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REVIEW OF SCIENTIFIC INSTRUMENTS 77, 115101 共2006兲

Design and nonlinear modeling of a large-displacement XYZ flexure parallel mechanism with decoupled kinematic structure Xueyan Tang,a兲 I-Ming Chen, and Qing Li School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore

共Received 23 June 2006; accepted 25 September 2006; published online 10 November 2006兲 This article proposes an XYZ flexure parallel mechanism 共FPM兲 with large displacement and decoupled kinematic structure. The motion range of the large-displacement FPM using notch hinges is more than 1 mm and is achieved by assembling the typical large-displacement prismatic joints and a new large-displacement prismatic joint proposed in this article. Moreover, the decoupled XYZ stage has small cross-axis error and small parasitic rotation. An exact nonlinear modeling method for the typical prismatic joint and the new type of prismatic joint is presented. Using the proposed nonlinear method, the stiffness and dynamics of the XYZ FPM are studied. Comparison between the pseudo-rigid-body and the proposed nonlinear methods is performed on the prismatic joints and the XYZ FPM. To verify the proposed nonlinear modeling method, experiments are conducted to study the linear stiffness of the prismatic joint. The stiffness and the natural frequency of the XYZ FPM are also obtained from the experimental data. The cross-axis error and the parasitic rotation of the XYZ FPM are measured. The experimental results show that the proposed XYZ FPM achieves large displacement and decoupled motion, and the proposed nonlinear method is valid for the flexure mechanism. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2364132兴

I. INTRODUCTION

In many applications such as chip assembly in semiconductor industry and cell manipulation in biotechnology, there is an urgent need for manipulators with high precision and rapid response. Conventional mechanisms with assembled joints and rigid links cannot meet this demand due to their coarse precision caused by friction, backlash, etc. Flexure mechanisms of monolithic and miniature workpiece can provide highly accurate microscale motion because they have less wear, no backlash, and no friction. Moreover, advanced electrodischarge machining 共EDM兲 makes flexure mechanisms feasible. In most flexure mechanisms, two fundamental flexible components have been used extensively, leaf springs and notch hinges. However, both of them have limitations. Leaf springs can reach large displacement but are prone to buckling under compressive axial load and stiffening in the presence of tensile loads. Thickness and material used are also limited due to manufacturing technology.1 Notch hinges are also used extensively because of its easy manufacturability and high off-axis stiffness. However, stress concentration near the thinnest portion of the notch hinge results in limitation on the motion range of flexure mechanisms. For example, Kim et al.2 worked out a single-degree-of-freedom 共DOF兲 flexure mechanism with the motion range of 200 ␮m. The XY ␪Z stage developed by Ryu et al.3,4 worked within an area of 40⫻ 40 ␮m2. The workspace of the 6-DOF compliant manipulator by Chao et al.5 is 120 ␮m ⫻ 130 ␮m a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

0034-6748/2006/77共11兲/115101/11/$23.00

⫻ 18 mrad. The XYZ FPM studied by Li and Xu6 has the workspace of 140⫻ 140⫻ 140 ␮m3. In our project, the proposed XYZ FPM using notch hinges has the advantage of large displacement more than 1 ⫻ 1 ⫻ 1 mm and easy manufacturability. Similar to rigid-body mechanisms, flexure mechanisms have several common structures such as 6-DOF, XY ␪Z, and XYZ. Yamakawa et al.,7 McInroy and Hamann,8 Liu et al.,9 etc., have extensively studied 6-DOF motion stages. Lee and Kim,10 Chang et al.,11 Ryu et al.,3 Yi et al.,12 etc., focused the study on XY ␪Z stages. Most recent research focuses on XYZ stages. Tsai et al.13 proposed a three-limb parallel structure, but the motion range is not large and the three translational motions are coupled. Arai et al.14 designed a decoupled serial stage with unsatisfactory stiffness. The parallel and decoupled XYZ stage is advantageous because of satisfactory stiffness and decoupled characteristics. The modeling of flexure mechanisms is a critical issue. For leaf-type flexure mechanisms, the deformed leaf beams under different loads and constraints have been modeled by Smith.1 For notch hinge-type flexure mechanisms, the first study was by Paros and Weisbord.15 Lobontiu16 derived the exact compliances for bending, compression, tension, and shear of notch hinges. Nevertheless, it is complicated to directly apply these exact formulations of notch hinges to stiffness and dynamics analyses of entire flexure mechanisms. For simplicity, the pseudo-rigid-body 共PRB兲 method was proposed by Howell17 to approximately model flexure mechanisms. However, the maximum difference between PRB prediction and the actual value reaches 9%.18 This article proposes a nonlinear method which can be used for

77, 115101-1

© 2006 American Institute of Physics

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Tang, Chen, and Li

FIG. 1. Conventional prismatic joint.

exact modeling of stiffness for notch hinges and entire flexure mechanisms, as well as dynamics of flexure mechanisms. In this article, the new type of prismatic joint and the conceptual design of the XYZ FPM are proposed in Sec. II. Section III introduces the PRB modeling of the XYZ FPM. Dimension optimization of the XYZ FPM based on PRB is investigated in Sec. IV. Nonlinear modeling of the XYZ FPM is studied in Sec. V. The experiments on stiffness tests and hammer test conducted to verify the FPM design and the nonlinear method are described in Sec. VI. II. CONCEPTUAL DESIGN OF XYZ FPM A. Large-displacement pripsmatic joints

For the XYZ FPM, 1-DOF prismatic joints are fundamental components. The motion range and decoupled characteristics of the XYZ FPM are determined by performance and arrangement of the prismatic joints. The conventional prismatic joint using notch hinges is shown in Fig. 1. The drawbacks of this type of joints are limited motion range and cross-axis error. When a force FX along the X axis is applied to the motion stage which is fixed along the Y axis, there will be tensile load along the axis of the notch hinge. The angular stiffness of the notch hinge will increase significantly. As a result, the motion range will be reduced. If there is no constraint on the motion stage, the cross-axis error along the Y axis will occur when FX is exerted. In order to solve these two problems in the conventional prismatic joint, a large-displacement prismatic joint 共Fig. 2兲 was designed. When FX is applied to the motion stage, the

FIG. 2. Large-displacement prismatic joint.

Rev. Sci. Instrum. 77, 115101 共2006兲

FIG. 3. New large-displacement prismatic joint.

secondary stage moves only half of the distance at the motion stage along the X axis and also compensates for the slight motion along the Y axis. Compensation at the secondary stage for the Y-axis motion nullifies the tensile axial load of the notch hinges. Thus, no stiffening occurs and the motion range of this joint is large. However, the drawback of such design is the asymmetric structure without uniform thermal expansion. To achieve uniform thermal expansion, a largedisplacement prismatic joint with an asymmetric structure is designed, as shown in Fig. 3. It is derived from the foldedbeam suspension19 共Fig. 4兲 applied in microelectromechanical system 共MEMS兲. In Fig. 4, the leaf springs are utilized and fabricated using the standard surface micromachining technology, multiuser MEMS processes 共MUMPS兲. However, in flexure mechanisms, fabrication of notch hinges is easier and more precise than that of leaf springs based on EDM. Hence, the notch-hinge-based prismatic joint 共Fig. 3兲 is needed. Its working principle is identical to that of the large-displacement prismatic joint 共Fig. 2兲. The difference is that the secondary stage in Fig. 2 is divided into two pieces. Since the structure is symmetry, two secondary stages in Fig.

FIG. 4. Folded-beam suspension applied in MEMS.

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Large-displacement and decoupled XYZ FPM

Rev. Sci. Instrum. 77, 115101 共2006兲

FIG. 5. Prismatic joint by Lobontiu and Garcia.

3 behave like that of one monolithic workpiece. When a force along the Y axis FY is exerted, the motion stage produces a linear displacement along the Y axis. Meanwhile, two secondary stages produce half of the displacement at the motion stage and move slightly along the X axis to nullify the axial load of the notch hinges. No tensile or compressive axial loads at the notch hinges means that no buckling or stiffening occurs, and in this case modeling of the deformation under the axial load can be omitted. The motion range of the prismatic joint will be analyzed in Sec. III A. The structure of this prismatic joint is similar to that proposed by Lobontiu and Garcia20 共Fig. 5兲. However, the working principles are different. For the prismatic joint by Lobontiu and Garcia, the forces are applied on the two secondary stages, and the linear output is generated on the motion stage. The notch hinges will bear axial compressive loads, which necessitates the modeling of axial deformation of notch hinges. B. XYZ FPM

One requirement of the XYZ FPM is to achieve a large workspace more than 1 ⫻ 1 ⫻ 1 mm3. In order to meet this demand, the large-displacement prismatic joints mentioned in Sec. II A are used to configure the XYZ FPM. The expected positioning precision of the closed-loop system is 0.1 ␮m, which will be achieved through the control algorithm design in the future work. To simplify the control algorithm, the three translational motions along the X, Y, and Z axes are decoupled. To obtain satisfactory stiffness, the structure of the XYZ FPM is designed to be a parallel combination of three orthogonal limbs. Each limb has three DOFs, translational along the X, Y, and Z axes. The serial structure of the limb combines three prismatic joints, which means that each limb has a prismaticprismatic-prismatic 共P-P-P兲 structure. As shown in Fig. 6, each Ui and Wi joint represents the prismatic joint in Fig. 2, and each Vi joint represents the prismatic joint in Fig. 3. The DOFs of the three limbs are

FIG. 6. Schematic diagram of XYZ FPM.

共Y,Z,X兲 艚 共X,Z,Y兲 艚 共Y,X,Z兲 = 共X,Y,Z兲. The working principle can be explained based on Fig. 6. When a force FX along the X axis is directly applied to W1 joint, W1 joint deforms to generate the X-axis motion and transfers the force to U1 and V1 joints. Theoretically, U1 joint and V1 joint move along the X axis without deformation and directly transfer the force to U2 and V3 joints through the end effector. At the same time U2 joint and V3 joint deflect to create the X-axis motion. Therefore, the end effector can move along the X axis. Briefly, when FX is exerted, W1, U2, and V3 joints deflect to generate the X-axis motion, U1 and V1 joints move the same displacement as the end effector without deformation, and other prismatic joints remain stationary. The motions along the Y and Z axes are identical. The prototype of the XYZ FPM has been manufactured, as shown in Fig. 7. Determination of the XYZ FPM’s dimensions will be introduced in Sec. IV. The material Al7075 aluminum is selected because of its good elasticity, low internal stress, and high strength. Each prismatic joint is a

共U1,V1,W1兲 = 共Y,Z,X兲, 共U2,V2,W2兲 = 共X,Z,Y兲, 共U3,V3,W3兲 = 共Y,X,Z兲. Based on the screw theory,21 the DOFs of the parallel combined structure are the intersection of those of the three limbs. Therefore, the DOFs of the XYZ FPM are

FIG. 7. Prototype of XYZ FPM.

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Tang, Chen, and Li

FIG. 10. Notch hinge.

LITE control card is integrated into this mechatronic system for the future high-precision position control. III. PRB MODELING OF XYZ FPM

The PRB model of the XYZ FPM is presented here for determination of the optimal dimensions of the mechanism. A. PRB modeling of prismatic joints FIG. 8. PRB model of W / U joint.

monolithic workpiece cut by wire EDM, and thinnest portion of the notch hinges has the width of 0.4 mm. The actuation is accomplished by three linear voice coil actuators MGV 52-25 共miniature guide voice coil module兲 connected directly to Wi joints, and the precision reaches 20 nm with Mercury 3000 encoder as actuation-loop feedback. The MC4000

In the PRB theory, the notch hinge can be simplified as a revolute joint when there is no axial compressive or tensile load. In our design, the notch hinges do not support any axial load, and a revolute joint with constant angular stiffness is equivalent to the notch hinge when there is no stress concentration. The links connecting every two notch hinges are regarded as rigid, and the length is the distance between the centers of every two notch hinges. The equivalent rigid models of the prismatic joints are shown in Figs. 8 and 9. Each equivalent rigid model can be further simplified to a secondorder system with a mass and a linear spring. Based on the assumption that kinematic energy of the flexure-based prismatic joint is equal to that of the second-order system, the equivalent mass of the second-order system can be derived. The values of potential energy are also assumed to be identical, and the equivalent stiffness can be obtained, M W/U =

KW/U =

MV =

2 共5M W/UA + M W/UB + 4M W/UC兲LW/U + 4JW/UA 2 4LW/U

2KW/Ur 2 LW/U

, 共1兲

共2兲

,

共4M VA + 5M VB + 2M VC兲LV2 + 4JVA 4LV2

共3兲

,

TABLE I. Results of dimension optimization. R 3.5 mm

t 0.4 mm

b 10 mm

L 25 mm

FIG. 9. PRB model of V joint.

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Large-displacement and decoupled XYZ FPM

KV =

2KVr LV2

共4兲

.

the angular displacement of the notch hinge is ␪, the estimated stress ␴ at the surface of the thinnest portion can be given as1

In order to make the two prismatic joints possess the same stiffness, dimensions of the notch hinges are identical, and lengths of the connection links are also identical, thus

␴= where

KU/Wr = KVr = Kr ,

f共␤兲 =

LU/W = LV = L, where Kr is the angular stiffness of the notch hinge. The design of notch hinges is critical, because the notch hinges’ stiffness ratio of the sensitive axis to the off axis affects the parasitic motions of flexure mechanisms, and permissible maximum stress in notch hinges limits the motion range of the prismatic joints. Therefore, the stiffness ratio of the sensitive axis to the off axis and the permissible maximum stress are two important parameters in the design of the prismatic joints. The notch hinge shown in Fig. 10 is used in the prismatic joints. The rotation about the Z axis is the sensitive motion. The angular stiffness of the notch hinge is given as15 2Ebt5/2 Kr = K␪Z−M Z ⬇ . 9␲R1/2

共5兲

The notch hinge bears the moment about the Y axis in the proposed XYZ FPM. The rotation about the Y axis is off-axis motion. The angular stiffness about the Y axis can be calculated as15 1 K␪Y −M Y

=

4共2R + t兲冑关t/共4R + t兲兴 arctan兵冑关t/共4R + t兲兴其 − ␲t . Eb3t/6 共6兲

To reduce the parasitic motion, the notch hinge should be sensitive to the moment about the Z axis and immune to the moment about the Y axis. The ratio ␭ should be small and 共7兲

␭ = K␪Z−M Z/K␪Y −M Y .

When the notch hinge is under a bending moment, the maximum stress occurs at each surface of the thinnest part. If

冤冥

FX FY = M FZ

M=

冤

E共1 + ␤兲9/20 ␪, ␤2 f共␤兲

␤=

3 + 4␤ + 2␤2 6共1 + ␤兲 + tan−1 1+␤ 共2␤ + ␤2兲1/2

共8兲

冉冑 冊

2+␤ , ␤

t . 2R

The yielding limit ␴b limits the maximum bending of the notch hinge and then limits the maximum translational motion of the prismatic joints. When ␴ increases to the maximum value of ␴b, the angular deformation ␪ reaches the maximum value of ␪max,

␪max =

umax , 2L

共9兲

where umax is the maximum linear displacement of the prismatic joint, that is, the motion range. Dimensions of the notch hinges and the rigid links of two types of the largedisplacement prismatic joints are equal. Therefore, the motion ranges of them are the same as umax =

2L␴b␤2 f共␤兲 . E共1 + ␤兲9/20

共10兲

B. PRB modeling of XYZ FPM

In this section, the dynamic equation of the assembled XYZ FPM is presented. From Fig. 6, it can be seen that the XYZ FPM is a hybrid combination of the prismatic joints. Each prismatic joint can be regarded as a second-order system. Therefore, the XYZ FPM is a hybrid combination of these second-order systems. Under the assumption that the three translational motions are decoupled, the dynamic equation can be derived based on the Lagrange method,

冤冥 冤冥 X¨

X +K Y , Z

Y¨

Z¨

M W1 + M U2 + M V3

0

0

M W2 + M U2 + M U3

0

m U1 + m V1 0

m U1 + m V2 0

0

M W3 + M V1 + M V2 m U3 + m V3

冥

,

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K=

Rev. Sci. Instrum. 77, 115101 共2006兲

Tang, Chen, and Li

冤

K W1 + K U2 + K V3

0

0

0

K W2 + K U1 + K U3

0

0

0

K W3 + K V1 + K V2

Vector 关XYZ兴T represents the displacements of the end effector, which is the central point in Fig. 7. Vector 关FXFY FZ兴T represents the actuation forces exerted on the prismatic joints of W1, W2, and W3, respectively. M and K are the equivalent mass and the equivalent stiffness matrices of the XYZ FPM, and mWi, mUi, and mVi are the actual masses of Wi, Ui, and Vi joints. Note that in 共11兲, the damping ratio is omitted, because damping in flexure mechanisms due to energy dissipation during deflection is difficult to be modeled analytically,1 and the result of the hammer test of the XYZ FPM indicated that the damping ratio is far less than 1, which is negligible compared to the values of the stiffness and inertia of the mechanism.

冥

共11兲

.

motion range, and the natural frequency for the translational mode shape of the XYZ FPM based on PRB method can be calculated as KW/U = KV = 10.4 N/mm, K = 31.3 N/mm, umax = 2.3 mm, f=

1 2␲

冑

K ⬇ 78 Hz. M

V. NONLINEAR METHOD IV. DIMENSION OPTIMIZATION

Flexure mechanisms depend on the deflection of flexible components to achieve desired motion. The dimensions of flexible components are critical to static and dynamic performances of flexure mechanisms, such as stiffness, motion range, parasitic motion, natural frequency, etc. Ability of actuation, possibility of manufacturing, and permissible volume usually limit the achievable dimensions. It is necessary to consider all factors in the determination of the dimensions of the flexure mechanism to achieve optimal design. Based on the above PRB model, the design of the mechanism can be formulated as an optimization problem as follows. • Objective: minimize ␭. • Design variables: R, t, b, L. • Constraints: 0.3 mm艋 t 艋 1 mm, 10 mm艋 b 艋 20 mm, 20 mm艋 L 艋 30 mm, 1 mm艋 umax, and K 艋 50 N / mm. The objective of the optimal design is to develop an XYZ FPM with minimum parasitic motion. The design variables are the dimensions of the notch hinges and the distance between every two notch hinges. The constraints consider the factors as follows. The flexure mechanism is cut by wire EDM. The thinnest portion of the notch hinge is not less than 0.3 mm, because the tolerance of ±0.01 mm cannot be ensured when the thickness is smaller than 0.3 mm. For compactness, the size of the XYZ FPM should not be large, thus the distance between every two notch hinges is limited. In order to achieve the desired motion range without plastic deformation, the maximum displacement should be larger than the required motion range of 1 mm. With limitation of the output of the voice coil actuators, the linear stiffness of the XYZ FPM cannot exceed 50 N / mm. The gradient projection method 共GPM兲 is used to search the optimal points in the workspace. The final values of the design variables are listed in Table I. The linear stiffness, the

The prismatic joint can be regarded as an equivalent second-order system. The assembled XYZ FPM is a hybrid combination of the prismatic joints, and it can also be seemed as a second-order system. The common practice to derive the equivalent second-order system is the PRB modeling. The PRB modeling is an approximate method for analytical formulations. In the PRB theory, the notch hinge is treated as a center-fixed revolute joint with constant angular stiffness when there is no axial load, and the connection link between the centers of every two notch hinges is regarded as a rigid link. Presumably, the potential energy is caused by deformations of notch hinges, and the kinematic energy is from rigid motions of connection links. However, the equivalent second-order system obtained from PRB is not accurate enough. In reality, the notch hinge deforms more like an end-fixed beam than a center-fixed revolute joint. The notch hinge also contributes to the kinematic energy, especially when the mass of the notch hinge cannot be neglected compared to that of the connection link. In the proposed method, both notch hinges and connection links are regarded as deformable beams. Deformation of each part is represented exactly with nonlinear integral equations. The deformation equations can be used to formulate the exact model of the equivalent second-order system. A. Nonlinear modeling of prismatic joint

It has been mentioned in Sec. III A that the two prismatic joints used in the proposed XYZ FPM have the same stiffness and use the same analytical method. Only analysis of the prismatic joint in Fig. 11 is presented here. As illustrated in Fig. 11, the prismatic joint is composed of one motion stage, two secondary stages, and four limbs. The motion stage and the secondary stages are regarded as rigid, and the limbs are regarded as flexible. Each limb 共Fig. 12兲 consists of three parts, two notch hinges 共part I and part III兲 and one connection link 共part II兲. Points P0 – P3 are the

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Rev. Sci. Instrum. 77, 115101 共2006兲

Large-displacement and decoupled XYZ FPM

TABLE II. Specifications of force sensor. Sensing range FX, FY , FZ 共±N兲 TX, TY , TZ 共±N m兲

36 0.5 Resolution

FX, FY , FZ 共±N兲 TX, TY , TZ 共±N m兲

FIG. 11. Deformed prismatic joint.

I 3共 ␤ 兲 = ends of parts I–III. When the force F is applied to the motion stage, the motion stage produces a linear displacement without any parasitic motion, and the secondary stages produce half of the displacement at the motion stage. Limb 1 is connected to the ground at one end and to the secondary stage at the other end. Limb 2 is connected to the secondary stage at one end and to the motion stage at the other end. It can be seen that limb 1 and limb 2 deform in the same manner. Both of them deform to generate half of the linear displacement at the motion stage. Therefore, each limb can be regarded as a fixed-free beam in derivation of deformations shown in Fig. 12. At the free end of each beam, 41 F is transferred, and an internal moment M 3 is generated to nullify the angular deformation at P3. A1共␣兲, A2共x兲, and A3共␤兲 are cross-section areas at any points of the three parts in Fig. 12, hence A1共␣兲 = 2共h − R cos ␣兲b,

−

␲ ␲ 艋␣艋 , 2 2

A2共x兲 = 2hb,

共12兲 共13兲

A3共␤兲 = 2共h − R cos ␤兲b,

−

␲ ␲ 艋␤艋 . 2 2

共14兲

I1共␣兲, I2共x兲, and I3共␤兲 are moments of inertia, hence I 1共 ␣ 兲 =

2b共h − R cos ␣兲3 , 3

I2共x兲 =

bh3 , 12

−

␲ ␲ 艋␣艋 , 2 2

共15兲 共16兲

1 512 ⫻ 36 1 40 000 ⫻ 0.5

2b共h − R cos ␤兲3 , 3

−

␲ ␲ 艋␤艋 . 2 2

共17兲

M 1共␣兲, M 2共x兲, and M 3共␤兲 are moments, hence F共L + R − R sin ␣兲 + M 3, 4

M 1共 ␣ 兲 =

−

␲ ␲ 艋␣艋 , 2 2 共18兲

M 2共x兲 =

F共L + 2R − x兲 + M 3, 4

M 3共 ␤ 兲 =

F共R − R sin ␤兲 + M 3, 4

2R 艋 x 艋 L,

−

␲ ␲ 艋␤艋 . 2 2

共19兲 共20兲

The angular deformations are calculated as

␪ 1共 ␣ 兲 = ␪2共x兲 = ␪ 3共 ␤ 兲 =

冕 冕 冕

M 1共 ␣ 兲 R cos ␣d␣ , EI1共␣兲

共21兲

M 2共x兲 dx, EI2共x兲

共22兲

M 3共 ␤ 兲 R cos ␤d␤ , EI3共␤兲

共23兲

where E is Young’s modulus of 7.1⫻ 104 N / mm2. Because of no angular deformations at the two ends of the beam in Fig. 12 and continuous beam, 共21兲, 共22兲, and 共23兲 are solved under the constraints

冉 冊

␪1 −

␲ = 0, 2

␪2共2R兲 = ␪1

冉 冊 冉冊

␪3 − ␪3

共24兲

冉冊

␲ , 2

共25兲

␲ = ␪2共L兲, 2

␲ = 0. 2

共26兲

The internal moment M 3 can be derived as TABLE III. Specifications of laser sensor.

Measuring range Resolution FIG. 12. One Limb.

LC 2430

LC 2440

±0.5 mm 0.02 ␮m

±3 mm 0.2 ␮m

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Tang, Chen, and Li

TABLE IV. Results of linear stiffness of prismatic joint.

Value Percentage error

Y 2共2R兲 = Y 1 Y 3共 ␤ 兲 =

冕

Expt.

PRB

7.79 N / mm

10.4 N / mm 33.5%

Nonlinear method 7.31 N / mm 6.2%

冉冊

␲ , 2

共31兲

␪3共␤兲R cos ␤d␤ ,

共32兲

冉 冊

Y3 − FIG. 13. Experimental setup of prismatic joint.

共27兲

M 3 = − FR/4.

冕

␪1共␣兲R cos ␣d␣ ,

共28兲

冉 冊

T=

冕

冕

␲/2

−␲/2

共29兲

␪2共x兲dx,

␳ A 1共 ␣ 兲

F/4 ⬇ 7.31 N/mm. Y 3共␲/2兲

KY−FY =

共34兲

The equivalent linear stiffness of the prismatic is identical to that of the single limb, KV* = KY−FY ⬇ 7.31 N/mm.

␲ = 0, Y1 − 2 Y 2共x兲 =

共33兲

By solving 共28兲–共33兲, the linear deformation Y 3共␲ / 2兲 at P3 can be obtained. Then the linear stiffness of the single limb can be calculated using the dimensions listed in Table I as

The linear deformations and the constraints can be represented as Y 1共 ␣ 兲 =

␲ = Y 2共L兲. 2

再

共30兲

d关Y 1共␣兲兴 dt

+ 共m11 + m12 + m13 + mS兲

再

冎

2

R cos ␣d␣ +

d关Y 3共␲/2兲兴 dt

冎

冕

L

2R

2

␳A2共x兲

再

再

After calculation of the equivalent linear stiffness of the prismatic joint, the equivalent mass is derived. The kinetic energy of the prismatic joint is needed for derivation and can be calculated as

冎 冎

d关Y 2共x兲兴 dt

1 d关2Y 3共␲/2兲兴 + mM 2 dt

FIG. 14. Experimental result of linear stiffness of prismatic joint.

共35兲

2

dx +

冕

␲/2

−␲/2

␳ A 3共 ␤ 兲

再

d关Y 3共␤兲兴 dt

冎

2

R cos ␤d␤

2

共36兲

,

FIG. 15. Experimental setup of XYZ FPM.

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115101-9

Rev. Sci. Instrum. 77, 115101 共2006兲

Large-displacement and decoupled XYZ FPM

TABLE V. Results of linear stiffness of XYZ FPM.

Value Percentage error

Expt.

PRB

25.4 N / mm

31.3 N / mm 23.2%

Nonlinear method 21.93 N / mm 13.7%

where ␳ of 2.7⫻ 103 kg/ m3 is the density. Therefore, the equivalent mass of the prismatic joint is M V* =

T . 2兵d关Y 3共␲/2兲兴/dt其2

共37兲

B. Nonlinear modeling of XYZ FPM

The dynamic equation formulated using the nonlinear method is the same as that from the PRB model. The equivalent mass and the equivalent stiffness matrices of the XYZ FPM are obtained as

FIG. 16. Experimental result of linear stiffness of XYZ FPM.

*

M =

K* =

冤

冤

* * MW + MU + M V* 1

2

0

3

0

m U1 + m V1 * * * + MU + MU MW

0

2

2

0

3

m U1 + m V2 0

* + M V* + M V* MW

0

3

1

m U3 + m V3

* * KW + KU + KV* 1

2

0

0

3

* KW 2

+

0

* KU 1

0

0 +

* KU 3

0 * KW 3

+

KV* 1

+

KV* 2

冥

2

冥

,

.

Based on the above matrices of mass and stiffness, the linear stiffness and the natural frequency for the translational mode shape of the XYZ FPM are obtained as K* = 21.93 N/mm, f* =

1 2␲

冑

K* ⬇ 66.3 Hz. M*

VI. EXPERIMENTAL VERIFICATION A. Static test of prismatic joint

To verify the proposed nonlinear method, the static test was conducted to measure the linear stiffness of the prismatic joint. The force exerted on the prismatic joint and the output displacement are needed to calculate the stiffness. The actuation force is measured by the six-axis force sensor FT05270 共ATI兲, and the displacement is detected by the laser

FIG. 17. Experimental result of cross-axis error of XYZ FPM.

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Rev. Sci. Instrum. 77, 115101 共2006兲

Tang, Chen, and Li

FIG. 19. Natural frequencies of XYZ FPM.

FIG. 18. Schematic diagram of hammer test.

sensor LC2430 共Keyence兲. The specifications of the force sensor and the laser sensor are listed in Tables II and III. The experimental setup is shown in Fig. 13. The experimental data, the forces and the displacements, are plotted in Fig. 14. Based on these data, a straight line is fitted with the slope of 128.3 ␮m / N. The reciprocal of the slope is the linear stiffness of the prismatic joint with a value of 7.79 N / mm. Table IV lists the results of the stiffness of the prismatic joint. It is shown that the nonlinear method is more accurate than the PRB method. The percentage error between the value from the nonlinear method and the experimental data may be due to manufacturing error and sensing error. B. Static test of XYZ FPM

Two experiments were conducted to measure the linear stiffness and the parasitic motions of the XYZ FPM. The six-axis force sensor FT05270 共ATI兲 is mounted to measure the actuation force, and the laser sensors LC2430 and LC2440 共Keyence兲 are used to measure the displacement of the end effector located at the center of the XYZ FPM. Figure 15 shows the experimental setup. The first experiment was carried out for determining the linear stiffness of the XYZ FPM. The data of the input force TABLE VI. Specifications of accelerometer. Measuring range Sensitivity Resonant frequency

±500 g 10.26 mV/ g 70.0 kHz

and the output displacement are shown in Fig. 16. Curve fitting is used to calculate the linear stiffness. It can be seen that the slope of the fitting line is 39.4 ␮m / N, and its reciprocal is the linear stiffness of 25.4 N / mm. Table V summarizes the stiffness obtained in different ways. The result from the nonlinear modeling method is closer to the actual value than that from the PRB method. The difference between the analytical value and the experimental result may be caused by manufacturing and assembly errors. The second experiment aims at verifying the decoupled characteristics. When the X-axis motion was activated, the displacement along the X and Y axes was simultaneously tested to study the cross-axis error. The experimental results show that the cross-axis error is smaller than 1.9%, as shown in Fig. 17. At the same time, the parasitic rotation ␪Z was calculated based on the measured displacement along the Z axis. It is shown that the parasitic rotation is smaller than 1.5 mrad. Therefore, it can be verified that the XYZ FPM has a decoupled kinematic structure. C. Modal test of XYZ FPM

Hammer test was conducted for measuring the natural frequency for the translational mode shape of the XYZ FPM. The experimental setup is shown in Fig. 18. The impact excitation was input by an impact hammer. The accelerometer in Table VI was attached on the end effector and was used to measure the vibration under the impact excitation. Polytech scanning vibrometer was used for signal conditioning. The converted signals from the vibrometer are fed into the Fourier analyzer for calculating the measured frequencies. From the modal analysis by finite-element simulation software ANSYS, it is known that the first mode shape describes the translational motion of the XYZ FPM. The modal analysis of the XYZ FPM is explained in detail in the previous paper.22 The measured natural frequency for the first mode shape is about 56 Hz shown as in Fig. 19. VII. SUMMARY

An XYZ FPM with large displacement and decoupled kinematic structure has been developed. The experimental results show that the cross-axis error is smaller than 1.9% and the parasitic rotation is smaller than 1.5 mrad. A new type of large-displacement prismatic joint is designed. An exact nonlinear method for the large-displacement prismatic

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joints is proposed and applied to the proposed prismatic joint and the XYZ FPM. Comparison based on the experimental data shows that the nonlinear method for the largedisplacement prismatic joints is more accurate than the conventional PRB method. ACKNOWLEDGMENTS

This project is financially sponsored by the Ministry of Education, Singapore, under ARP RG 06/02, and Manufacturing Science and Technology 共MST兲 program of Singapore-MIT Alliance. T. S. Smith, Flexures: Elements of Elastic Mechanisms 共Gordon and Breach, New York, 2000兲. 2 J. H. Kim, S. H. Kim, and Y. K. Kwak, Rev. Sci. Instrum. 74, 2918 共2003兲. 3 J. W. Ryu, D. G. Gweon, and K. S. Moon, Precis. Eng. 21, 18 共1997兲. 4 J. W. Ryu, S. Q. Lee, D. G. Gweon, and K. S. Moon, Mechatronics 9, 657 共1999兲. 5 D. H. Chao, G. H. Zong, and R. Liu, Proceedings of the 2005 IEEE/ ASME International Conference on Advanced Intelligent Mechatronics, 2005 共unpublished兲, p. 765. 6 Y. M. Li and Q. S. Xu, Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2005 共unpublished兲, p. 93. 7 K. Yamakawa, K. Furutani, and N. Mohri, Proceedings of the 1999 ASME Design Engineering Technical Conferences, 1999 共DETC/MOVIC 8425兲, 1

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Large-displacement and decoupled XYZ FPM

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