that govern the motion behavior can be varied and affected by different scales of geometrical size 关5,7兴. In addition to the issues in system miniaturization, it is also noted that the fabrication and test of a micromachine usually are highly expensive in the development process. A cost-effective method for validating the miniature realization is required before a micromachine is physically implemented and tested. In an engineering approach, a method of similitude for prototype development and a modeling test is usually employed. The method of similitude, in general, includes the modeling schemes based on geometry, kinematic, and dynamic similitude between a prototype and system. In contrast to developing large-scale machines by employing a scaled-down prototype, a scaled-up prototype is required and utilized for validating the final system design in miniaturization. The role and use of scaling rules for the effective development of miniature machines are depicted in Fig. 1. As illustrated by Fig. 1, fabrication and testing of a scaled-up prototype are essential to ensure the satisfaction of design specifications in the later scaled-down realization. For exploiting the deformation of compliant joints in the miniature realization of four-bar polymer machines, the analysis of a

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OCTOBER 2007, Vol. 129 / 951

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Fig. 1 Role and use of scaling rules in the development of miniature machines

kinematic down-scaled rule and the development of a physical scaled-up test for the compliant machine are required. The ideal motion behavior of a four-bar compliant machine governed by a stress-strain field usually can be analyzed through utilizing the techniques of finite element analysis 共FEA兲. However, a lumpedmodel is usually needed for efficient analysis and synthesis, especially if the uncertainty of material properties and manufacturing error exist. For the precision design and realization of miniature four-bar compliant machines, a lumped model of pseudolinkages model 共PLM兲 that is different from the pseudo-rigid-body model 共PRBM兲 was developed 关9兴. For a compliant four-bar mechanism, a lumped model by PRBM has been developed and effectively used for kinematic analysis and synthesis 关3兴. The development of PRBM is essentially based on the kinematic equivalence in linkage mechanism, whereas for the method of PLM, it is based on the invariant strain energy in modeling the compliant mechanism. Actually, a compliant mechanism is a compliant structure instead of a linkage mechanism, real linkages do not exist in the mechanism. Therefore, in contrast to the approach of employing rigidbody linkage motion a priori in the PRBM formulation, the PLM of an equivalent mechanism is constructed with the linkage motion estimated a posteriori. In addition to the different modeling schemes in the PLM and PRBM, the PLM was developed for accurate small linear operations while the PRBM was employed for large deflection applications. As a result, for the requirement of an accurate and efficient model in miniature realization, the PLM will be selected and employed for developing the present scaling analysis. In this paper, a precision-scaling kinematic rule with the effects of physical error is formulated for the realization of planar fourbar polymer machines with compliant joints. A PLM is extended and developed first for the multiple-links compliant machine through the field information obtained by employing FEA 关9兴. Then, an ideal kinematic scaled-down analysis is undertaken on both stress-strain field and lumped PLM of the compliant machine. The effects of physical errors of fabrication processes, material properties, and experimental tests in scaled-down realization are explored and modeled. Two scaled-up prototypes of miniature four-bar compliant polyethylene 共PE兲 machines, along with one actual-scale miniature, are fabricated, tested, and compared to illustrate the present analysis in miniature realization. Finally, a signal-to-noise 共S/N兲 ratio is proposed and simulated for evaluating the performance of further miniaturization of a four-bar compliant polymer machine.

2

Multiple-Links Polymer PLM

The stress-strain field of multiple-links polymer machines with compliant joints in a planar motion and under the environment of uniform and constant temperature can be analyzed by performing FEA with a viscoelastic constitutive law 关12–15兴. A constitutive law of stress and strain for solid polymer is rather complicated. 952 / Vol. 129, OCTOBER 2007

However, it is realized that an average linear model with bias compensation after preconditioning loops can be employed as an equivalent linear elastic model for constructing the input-output PLM of compliant polymer micromachines 关9兴. For a four-bar polymer micromachine, an input-output PLM was developed by employing the field information of displacement and strain energy. Under the kinematic constraints employed in a mechanism, the angular displacement, torsion stiffness, and equivalent moment and force can be estimated. By utilizing an equivalent linear elastic model in FEA, the present formulation carries out an extension of modeling and analysis from the four-bar to multiple-links polymer machines. The material properties of polymer employed in the present miniature scale are assumed to be isotropic and homogeneous. For the derivation of a multiple-links PLM, a finiteelement formulation in plane-stress analysis is described briefly as follows 关9,12,13兴. In the FEA formulation, the relation between a load and displacement vector of each element is expressed as 关K兴共e兲关a兴共e兲 = 关F兴共e兲 共e兲

共1兲

共e兲

where 关K兴 , 关a兴 , and 关F兴共e兲 represent the stiffness matrix, the displacement vector, and the load vector, respectively. The stiffness matrix can be derived as an integral of geometrical shapes 关B兴共e兲, thickness t, and material properties 关C M 兴共e兲 to give 关K兴共e兲 =

冕冕

T

关B兴共e兲 关C M 兴共e兲关B兴共e兲tdxdy

共2兲

共e兲

The integrand in Eq. 共2兲 can be expressed further as functions of ␾i, E, and ␯, which represent shape functions, Young’s modulus, and Poisson’s ratio, respectively. The load vector of each element is a sum of two load vectors as 关F兴共e兲 = 关F兴共␶e兲 + 关F兴共fe兲 with

冖 冕冕

关F兴共␶e兲 =

关F兴共fe兲 =

共3a兲

T

共3b兲

T

共3c兲

关⌽兴共e兲 关␶兴共e兲tds

共e兲

关⌽兴共e兲 关f兴共e兲tdxdy

共e兲

Here, the 关␶兴共e兲 and 关f兴共e兲 in Eqs. 共3b兲 and 共3c兲 are boundary stresses and body force, respectively. The stress of each element is 关␴兴共e兲 = 关C M 兴共e兲关B兴共e兲关a兴共e兲

共4兲

An assembly of displacement, stiffness, and load vector of all elements gives the system matrix of a displacement as ¯ 兴 = 关K ¯ 兴−1关F ¯兴 关A

共5兲

¯ 兴 and 关F ¯ 兴 are system stiffness and load vector, respecwhere 关K tively. The strain energy of the system is written as ne

⌳=

1

兺2 e=1

冕冕

T

T

关a兴共e兲 关B兴共e兲 关C M 兴共e兲关B兴共e兲关a兴共e兲tdxdy

共6兲

共e兲

with ne representing the number of elements. By following the method in modeling a four-bar polymer micromachine, an input-output model of the multiple-links compliant machines can be derived through the equivalent PLM, as shown in Fig. 2. In Fig. 2, the PLM consists of equivalent torsion springs, linkages, and external loads. The geometrical relations and constraints of PLM will be derived through the model as shown in Fig. 3. By referring to Fig. 3, the notations are defined as ri as the length of the ith link, ␺i as a fraction of length of ri measured from 共xi , y i兲 to the center of the ith joint, ⌬␪i as angular displacement of the ith joint. For the linkages machine shown in Fig. 3, the closed-chain kinematics can be derived through disconTransactions of the ASME

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Fig. 2 Equivalent PLM model of planar multiple-links compliant machines Fig. 3 Notations of geometrical configuration and angular displacement of an equivalent multiple-links PLM

necting the final joint virtually to form an open-chain mechanism and employing the constraints of kinematic relations and a topological angular relation. For the miniature compliant machine, it is treated as an assembly of n p links with the n p joint virtually disconnected. The kinematic relations with ui and vi as the output displacements of xi and y i component, respectively, in Fig. 3, and under the assumption of small angular deformation can be derived to give 关A兴d = 关X兴关⌬␪兴

关⌬␪兴 = 关⌬␪1 ⌬␪2 ¯ ⌬␪np−1兴T 关A兴d = 关u1 v1 u2 v2 ¯ unp−1 vnp−1兴T with

共7兲

关I兴d =

where 关X兴 = 关R兴d关I兴d

关R兴d =

冤

0 0 0 0 ¯ 1 1

冥

0

0

0

¯

0

0

0

r1␺1 cos ␪1

0

0

¯

0

0

− r1 sin ␪1 − r2␺2 sin ␪2

0

− r2␺2 sin ␪2

0

¯

0

0

0

r1 cos ␪1 + r2␺2 cos ␪2

0

r2␺2 cos ␪2

¯

0

0

]

]

]

]




0

0

0

¯ − rnp−1␺np−1 sin ␪np−1

冢兺

ri sin ␪i + rnp−1␺np−1 sin ␪np−1

i=1

0

冣

冢兺 n p−2

0

冢兺 n p−2

ri cos ␪i + rnp−1␺np−1 cos ␪np−1

i=1

−

冣

ri sin ␪i + rnp−1␺np−1 sin ␪np−1

i=2

¯ 兴 − 关X兴关⌬␪兴 关e兴 = 关A

ei2 = 关e兴T关e兴

共9兲

=0

共10兲

i=1

冏 冏 ⳵J ⳵关⌬␪兴

关⌬␪兴=关⌬␪ˆ 兴

then ⌬␪i, i = 1 , 2 , 3 , . . . , n p − 1 are optimally estimated through a pseudoinverse to give ¯兴 关⌬␪ˆ 兴 = 共关X兴T关X兴兲−1关X兴T关A

冣

¯

0

rnp−1␺np−1 cos ␪np−1

冥

PLM, further constraint equations need to be included. For the multiple-links compliant machine, the equivalent PLM gives a closed multiple-angular shape. Therefore, an invariant topological angular relation gives 关4兴 np

兺 ⌬␪ = 0

共12兲

i

i=1

In addition to the constraint of angular displacements of a virtually open chain of the PLM given by Eq. 共12兲, the actual relations among angular displacements ⌬␪i are also constrained in the equivalent closed-chain PLM. For the multiple-links compliant machine, as shown in Fig. 2, the relations of angular displacements were derived for the applications in micro operations 关4,9兴. Since the n pth joint in the PLM is grounded, the sum of all displacements due to each angular displacement ⌬␪i has to be zero. The constraint equations of planar displacements are written as

冉兺 冊 冉兺 冊 n p−1

n p−1

ri cos ␪i ⌬␪1 +

i=1

共11兲

The angular displacement and kinematic relations of a virtually open chain for the PLM are derived as Eqs. 共7兲 and 共11兲. For a closure compliant mechanism that is modeled as a closed-chain Journal of Manufacturing Science and Engineering

ri cos ␪i + rnp−1␺np−1 cos ␪np−1

0

共8兲

2共n p−1兲

兺

冢兺 n p−2

i=2

T ¯ 兴 = 关u v ¯ v where 关A F1 F1 Fn −1兴 is the displacement of the positions, p on the longitudinal centerline of linkages of the miniature compliant machine, which are located on those of the corresponding PLM in Eq. 共7兲. By employing the criterion of mean-square error as

J=

冣

0

The displacement error between the numerical results of FEA, ¯ 兴, and the corresponding PLM of the miniature compliant ma关A chine is

and utilizing

0 0 1 1 ¯ 0 0 ] ] ] ]
] ]

T

− r1␺1 sin ␪1

n p−2

−

冤

1 1 0 0 ¯ 0 0

冉兺 冊

ri cos ␪i ⌬␪2

i=2

n p−1

+

i=3

ri cos ␪i ⌬␪3 ¯ + rnp−1 cos ␪np−1⌬␪np−1 = 0 共13a兲 OCTOBER 2007, Vol. 129 / 953

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冉兺 冊 冉兺 冊 冉兺 冊 n p−1

n p−1

ri sin ␪i ⌬␪1 +

i=1

ri sin ␪i ⌬␪3 ¯

i=2

i=3

+ rnp−1 sin ␪np−1⌬␪np−1 = 0

冤

共13b兲

1

1

1

n p−1

兺

ri cos ␪i rnp−1 cos ␪np−1 0

i=n p−2 n p−1

兺

In order to obtain the constraint equations in a form ⌬␪i / ⌬␪1 for the present analysis, Eqs. 共12兲, 共13a兲, and 共13b兲 are rewritten in a matrix form as

n p−1

ri sin ␪i ⌬␪2 +

ri sin ␪i rnp−1 sin ␪np−1 0

i=n p−2

冥冤 冥 冤 ⌬␪np−2

⌬␪np−1 = ⌬␪np

−1

−

冉兺 冊 冉兺 冊 冉 兺 冉兺 冊 冉兺 冊 冉 兺 ri cos ␪i

冤 冥冤 ⌬␪3

i=3

−1

]

冤 冥冤

⌬␪np−1 = cnp−1,2 cnp−1,3 ¯ cnp−1,np−3 ⌬␪np

cnp,2

cnp,3

¯

cnp,np−3

冤 冥

冥冤

⌬␪3 ]

⌬␪np−3

冉兺 冊 冉兺 冊 n p−1

−

+

ri cos ␪i

i=1

n p−1

−

ri sin ␪i

i=1

冤冥

⳵⌳ ⳵⌬␪2 k2 ⳵⌳ 0 ⳵⌬␪3 = ] ] 0 ⳵⌳ ⳵⌬␪np−3

By assuming that the inversion of a matrix in the left-hand side of Eq. 共14a兲 is nonsingular, the relation of angular displacements is further expressed to give

cnp−2,2 cnp−2,3 ¯ cnp−2,np−3

冥

cnp−2,1

冤

+

+ cnp−1,1 ⌬␪1

n p−1

ri sin ␪i

−

i=2

⌬␪np−3

⌬␪np−2

i=n p−3

n p−1

ri sin ␪i

共14b兲

cnp,1

冤

i=n p−3

共14a兲

k3 ¯

0

]

]




共15兲

According to the method of virtual work, by taking the partial derivatives of Eq. 共15兲 with respect to ⌬␪i for i = 2 to n p − 3 and equating them to zero, respectively, the Eq. 共16兲 can be derived to yield 954 / Vol. 129, OCTOBER 2007

冥冤 冥 ⌬␪2

0

⌬␪3 ]

0 ¯ knp−3

⌬␪np−3

knp−2cnp−2,2

knp−1cnp−1,2

knpcnp,2

knp−2cnp−2,3

knp−1cnp−1,3

knpcnp,3

]

]

]

knp−2cnp−2,np−3 knp−1cnp−1,np−3 knpcnp,np−3

冤 冥冤冥

冥

0 0

⫻ ⌬␪np−1 =

共16兲

]

⌬␪np

0

By substituting Eq. 共14b兲 into Eq. 共16兲, a linear proportional relation can be obtained as ⌬ ␪ i = ␤ i⌬ ␪ 1,

␤1 = 1

共17兲

If an equivalent moment is applied at the first link, the magnitude of an equivalent moment can be derived by employing Eqs. 共15兲 and 共17兲 to obtain

np

兺

冥

⌬␪1

0 ¯

M= 1 ki⌬␪i2 2 i=1

ri sin ␪i

¯ −

⌬␪np−2

It is noted that if n p = 4 is in the constraint equation 共14b兲, the angular relations become a pure kinematic relation and can be expressed in proportional relations as ⌬␪i / ⌬␪1 关9兴. If n p ⬎ 4, the machine has redundant degrees of freedom and more information is required to deal with this statically indeterminate problem. Since the compliant machine is a kind of compliant structure, the method of potential energy can be utilized to find n p − 4 more equations. For the PLM of a multiple-links compliant machine, total strain energy is obtained as

⌳=

冥

冊 冊

ri cos ␪i

¯ −

i=3

⌬␪2

⌬␪2

n p−1

ri cos ␪i

−

n p−1

⫻

−1

n p−1

i=2

−

¯

−1

n p−1

⳵⌳ = ⳵⌬␪1

冉兺 冊 np

ki␤i2 ⌬␪1

共18a兲

i=1

or M = K⌬␪1

共18b兲

The equivalent force in Fig. 2 is obtained as

ជ= F

np

兺 ជf

i

共19兲

i=1

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With the displacement and strain energy obtained by the FEA, an equivalent PLM with the angular deformation can be derived and estimated. The stiffness in Eq. 共18b兲 gives an equivalent system stiffness of a compliant machine. The PLM consists of pseudorigid linkages and lumped torsion springs in each joint. However, for a four-bar compliant mechanism, it is usually not necessary to derive the equivalent stiffness of each joint in the further scaled-down analysis. If the multiple-links compliant mechanism is with more than four links, the equivalent stiffness of each joint needs to be identified. By setting an equivalent force system between a local free-body diagram and the corresponding PLM in the neighborhood of each compliant joint, the equivalent stiffness ki, i = 1 to n p can be numerically obtained through the results of FEA and ⌬␪ˆ i.

n

yp =

兺y

xp =

兺x

冕冕 1

−1

1 T

关B兴共pe兲 关C M 兴共pe兲关B兴共pe兲t p兩J p兩共e兲d␰d␩

−1

e兲 关F兴共pe兲 = 关F兴共␶e,p兲 + 关F兴共f,p =

冕冕 1

+

共20兲

−1

冕

where y pi and x pi are planar coordinates of a prototype. If the planar coordinates between a scaled-down mechanism and a prototype are scaled as y si = nsy pi and xsi = nsx pi, the planar position of a scaled-down mechanism can be obtained as

si

xs =

兩Js兩共e兲 =

1

=

1

共21兲

−1

1 T

关B兴s共e兲 关C M 兴s共e兲关B兴s共e兲ts兩Js兩共e兲d␰d␩

−1

e兲 = 关F兴s共e兲 = 关F兴␶共e,s兲 + 关F兴共f,s

+

共22兲

冕冕

冕

1

−1

−1

n

兺x ␾ =兺nx si

= n sx p

共27兲

冉

冊

冑冋 冑冋

冑冋

⳵xs ⳵xs d␰ + d␩ ⳵␰ ⳵␩

册 冋 册 冋

⳵ n sx p ⳵ n sx p d␰ + d␩ ⳵␰ ⳵␩

= ns

⳵x p ⳵x p d␰ + d␩ ⳵␰ ⳵␩

2

+

2

+

册 冋 2

+

⳵ys ⳵ys d␰ + d␩ ⳵␰ ⳵␩

⳵nsy p ⳵nsy p d␰ + d␩ ⳵␰ ⳵␩

⳵y p ⳵y p d␰ + d␩ ⳵␰ ⳵␩

册

共23兲

册

册

2

2

2

共29a兲

= nsds p

On the boundary ␩ = 0 and with d␩ = 0 in Eq. 共29a兲, the boundary 共e兲 line segment can be simplified to give dss = J⌫,sd␰ and ds p 共e兲 = J⌫,pd␰. Hence, the Jacobian on the boundary is scaled to give 共e兲 共e兲 = nsJ⌫,p J⌫,s

By comparing Eq. 共20兲 with 共22兲, and Eq. 共21兲 with 共23兲, the scaled-down FEA will be formulated by the following approach. First, it is assumed that both prototype and scaled-down mechanism are fabricated with the same material and operated under the 共e兲 共e兲 共e兲 共e兲 same body force to give 关C M 兴 p = 关C M 兴s = 关C M 兴 and 关f兴 p = 关f兴s = 关f兴. The assumption of the same material is not necessary; different materials may be used for fabricating a cost-effective prototype. The scaling factors of thickness and boundary stress are 共e兲 共e兲 defined as ts = ntt p and 关␶兴s = n␶关␶兴 p , respectively. Now, the di共e兲 共e兲 共e兲 mensional effects on 关B兴 , 兩J兩 , and J⌫ will be analyzed. By selecting the mapping functions as the shape functions employed in the FEA for an isoparametric transformation, the planar position of a prototype can be expressed as Journal of Manufacturing Science and Engineering

s pi␾i

i

⳵xs ⳵ y s ⳵xs ⳵ y s ⳵x p ⳵ y p ⳵x p ⳵ y p − = ns2 − = ns2兩J p兩共e兲 ⳵␰ ⳵␩ ⳵␩ ⳵␰ ⳵␰ ⳵␩ ⳵␩ ⳵␰

共e兲 关⌽兴共e兲关␶兴s共e兲J⌫,s t sd ␰

兩⌽兩共e兲T关f兴s共e兲ts兩Js兩共e兲d␰d␩

共26兲

i=1

1

−1

1

n

= nsy p

By using Eqs. 共26兲 and 共27兲, a line segment is derived as

−1

冕冕

i=1

共28兲

For the scaled-down mechanism, the stiffness matrix and load vectors of Eqs. 共2兲 and 共3兲, respectively, are written as 关K兴s共e兲 =

i=1

By utilizing Eqs. 共26兲 and 共27兲, the Jacobian is scaled and written as

共e兲 关⌽兴共e兲关␶兴共pe兲J⌫,p t pd ␰

T

s pi␾i

i

i=1

−1

兩⌽兩共e兲 关f兴共pe兲t p兩J p兩共e兲d␰d␩

n

兺y ␾ =兺n y

and

dss = 冑dxs2 + dy s2 =

1

共25兲

i=1

Precision-Scaling Model in Miniaturization

关K兴共pe兲 =

pi␾i

n

n

3.1 Kinematic Scaled-Down FEA. The ideal kinematic scaling rule of the stress-strain field between a prototype and a scaleddown mechanism of a compliant machine will be derived. The finite-element formulation is under the constraints of the same linear-elastic law, numbers of elements, numbers of nodes, shape functions, and types of boundary conditions. In the formulation of a kinematic scaling FEA, the equations for the prototype and scaled-down mechanism are denoted by subscripts p and s, respectively. Mapping the real elements onto the parent elements in ␰, ␩-space through coordinate transformations first precedes the formulation of the scaling FEA. The stiffness matrix of Eq. 共2兲 and load vectors of Eq. 共3兲 of the prototype are expressed, respectively, as

共24兲

and

ys =

3

pi␾i

i=1

共29b兲

By using Eqs. 共26兲 and 共27兲 in 关B兴共e兲, the following equations can be derived: 关B兴s共e兲 =

1 关B兴共e兲 ns p

共30兲

where

关B兴s共e兲 =

with

冤

⳵␾1共e兲 ⳵xs

0

0

⳵␾1共e兲 ⳵ys

⳵␾2共e兲 ¯ ⳵xs 0

¯

0

⳵␾n共e兲 ⳵ys

⳵␾1共e兲 ⳵␾1共e兲 ⳵␾2共e兲 ⳵␾n共e兲 ¯ ⳵ys ⳵xs ⳵ys ⳵xs

冥

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⳵␾i共e兲 1 ⳵y s ⳵␾i共e兲 1 ⳵y s ⳵␾i共e兲 = − 共e兲 ⳵xs 兩Js兩 ⳵␩ ⳵␰ 兩Js兩共e兲 ⳵␰ ⳵␩ =

1 ns2兩J p兩共e兲

冉

lation of equivalent moment between a prototype and a scaleddown mechanism is given by

1 ns⳵y p ⳵␾i共e兲 ns⳵y p ⳵␾i共e兲 − 2 共e兲 ⳵␩ ⳵␰ ⳵␰ ⳵␩ ns 兩J p兩

1 1 ⳵y p ⳵␾i共e兲 1 ⳵y p ⳵␾i共e兲 = − ns 兩J p兩共e兲 ⳵␩ ⳵␰ 兩J p兩共e兲 ⳵␰ ⳵␩

Ms =

冊

As a result, the formulation of kinematic-precision scaled-down ¯ 兴 / 关A ¯兴 FEA and PLM derived as above gives scale factors of 关A s p 共e兲 共e兲 = n n , ⌳ / ⌳ = n2n n2, 关␴兴 / 关␴兴 = n , ⌬␪ˆ / ⌬␪ˆ = n , M / M s ␶

⳵␾i共e兲 1 ⳵xs ⳵␾i共e兲 1 ⳵xs ⳵␾i共e兲 1 ⳵␾i共e兲 =− + = ⳵ys 兩Js兩共e兲 ⳵␩ ⳵␰ 兩Js兩共e兲 ⳵␰ ⳵␩ ns ⳵ y p

关K兴s共e兲 = nt关K兴共pe兲

共31兲

The scaled-down effect in load vectors by Eqs. 共21兲 and 共23兲 is e兲 关F兴s共e兲 = nsn␶nt关F兴共␶e,p兲 + ns2nt关F兴共f,p

共32兲

By assembling the local matrices to obtain a global matrix and utilizing Eq. 共5兲, the displacements in the scaled-down mechanism can be written as 2 ¯ ¯ 兴 = 关K ¯ 兴−1关F ¯ 兴 = 关K ¯ 兴−1共n n 关F ¯ 关A s s s ␶ 兴␶,p + ns 关F兴 f,p兲 s p

共33a兲

If the compliant machine is operated horizontally, which is perpendicular to the vertical body force, Eq. 共33a兲 can be simplified to give ¯ 兴 = 关K ¯ 兴−1n n 关F ¯ ¯ 关A s p s ␶ 兴␶,p = nsn␶关A兴 p

共33b兲

The relation of strain energy between a prototype and scaleddown mechanism can be derived as 1

e=1

冕冕

共e兲

T

T

关a兴s共e兲 关B兴s共e兲 关C M 兴共e兲关B兴s共e兲关a兴s共e兲tsdxdy = ns2ntn2␶ ⌳ p 共34兲

The relation of stress between a prototype and scaled-down mechanism is 1 关␴兴s共e兲 = 关C M 兴共e兲 关B兴共pe兲nsn␶关a兴共pe兲 = n␶关␴兴共pe兲 ns

共35兲

关X兴s = ns关X兴 p

共36兲

From Eq. 共11兲 and with Eqs. 共33b兲 and 共36兲, the relation of angular displacement between a prototype and a scaled-down mechanism is obtained as ¯ 兴 = 1 共关X兴T关X兴 兲−1关X兴Tn n 关A ¯ 关⌬␪ˆ 兴s = 共关X兴sT关X兴s兲−1关X兴sT关A s p p p s ␶ 兴p ns

s t ␶

s

␶

p

4

si

pi

␶

s

p

Physical Scaling Errors in Miniaturization

The derivation of a kinematic scaling analysis on the stressstrain field and PLM in the previous sections is under the assumption of an ideal kinematic scaling rule. Actually, even if the dominant governing rules are not changed, the motion behavior of a scaled-down mechanism can be affected by the physical scaling errors due to fabrication processes, material properties, and experimental tests. For the effective and reliable use of the scaling analysis in the miniaturization of the polymer machine, the physical scaling errors need to be analyzed, modeled, and estimated. The uncertainties due to fabrication processes, such as geometrical error, and material properties, such as hysteresis, in compliant polymer machines will cause errors of strain energy, torsion stiffness, and angular displacement in the PLM. Although the error sources are different, the overall effect of these errors will be reflected in the output response. The operational error in the output displacement, in general, can be decomposed into distribution and biased errors 关16兴. For the distribution error, it can be modeled through the equivalent strain energy between the FEA and PLM as np

⌳ + e⌳ =

1

兺 2 共k + e 兲共⌬␪ + e ␪ 兲 i

i=1

ki

共37兲

⌬ i

i

2

共40兲

where e⌳, eki, and e⌬␪i denote the errors of strain energy, joint stiffness, and angular displacement, respectively. By neglecting the second-order terms and employing Eq. 共15兲 in Eq. 共40兲, e⌳ can be expressed as e⌳ =

兺 冉 k ⌬␪ e ␪ + 2 ⌬␪ e 冊 1

i

i=1

i ⌬ i

2 i ki

共41兲

It is observed from Eq. 共41兲 that the distribution error in a stressstrain field will cause combined and coupled effects in the angular displacement and joint stiffness. Since both ki and ⌬␪i are estimated from FEA under the kinematic constraint of PLM, the e⌬␪i and eki are very difficult to be modeled and analyzed accurately. Hence, a different and practical approach to model the output distribution error by PLM is proposed. By assuming that the output error is due to an ideal PLM under the uncertainty of input loading, one has ⌳ + e⌳ = 共M + e M 兲⌬␪1

共42兲

where e M is an error distribution in the equivalent input moment. From using Eq. 共42兲 and ⌳ = M⌬␪1, an equivalent error moment is modeled as eM =

From Eq. 共18a兲 and employing Eqs. 共17兲, 共34兲, and 共37兲, the re956 / Vol. 129, OCTOBER 2007

p

np

3.2 Kinematic Scaled-Down PLM. The derivation of scaling rule on the equivalent load, torsion stiffness, and output displacement of the PLM between a prototype and a scaled-down mechanism will be undertaken. With the kinematic scaled-down FEA of strain energy, stress, and displacement, the formulation of the scaled-down PLM is derived for the miniature realization. The derivation of a kinematic scaled-down analysis on the PLM is given as follows. First, a geometrical relation of the PLM between a prototype and a scaled-down mechanism is expressed as

= n␶关⌬␪ˆ 兴 p

s

= ns2ntn␶, and ksi / k pi = ns2nt by Eqs. 共33b兲, 共34兲, 共35兲, and 共37兲–共39兲, respectively.

From the derivation as above, the relation of local stiffness matrices between Eqs. 共20兲 and 共22兲 can be obtained as

兺2

共39兲

ksi = ns2ntk pi

and

ne

共38兲

The relation of equivalent stiffness between a prototype and a scaled-down mechanism can be obtained by utilizing Eqs. 共37兲 and 共38兲 in 共18b兲 to give

1 ⳵␾i共e兲 = ns ⳵x p

⌳s =

⳵⌳s = ns2ntn␶M p ⳵⌬␪1,s

e⌳ ⌬␪1

共43兲

In Eq. 共43兲, it is noted that the distribution error of Eq. 共41兲 can be modeled as solely attributed to the input moment. As for the biTransactions of the ASME

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Fig. 4 An input-output model of equivalent PLM with error effects for a multiple-links polymer machine

ased error in the output displacement, which is denoted as 关e¯A兴, it can be obtained from the loaded-recovery test on a compliant polymer micromachine. The biased error is estimated as a residual offset from the average regression line of a loaded-recovery curve after the preconditioning loops 关9兴. By calibrating the bias error of the output displacement at the midpoint of the final link and employing the geometrical relation of PLM, the components of 关e¯A兴, as denoted by e¯Ai, can be obtained. The combined effects of fabrication error and material uncertainties on the PLM can be formulated through a nominal PLM by Eqs. 共7兲, 共17兲, and 共18b兲 and with the model of distribution and biased errors. The output displacement of the components xi and y i of the PLM, as shown in Fig. 3, with physical errors can be expressed by 关A兴 M = 关X兴关␤1 ␤2 ¯ ␤np−1兴T

M + eM + 关e¯A兴 K

共44a兲

The input-output formulation of 共44a兲 can be expressed as shown in Fig. 4. By using the component expression in 共44a兲, one has A M,i = 冑u2M,i + v2M,i + e¯Ai

共44b兲

for a specific position 共xi , y i兲. By combining the measuring and testing error, Ni, in Eq. 共44b兲, the equation of output displacement will be modified to yield A M,i = 冑u2M,i + v2M,i + e¯Ai + Ni

共45a兲

By taking the scaling effects into consideration, the scaling factors ns, n␶ are employed in Eq. 共45a兲 to give a formulation of kinematic scaled-down PLM with biased error as A M,i =

冑u2M,i + v2M,i n sn ␶

+

e¯Ai n sn ␶

+ Ni

共45b兲

For validating the final design specifications in miniature realization, the improvement of experimental accuracy, precision, and resolution is also required. The scaling factor, denoted as nm, on the experimental test is taken into consideration in Eq. 共45b兲. As a result, one obtains A M,i =

冑u2M,i + v2M,i n sn ␶

e¯Ai

Ni + + n sn ␶ n m

Monte Carlo method is then undertaken iteratively for obtaining the statistics of a uniform distribution. The range of distribution between a and b, as shown in Fig. 5, is finally obtained by checking if the experimental results from the measurement and test are within three times of the standard deviation of the scaled-down PLM of a fabricated miniature machine. As a result, with the proper statistics obtained from the Monte Carlo method and by employing the experimental data in the error analysis, the kinematic scaled-down PLM with an error bound can be estimated. In a kinematic scaled-down realization with the effects of physical error, a systematic measure is desired for the evaluation of the operational performance. An index of S/N ratio may be defined and used as ¯A2 S M,i = 10 log10 2 N S M,i

共45c兲

The physical scaling errors due to the fabrication processes, material properties, and experimental tests are analyzed and modeled in the given formulation. These errors need to be estimated further quantitatively. A procedure that extends the integration method for input-output modeling and error analysis of four-bar polymer compliant micromachines 关9兴 is developed. By integrating the scaling effects into the PLM modeling, fabrication error, and experimental test, a procedure to realize the miniaturization scheme in Fig. 1 is proposed, as depicted in Fig. 5. From Fig. 5, a kinematic scaled-down PLM with error bound on the output displacement in the scaled-down miniaturization can be derived and estimated through utilizing the experimental data and Monte Carlo method. For obtaining the error bound, some error distributions from the experimental data are required before the Monte Carlo method can be preceded. When the input load is not applied, the error is only due to measurement and testing system. The measurement and testing system is well calibrated to give a zero-mean Gaussian distribution. The offset of the output displacement of the polymer machine is measured and denoted as biased error. The Journal of Manufacturing Science and Engineering

Fig. 5 Method of scaled-down PLM in the miniature design, fabrication, and test of multiple-links polymer machines

共46兲

where ¯A M,i and S M,i are the mean value and the standard deviation, respectively, of a specific output displacement A M,i, which are obtained from the results as given by the procedure in Fig. 5.

5

Scaled-Down Analysis and Scaled-Up Test

The precision scaled-down mechanism with the physical error effects on the distributed stress-strain field and lumped PLM of the multiple-links polymer machines has been formulated. With the same material properties and dominant governing rules on both scaled-up prototype and actual-scale mechanism, various numerical analyses and experimental tests are undertaken. The applications of the formulated scaling rules in the miniaturization of four-bar compliant polymer machines are numerically simulated and experimentally tested in Secs. 5.1–5.4. 5.1 Four-Bar Formulation and Analysis. The four-bar compliant polymer machine under investigation belongs to mesoscale machines of the secondary category, as defined in the IntroducOCTOBER 2007, Vol. 129 / 957

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tion. For the analysis of kinematic scaled-down realizations of the miniature four-bar machines, the following equations can be obtained for investigations. From Eq. 共14兲, one has 关T兴␪关⌬␪兴⬘ = 关L兴␪⌬␪1

共47兲

where 关⌬␪兴⬘ = 关⌬␪2 ⌬␪3 ⌬␪4兴T

冤 冤

1

1

1

关T兴␪ = r2 cos ␪2 + r3 cos ␪3 r3 cos ␪3 0 r2 sin ␪2 + r3 sin ␪3 r3 sin ␪3 0 −1

冥 冥

Fig. 6 FEA for the ideal prototype of a miniature four-bar polymer machine

关L兴␪ = − 共r1 cos ␪1 + r2 cos ␪2 + r3 cos ␪3兲 − 共r1 sin ␪1 + r2 sin ␪2 + r3 sin ␪3兲

A linear relation of angular displacements is obtained from Eq. 共17兲 as ⌬ ␪ i = ␤ i⌬ ␪ 1,

␤1 = 1

共48兲

where 关␤2 ␤3 ␤4兴T = 关T兴␪−1关L兴␪ From Eqs. 共15兲 and 共48兲, the strain energy with equivalent torsion stiffness ki is 1 1 1 1 ⌳ = k1⌬␪21 + k2共␤2⌬␪1兲2 + k3共␤3⌬␪1兲2 + k4共␤4⌬␪1兲2 2 2 2 2 1 = K⌬␪21 2

共49兲

where K = k1 + k2␤22 + k3␤23 + k4␤24 The system stiffness in Eq. 共49兲 can be estimated directly from PLM or derived through the estimation of individual joint stiffness. From Eqs. 共18b兲 and 共49兲, an input-output behavior of the PLM is governed by ⌬␪1 =

M K

共50兲

where M is an equivalent input moment. With the formulation of a four-bar polymer machine, the ideal and physical scaling analyses on the four-bar miniature machine will be investigated numerically and experimentally. 5.2 Numerical Scaled-Down Analysis. For the verification of the present scaling analysis on the FEA and PLM models, numerical scaled-down analyses on a four-bar miniature machine are undertaken. The numerical analysis is illustrated for the geometrically scaled-down effects on both FEA and PLM of the miniature machine. The numerical results by the FEA are verified by employing a commercial package MARC. The numerical analysis on the four-bar polymer machine is undertaken for investigating the input-output behavior of a twice geometrically up-scaled prototype, as shown in Fig. 6, and an actual-scale mechanism. By applying the distributed load of 0.981 Mpa with uniform distribution over 0.4 mm on the prototype and over 0.2 mm on the corresponding actual-scale mechanism, numerical results by utilizing Eqs. 共5兲, 共6兲, 共11兲, 共12兲, and 共48兲–共50兲 for the scaled-up prototype and the actual-scale mechanism are obtained and listed in Table 1. The equivalent lumped parameters for the scaled-up prototype and the actual-scale mechanism are listed in Table 2. From the numerical results in Tables 1 and 2, it is realized that the scaling results by PLM are almost exactly the same as employing the scaleddown factors in FEA with n␶ = 1, nt = 1, ns = 1 / 2 to give dsi / d pi = 1 / 2, ⌬␪ˆ si / ⌬␪ˆ pi = 1, ksi / k pi = 1 / 4, M s / M p = 1 / 4, and ⌳s / ⌳p = 1 / 4. 958 / Vol. 129, OCTOBER 2007

5.3 Experimental Scaled-Up Test. The experimental test is undertaken on realizing the physical effect of fabrication error in prototype. For the experimental realization, both the scaled-up prototype and actual-scale mechanism are to be made of the same material and thickness from a PE plate. The experimental four-bar polymer machines are fabricated by using numerically controlled machine and hand tools. The available PE plate is 1 mm in thickness. The material properties of the PE plate are estimated under the loaded-deformation test to give E = 430.15 MPa and ␯ = 0.344. The input load is applied and distributed about the center of the first link of both the scaled-up prototypes and actual-scale mechanism. To facilitate the comparison of loaded-deflection tests on the scaled-up prototypes and actual-scale mechanism, the distributed load of the scaled-up prototype is selected as one-half of that of the actual-scale mechanism. In experiment, the output displacements at the star-mark positions, as shown in Fig. 6, of the scaled-up prototype and the corresponding actual-scale mechanism will be tested and compared. An experimental test is undertaken for investigating the inputoutput behavior of two scaled-up prototypes and one actual-scale mechanism. Two physical scaled-up prototypes of the miniature four-bar machine fabricated from the same PE plate but with different geometrical errors are shown in Fig. 7. One prototype M-A is fabricated with a geometrical shape that is almost exactly twice planar geometry of the actual-scale mechanism. Another prototype M-I is fabricated with twice nominal shape, as the prototype M-A, but with large scaling error in the fourth joint. For the investigation of the physical scaling effect by the fabrication error only, the effects of material property, applied load, and experimental error are controlled to be the same for both the scaled-up prototype and actual-scale mechanism. The experimental testing system, including three subsystems: loading platform, image system, and data processor with display, is shown in Fig. 8. The testing and measurement system is calibrated to give a zero-mean Gaussian noise with standard deviation of 0.04 mm. The applied forces on the prototypes and the actual-scale mechanism by the standard weight are ranging from 0 gw to 80 gw. The input-output behavior of the scaled-up prototype and the actual-scale mechanism is obtained from experimental tests. The offset of output displacement of the third link is tested and measured to give eA3 = 0.025 nm. Experimental results with numerical calculation by Eq. 共45b兲 are shown in Fig. 9. From Fig. 9, it is observed that the results obtained by the scaled-up prototype and the actual-scale mechanism are in good agreement with those predicted by the PLM if the physical scaling errors are well controlled. Next, an experimental test is undertaken for the prototype M-I. By employing the same testing conditions as those for the prototype M-A, the input-output behavior of the prototype and the PLM through utilizing Eq. 共45a兲 is obtained, as shown in Fig. 10. From Fig. 10, it is noted that the results of output displacement under 80 gw input load by the prototype with fabrication error are inconsistent with those predicted by the ideal PLM. By following the procedure given in Fig. 5, the error bounds of the second and Transactions of the ASME

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Table 1 System parameters and output displacements of scaled-up prototype and actual-scale mechanism d3 mm

⌬␪ˆ 1 rad

⌬␪ˆ 2 rad

⌬␪ˆ 3 rad

⌬␪ˆ 4 rad

k1 N-mm

k2 N-mm

k3 N-mm

k4 N-mm

0.584

0.257

0.026

−0.039

0.027

−0.027

70.39

15.27

61.97

25.67

0.273

0.128

0.026

−0.039

0.027

−0.027

17.60

3.82

15.49

6.43

0.48

0.5

1.0

1.0

1.0

1.0

0.25

0.25

0.25

0.25

d2 mm Scaled-up Prototype Actual-scale Mechanism Scaled-down Ratio

third link for the experimental tests are obtained through 1000 Monte Carlo runs. Here, the equivalent input load is a uniform distribution assigned with 40 gw mean load. Iterated computations are undertaken to ensure that the testing data are within the upper and lower error bounds. From the iterations, one obtains a uniform distribution ranging from 35 gw to 45 gw. The estimated error bounds of the displacement of prototype M-I are also shown in Fig. 10. From Fig. 10, it is realized that the output displacement of prototype M-I is deviated from that predicted by an ideal PLM. As shown in the Fig. 10, a PLM with error bound is required to provide a confident interval on the results predicted by an ideal PLM in physical realization.

Table 2 Equivalent input-output lumped parameters scaled-up prototype and actual-scale mechanism

Scaled-up prototype Actual-scale mechanism Scaled-down ratio

⌳ N-mm

K N-mm

M N-mm

0.066 0.017 0.26

201.324 50.331 0.25

5.100 1.275 0.25

of

5.4 Signal-to-Noise Ratio in Scaled-Down Realization. The formulation of kinematic scaling rule and physical error analysis on the input-output behavior of the miniature machine has been verified by performing both numerical analysis and experimental test. The miniature issue due to the physical errors in the scaleddown miniaturization of the four-bar polymer machine will be further investigated. With the four-bar polymer machine depicted in Fig. 6 as a precision prototype, the physical effect of measurement error, which is denoted as noise, on the scaled-down miniaturization will be investigated. The parameters, loaded conditions, and experimental results of the compliant polymer machine given in Fig. 6 are employed for simulations. When the scaled-down analysis on both PLM model and physical noise is according to Eq. 共45c兲 with nm = ns, the simulated results are denoted as a S/N invariant realization. A S/N decaying realization is also simulated by employing Eq. 共45c兲 with nm = 1 for the PLM model and physical noise under inappropriately scaled-down realization. The S/N ratio given in Eq. 共46兲 is computed for the simulated results. The simulated results of the scaled-down miniaturization with both S/N invariant and S/N decaying realizations are shown in Fig. 11. From Fig. 11, it is realized that the S/N ratio will decrease if the effect of physical noise is not scaled down proportionally in the miniaturization. As a result, the improper scaling errors in the development of a scaled-down compliant four-bar machine eventually should cause uncertain output motion, which is totally deviated from the design specifications predicted by an ideal downscaling prototype.

Fig. 7 Accurately and inaccurately fabricated scaled-up prototypes of four-bar PE machine for experimental tests

Fig. 8 A setup for the experimental test and measurement of miniature compliant machines

Journal of Manufacturing Science and Engineering

Fig. 9 Loaded-displacement analysis and test of accurately scaled-up prototype and actual-scale mechanism of four-bar PE machine

OCTOBER 2007, Vol. 129 / 959

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investigated. The input-output behavior of two scaled-up prototypes of miniature four-bar PE machine and a true-scale mechanism is numerically analyzed and experimentally tested. Numerical results reveal that, under proper modeling conditions, the derived scaling factors on a prototype can be employed to derive an ideal scaling PLM. Physical scaling analysis with the estimation of error bound is investigated through the Monte Carlo method associated with testing data. Experimental and simulated results reveal that the physical errors due to fabrication processes, material properties, and experimental tests should be considered concurrently in scaling PLM. With the precision-scaling analysis, a S/N invariant realization is required to ensure the satisfaction of design specifications in the further miniaturization of a four-bar compliant machine.

Acknowledgment The authors would like to thank the NSC of Taiwan for the support under Contract No. 共95兲-2221-E-006-157. The authors also appreciate valuable comments from the anonymous reviewers.

References Fig. 10 PLM with error bound for loaded-displacement relation of an inaccurately scaled-up prototype of four-bar PE machine

6

Conclusions

A precision-scaling kinematic model with physical error analysis is essential in the miniature design, fabrication, and test of four-bar polymer machines. A compliant polymer machine as a one-piece no-assembly system is particularly suited for the scaleddown realization. For miniature multiple-links polymer machines, the formulations of both input-output PLM and scaled-down analysis are developed. A precision-scaling analysis with the effects of physical error on the PLM of the polymer machine is

Fig. 11 S/N ratio of the third-link displacement of a four-bar polymer machine for the scaled-down miniaturization

960 / Vol. 129, OCTOBER 2007

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