Design and modeling of a novel monolithic parallel XY stage with

Special Issue Article

Design and modeling of a novel monolithic parallel XY stage with centimeters travel range

Advances in Mechanical Engineering 2017, Vol. 9(11) 1–17 Ó The Author(s) 2017 DOI: 10.1177/1687814017729624 journals.sagepub.com/home/ade

Hua Liu, Shixun Fan, Xin Xie, Zhiyong Zhang and Dapeng Fan

Abstract Since most of XY positioning stages with large travel range proposed by former researchers suffer from loose structure and low out-of-plane payload, this article presents a novel monolithic parallel XY stage based on spatial prismatic– prismatic joints with centimeters travel range, compact size, and high out-of-plane payload capacity. The novel parallel linear compliant mechanism of the stage is composed of four spatial prismatic–prismatic joints, which is two compound leaf spring parallelograms serially connected in cubical space to obtain large travel, compact size, and high out-of-plane payload capacity simultaneously. The theoretical static stiffness and resonant frequencies are obtained by matrix structural analysis. As a case study, a reified stage is presented and discussed in detail. Finally, theoretical models are comprehensively compared with finite element analysis models. It is shown that the stage in the case study has the following merits: large travel range up to 20 3 20 mm2, high-area ratio of workspace to the outer dimension of the stage about 2.26%, well-constrained cross-axis coupling motion less than 1.5 mm at the full primary motion, acceptable resonant frequencies of the two translational axes about 34 Hz, and large out-of-plane payload capacity more than 24 kg. Keywords Monolithic parallel XY stage, spatial prismatic–prismatic joint, large travel, compact structure, high out-of-plane payload, matrix structural analysis, finite element analysis

Date received: 18 February 2017; accepted: 8 August 2017 Handling Editor: Fen Wu

Introduction Compliant positioning stages possess many merits, such as reduced number of parts, no friction, no backlash, and free of lubrication,1,2 which make them widely used in the fields of micro-manufacturing, optical fiber alignment, biological engineering, scanning probe microscopy (SPM), and lithography.3–7 But in some special applications such as ultraviolet nanoimprint lithography (UV-NIL) and soft-contact lithography,8 it is expected that the XY centering positioning stage does not only act as a positioning device with large travel range but can also carry large out-of-plane payload without any other auxiliary supporting equipment. Moreover, restrained by space limitation, it is preferable for a compact positioning stage. Hence, this article

is concentrated on the design and development of a XY compliant positioning stage which has large travel, compact structure, and high out-of-plane payload capacity. Parallel XY stage (PXYS) is the most common layout of XY stages because of its advantages including identical dynamic features in working axes, low inertia, and low cumulative positioning errors.8 In order to

College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, China Corresponding author: Shixun Fan, College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha 410073, China. Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

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Table 1. Comparisons of typical PXYSs. Reference

Dimension (mm2)

Workspace (mm2)

Area ratio (%)

Out-of-plane payload capacity

Awtar9 Xu12 Yu et al.13 Xu18 Hao and Kong19 This work

385 3 385 410 3 410 311 3 311 320 3 320 540 3 540 133 3 133

10 3 10 10.5 3 10.5 535 11.75 3 11.66 20 3 20 20 3 20

0.067 0.069 0.026 0.133 0.137 2.26

Unknown Unknown Unknown 20 kg Unknown 24 kg

PXYS: parallel XY stage.

obtain PXYSs with large travel and compact size, leaf spring parallelograms are widely used as prismatic (P) joints. Normally, leaf spring parallelograms are used in couples forming a prismatic–prismatic (PP) joint to realize kinematically decoupling. According to whether the leaf springs are in the plane of motion or normal to the plane of motion, PP joints can be divided into two types: planar and spatial PP joints. PXYSs using planar or spatial PP joints are defined as planar parallel XY stages (PPXYSs) or spatial parallel XY stages (SPXYSs) in this article. PPXYSs are the most common XY stages because of its easy manufacturability. In order to obtain large travel of PPXYS, much interest has been focused on structural design. A large number of millimeter-scale PPXYSs were proposed by researchers. Awtar,9 Trease et al.,10 Tang et al.,11 Xu,12 and Yu et al.13 have designed and developed millimeter and even centimeter scale PPXYSs. However, there are two main defects of these PPXYSs: (a) the structures of the stages are relatively loose. It is challenging to design an XY stage with a large workspace and a compact physical dimension, simultaneously. The compactness of a stage is denoted by the modified area ratio12 whose denominators and the actuators’ dimensions should be added into. The area ratios of the stages proposed in Awtar,9 Xu12, and Yu et al.13 are only about 0.067%, 0.069%, and 0.026%, respectively; (b) the out-of-plane payload capacity is weak. When large out-of-plane payload is applied, the load-stiffening and elastokinematic phenomena14 emerge in PPXYSs, which would induce nonlinearity and cause the loss of travel range. Researchers have been trying to develop a large travel range PXYS with compact size and high out-ofplane payload simultaneously. There are several ways to enhance the out-of-plane stiffness such as magnetic levitation bearing, aerostatic bearing, and increasing the thickness of the leaf spring.15–17 Using either magnetic levitation or aerostatic bearing would increase the structural complexity and the cost of PXYS. By stacking two identical planar XY stages, Xu18 has tried to use spatial space to obtain large travel and compact size. The area ratio of the stage is significantly improved to 0.13% and the out-of-plane payload capacity is indeed increased. But the actuation stiffness

is also increased by 14% and the nonlinearity of the stage becomes obvious in the whole travel range. Besides, the assemble structure of the stage would deteriorate the reliability of the stage and induce the assemble error. Nevertheless, Xu pointed out to us a possible direction using the cubical space to meet these conflicting requirements. Since spatial PP joint has the remarkable supporting character in the length direction of the leaf spring, Hao and colleagues19,20 figured out using the combination of spatial parallelograms and planar compliant joints to enhance the out-of-plane stiffness and Shang et al.21 successfully followed the idea and developed a prototype. The out-of-plane stiffness is enhanced indeed, but the Shang’s stage suffers from relatively short travel range and low area ratio (only 0.002%) because the spatial parallelogram used in the stage is a lumped compliance parallelogram. The stage proposed by Hao meets the requirement of large travel range (20 3 20 mm2 motion range), compact structure (area ratio: 0.14%), and high out-of-plane payload capacity. But the structural configuration of Hao’s stage is too complex to be fabricated monolithically and has to be an assembly which would induce the assemble error. The performances of some typical PXYSs are tabulated in Table 1. Inspired by the aforementioned researcher’s work, a novel PXYS is presented in this article. It can deliver a large travel and sustain high out-of-plane payload, while maintain compact structure due to the usage of spatial PP joint and the conformal design of voice coil motors (VCMs). Specifically, the spatial PP joint is introduced to design the spatial parallel linear compliant mechanism (SPLCM), which has great superiority over the planar PP joint in terms of compact structure and high out-of-plane stiffness. Through symmetrical arrangement for four spatial PP joints and along with conformal-designed VCMs, compact design yields the novel PXYS with large travel range, high out-of-plane stiffness, and compact size. The major contribution of this article lies in the design and modeling of a novel SPXYS based on the spatial PP joint, which owns integrated merits of large travel range, compact size, and high out-of-plane payload capacity. Section ‘‘Conceptual design of the novel

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Figure 1. Schematic diagram of a 4-PP decoupled parallel mechanism.13,18,20

SPXYS’’ introduces the design of the SPXYS based on spatial leaf spring parallelogram and driven by conformal-designed VCMs. In section ‘‘Mathematical model of SPXYS,’’ the proposed SPXYS is modeled by means of multiple substructure approach of matrix structural analysis (MSA). In section ‘‘Case study,’’ the parameters of the SPXYS are determined based on theoretical analysis and design constraint. The precision and manufacturability is discussed in this section. Section ‘‘FEA comparison’’ carries out comprehensive comparisons between finite element analysis (FEA) simulation results and the theoretical results. The theoretical analysis is verified via FEA simulation. Finally, conclusions are drawn in section ‘‘Conclusion.’’

Conceptual design of the novel SPXYS Like the method used in designing PPXYS, the SPXYS can be constructed starting from a rigid body 4-PP decoupled parallel mechanism as shown in Figure 1. Each PP leg is composed of two P joints in series. The P joint connected to the base is actuated and the other is passive as a decoupler. The directions of two

adjacent P joints on the motion stage are perpendicular to each other to achieve kinematical decoupling, which is the necessary condition for parallel mechanisms. In order to increase the motion range, alleviate the cross-axis coupling, and load-stiffening effects, planar double leaf spring parallelograms are selected to construct compliant P joints. Since the leaf spring’s stiffness in both length and width directions is much larger than the thickness directional stiffness,22 leaf springs in both directions are often used as decouplers: the thickness direction of decoupling leaf spring is parallel to the decoupled direction to absorb the displacement of the decoupled axis, and one of the rest two directions can be set parallel to the guiding direction to transmit force and displacement in the guiding axis. According to the decoupling leaf springs’ direction used in the guiding axis, the PP joints can be divided into two categories: planar PP joints and spatial PP joints as shown in Figure 2. According to Figure 2, spatial PP joints can be evolved from planar PP joints by rotating the guiding and decoupling P joints 90° about the guiding and decoupled axes, which actually uses the exact

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Figure 2. Two types of PP joints: (a) planar PP joint and (b) spatial PP joint.

position-space reconfiguration approach.23,24 Hence, the spatial PP leg in this article can be regarded as a special configuration of the PP joints obtained by the position-space principle. It is a remarkable fact that the stiffness in the leaf spring’s length direction quadratically decreases as the increment of the deformation in the leaf spring’s thickness direction22 because of the elastokinematic effects.14 For the same reason given above, the stiffness reduction in the leaf spring’s length direction of the double parallelogram is much larger than that of the basic parallelogram.25 But the degree-of-freedom (DOF) directional stiffness of double parallelograms is less affected by axial force, which means double parallelogram has good linearity. Besides, by symmetric layout and multiple spatial PP legs connecting in parallel scheme, the supporting stiffness of the whole compliant mechanism could be still much larger than that of planar XY stage which uses the same double parallelograms and arrange them in the plane. More importantly, since the width of leaf spring is much smaller than the length, the outer dimensions of compliant mechanism utilizing spatial PP joints can be largely reduced, which is more suitable for space-limited applications. Therefore, in order to meet the requirements of space-limited applications, a novel SPXYS using spatial PP joints is proposed in this article. The proposed SPXYS mainly consists of two-axis SPLCM and four VCMs, as shown in Figure 3. The VCMs are specially designed according to the structure for the compact size of the stage, instead of using the existing commercial VCMs because it is hard to find one commercial VCM meeting the requirements of travel range, peak force, and compact size simultaneously.

Mathematical model of SPXYS Stiffness matrix derivation The stiffness characteristic of compliant mechanism has direct effects on the workspace, load-carrying capacity, dynamic behavior, and positioning accuracy of the entire mechanism.26 But stiffness modeling for spatial compliant mechanisms is relatively complex because of the spatial deformation of the leaf springs. Currently, the stiffness modeling methods mainly include the pseudo-rigid body model (PRBM),1 Castigliano’s theorem,2 beam constraint model (BCM),9,14,21,22,25 FEA, and the matrix of structural analysis (MSA).27 A simple and comprehensible method to predict the stiffness of the spatial compliant mechanisms is more attractive for designers. However, whether using PRBM, Castigliano’s theorem, or BCM, the modeling of spatial compliant mechanism results in a large number of equations to solve the internal forces and displacements on the end of the leaf springs. The uses of finite element method are reliable, but these FEA models have to be remeshed over again, involving very tedious and time-consuming routines. As the origin of FEA, MSA has been well developed and widely utilized in stiffness modeling of compliant parallel mechanisms.28 Hao20 utilized the matrix method in the kinetic analysis and stiffness modeling of a spatial XY micro-parallel mechanism. Besides, multiple substructure method based on MSA is capable of reducing the stiffness matrix order and simplifying the analytical model of the key structural parameters. Hence, MSA is utilized to establish the stiffness model of the SPLCM. The SPLCM can be divided into four identical legs (PP joints) as depicted in Figure 4(a). There are four guiding leaf spring flexures and four decoupling leaf

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Figure 3. Structure of the proposed SPXYS.

spring flexures in each leg, as shown in Figure 4(b). These leaf springs, guiding beams, and three platforms form two guiding parallelograms and two decoupling parallelograms, denoted as S1–S4. Details about each leg are shown in Figure 5. Each leg can be viewed as four basic parallelograms connected in series. Based on the linear modeling of compliant mechanisms in Hao20 and Jiang et al.,28 we can obtain the relationship of displacements between the basic parallelogram and the spatial PP leg. The output displacement of each leg can be obtained as

dli =

4 X

~sj Tlsj d

(i = 1, . . . , 4)

ð1Þ

j=1

~sj is the deformation of the jth basic parallelowhere d gram in the local coordinate and Tsj is the transformation matrix from jth basic parallelogram coordinate to the ith leg coordinate given as 

Rlsj Tlsj = 0

S(rlsj )Rlsj Rlsj

 ð2Þ

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Figure 4. Structure of (a) SPLCM and (b) SPLCM’s leg.

where Rlsj is the rotation matrix of the jth parallelogram’s local coordinate with respect to the leg coordinate, given as 2

0 6 Rls1 = 4 0 1 2 0 6 Rls3 = 4 0 1

1 0 0 0 1 0

3

2

3

0 0 1 0 7 6 7 1 5, Rls2 = 4 0 0 1 5, 0 1 0 0 3 2 3 1 0 0 1 7 6 7 0 5, Rls4 = 4 0 1 0 5 0 1 0 0

S(rlsj) is the skew-symmetric matrix in the form of



Fsj = TTasj Fai Fsj = 0

( j = 1, 2 ) ( j = 3, 4 )

ð4Þ

where Fai is the actuator force expressed in the ith leg’s local coordinate, Fai = [ fi, 0, 0, 0, 0, 0]T, and Tsjai is the transformation matrix of actuation force with respect to the jth parallelogram, there into 8   0 RTls1 > T > > < Tas1 = RT S( r ) RT ais1 ls1  ls1 T  > 0 R > ls2 > : TTas2 = RTls2 S( rais2 ) RTls2

ð3Þ

and raisj is the position vector expressed in the leg coordinate (    T ras1 = 0 bg 2  Dgd 2 L=2    T ras2 = 0 bg 2  Dgd 2 L=2

and rlsj is the position vector of the origin of the jth parallelogram with respect to the leg’s origin expressed in the leg coordinate, in particular

The output displacement dlai caused by actuation force Fai is

2

0 S(rlsj ) = 4 zlsj ylsj

8  > rls1 = 0 > >  < rls2 = 0 > > r =½0 > : ls3 rls4 = ½ 0

zlsj 0 xlsj

3

ylsj xlsj 5 0

T  bg 2  Dgd  3Dd =2  1:1Sa L T  bg 2  Dgd  3Dd =2  1:1Sa 0 Dd  1:1Sa L T 0 0 T

The load of the jth parallelogram caused by the actuation force Fai applied at point A is given as

dlai =

2 X j=1

~sj = Tlsj d

2 X

T Tlsj K1 sj Tasj Fai

ð5Þ

j=1

where Ksj is the stiffness of the jth compliant parallelogram. According to the same method illustrated above, the output displacement dlmi caused by the inner force Fmi between the legs and moving platform is

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Figure 5. Dimensions of the leg: (a) front view, (b) side view, and (c) isometric view.

dlmi =

4 X

T Tlsj K1 sj Tlsj Fmi

where Tlis is the transformation matrix from the moving platform to the ith leg, which takes the form of

j=1

So, the output displacement at point Di is dli =

2 X j=1

T Tlsj K1 sj Tasj Fai +

4 X

T Tlsj K1 sj Tlsj Fmi

ð8Þ

dli = Tlis ds

ð6Þ

 Tlis =

ð7Þ

j=1

Since the moving platform is regarded as a rigid body, the relationship between the deformation at point Di and the displacement of the center point Os of the moving platform satisfies

Rlis 0

  T S(rlis )Rlis R (u ) = z lis Rlis 0

S(rlis )RTz (ulis ) RTz (ulis )



ð9Þ

Rz(ulis) is the rotation matrix about the Z-axis, and ul1s = 0, ul2s = p/2, ul3s = p, ul4s = 2p/2. rl1s = rl2s = rl3s = rl4s= [0, rl, 0]T, where rl is the distance between the center of the moving platform and the output point of the leg, which is equal to (Dd/2 + Ds).

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3 c1 ð f1  f3 Þ  c2 ð f2  f4 Þ 6 7 c1 ð f2  f4 Þ + c2 ð f1  f3 Þ 6 7 6 7 c3 ð f1 + f2 + f3 + f4 Þ 6 ........................................... 7 Fae = 6 7 6 ðc4  c3 rl Þð f1  f3 Þ  c5 ð f2  f4 Þ 7 6 7 4 ðc4  c3 rl Þð f2  f4 Þ + c5 ð f1  f3 Þ 5 ðc1 rl + c6 Þð f1 + f2 + f3 + f4 Þ

According to the force equilibrium of the moving platform, the relationship between the inner force Fmi and the external force Fe can be obtained 4 X

TTlis Fmi = Fe

ð10Þ

i=1

Substituting equations (7) and (8) into equation (10) gives 2 4

4 X

TTlis

i=1 4 X

2

TTlis 4

i=1

1.

Tlis 5ds = Fe +

T Tlsj K1 sj Tlsj

j=1 4 X

According to equation (11), a few conclusions can be obtained as follows:

3

!1

4 X

!1 T Tlsj K1 sj Tlsj

j=1

2 X

3 T 5 Tlsj K1 sj Tasj Fai

Actually, the second item on the right-hand side of equation (11) is the equivalent external force Fae generated by actuation force Fai, which is in the form Fae =

2 TTlis 4

i=1

=

4 X

4 X

!1 T Tlsj K1 sj Tlsj

j=1

2 X

3 T 5 Tlsj K1 sj Tasj Fai

4 4 X X ∂Fe T = TTlis Tlsj K1 Ks = sj Tlsj ∂ds i=1 j=1

TTlis CFai

4 X

!1 T Tlsj K1 sj Tlsj

j=1

ð17Þ

2 X

T Tlsj K1 sj Tasj

Equation (17) can be used to analyze the influences of the external loads applied at the moving platform on the motion of the actuator.

ð13Þ

2. When the external forces Fe are equal to zero, the displacement of the stage under actuator forces can be written as

j=1

Denote Fai = fi  P = fi [1, 0, 0, 0, 0, 0]T. Then, the equivalent external force Fae generated by actuator forces can be given as + TTl2s

+ f3 TTl3s

f1  f3

+ f4 TTl4s



ds = K1 s Fae

CP

ð f 2  f 4 Þ

.................................

f1 TTl1s

2

ð16Þ

j=1

where C is a constant transformation matrix expressed as

Fae =

Tlis

 1 T 1 T dai = Tas2 T1 ls2 Tls1 Ks1 Tls1 + Ks2 Tls2 !1 4 X 1 T Tlsj Ksj Tlsj Tlis K1 s Fe

j=1

ð12Þ



!1

According to equations (7), (8), and (11), the displacement dasi of the ith leg at the actuator point caused by the external force Fe is

i=1

C=

When the actuator forces Fai are equal to zero, the stiffness of the stage related to the displacement ds at the center of the platform and the external force Fe is

j=1

ð11Þ

4 X

ð15Þ

ð18Þ 3

6 7 6 7 f2  f4 f1  f3 6 7 6 7 f1 + f3 + f2 + f4 6 7 6 = 6 ...................................................................................................................... 7 7CP rl ð f1  f3 Þ f1  f3 ð f2  f4 Þ 6 7 6 7 6 7 f1  f3 f2  f4 rl ð f2  f4 Þ 4 5 rl ð f1 + f3 + f2 + f4 Þ CP is a constant vector, denoted as CP = [c1, c2, c3, c4, c5, c6]T. So, the equivalent external force can be written as

ð14Þ

f1 + f3 + f2 + f4

The displacement under actuator forces at one force point is

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 1 dai ¼ Tas2 TTls2 Tls1 K1 s1 Ts1a þ Ks2 Ts2a Fai  1 þ Tas2 TTls2 Tls1 K1 s1 Tls1 þ Ks2 Tls2 Fmi

ð19Þ

ðL ðL 1 rAcv 2 4 2 mse = 2 rAv(x) dx = x (3L  2x)2 dx v0 v20 0

According to equations (17) and (19), and the unexpected displacement of one actuator caused by another actuator and external force, the input coupling in Li et al.24 can be obtained based on which the clearance between the coil and the stator of the VCM can be determined.

Modal analysis A high resonant frequency is desired to generate high bandwidth of the system, so it is necessary to analyze the resonant frequency of SPXYS. The modal analysis of the stage is based on the linear stiffness model. According to Lagrange’s equation, the dynamic model of SPXYS can be written as follows

 d ∂T ∂T ∂V  + =Q dt ∂d_ s ∂ds ∂ds

ð21Þ

The kinetic energy of SPXYS is calculated as 1 T = d_ Ts  M  d_ s 2

ð22Þ

where M is the mass matrices, M = diag{Mx, My, Mz, Jx, Jy, Jz} and Mx = My, Jx = Jy because of the symmetric structure of the stage. There are eight guiding leaf springs, eight decoupling leaf springs, two intermediate platforms, two connecting beams, and two VCMs’ mover moving together in the X (Y) directional translational motion. According to the approximate deformation shape function of parallelogram’s leaf spring, the velocity of the leaf spring along the length direction v(x) can be written as dy(x) = cv x2 (3L  2x) v(x) = dt

ð24Þ

13rAL 13 = ms 35 35

where r is the density of the material, and A and L are the cross-sectional area and length of the leaf spring, respectively. According to equation (24), the moving mass in the X- or Y-direction is Mx = 8mgse + 2mgc + 2ma + 8mds + 2mdc + 2mm + 0:5mdc + 8mdse + Mo 104 384 mgs + mds + 2mgc + 2ma + 2:5mdc = 35 35 + 2mm + Mo = My ð25Þ

ð20Þ

where T, V, t, and ds represent the kinetic energy, the potential energy, time, and displacement of the stage, respectively. Q is the sum of the actuators’ equivalent external forces and external force applied on the moving platform Q = Fe + Fae

=

0

ð23Þ

The end velocity of the parallelogram’s leaf spring is v0 = cvL3. The equivalent mass mse of the parallelogram’s leaf spring at the end of the leaf spring is

where mgs, mds, mgse, and mdse are the mass and equivalent mass of guiding and decoupling leaf springs; ma, mgc, mdc, mm, and Mo are the mass of the intermediate platform, guiding connecting beam, decoupling connecting beam, VCM’s mover, and moving platform, respectively. The Z-directional stiffness of the stage is much larger than other directions. The Z-directional moving mass is approximate to the mass of moving platform, Mz ’ Mo. The rotational inertia moment of the stage is given as 8  < Jx = 24mgs +mgc +4mds +ma +mdc +mm  lc2 +Jox J = 2 4mgs +mgc +4mds +ma +mdc +mm  lc2 +Joy : y Jz = 4 4mgs +mgc +4mds +ma +mdc +mm  lc2 +Joz ð26Þ where lc is the centroid distance of one leg’s moving mass with respect to the center of moving platform. Jox, Joy, and Joz are the moving platform moment about the X-, Y-, and Z-axis, respectively. The potential energy V can be computed by V=

1 T d  Ks  ds 2 s

ð27Þ

The free-motion vibration equation can be determined by substituting equations (22) and (27) into equation (20) € s + Ks  d s = 0 Md

ð28Þ

By solving equation (28), the resonant frequencies of the stage can be obtained as follows

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Table 2. Structural parameters of the stage. Guiding parallelogram

Decoupling parallelogram

Symbols

Description

Lg bg tg Dg tgg Dgd Ds

Leaf spring length 60 Ld Leaf spring width 10 bd Leaf spring thickness 0.8 td Leaf spring distance 33.8 Dd Guiding beam thickness 5 tdg Distance between guiding and decoupling parallelogram Distance between decoupling parallelogram and center of the stage

1 f= 2p

Value (mm)

rffiffiffiffiffi Ks M

ð29Þ

Since Ks is the function of structure parameters of the stage, the relationship between structural parameter and resonant frequencies can be revealed similarly by numerical method.

Case study Using the equations derived in section ‘‘Mathematical model of SPXYS’’ and the beam dimensions and arrangements defined earlier, we will present in this section a comprehensive study of an SPXYS, including the geometrical parameter determination, actuation force analysis, motion precision analysis, and monolithic manufacturability analysis.

Parameter determination According to equations (16) and (29), the static and dynamic performances of the stage primarily depend on the parameters of the leaf spring flexures. In order to maximize the bandwidth of the proposed stage while maintaining large travel range and a compact size, the parameter determination of the leaf spring flexures can be described as follows: 1.

2.

The objective of the parameter determination is to maximize the translational resonant frequency fT. The maximum deflection ymax of one flexure is equal to the one-side travel range Sa (a total motion range is 2Sa). According to maximal shear theory, the thickness t of the flexures is given as21

t

2sy  L2 2sy  L2 = 3S  Eymax 3S  ESa

ð30Þ

where sy is the yielding stress and S is the safe factor.

Symbols

Description

Value (mm)

Leaf spring length Leaf spring width Leaf spring thickness Leaf spring distance Guiding beam thickness

Ld = Lg 15 td = tg Dd = 5 tdg = tgg 22.8 12.7

According to the aforementioned constraints and manufacturability requirements, the travel range Sa is designed to 10 mm (a total motion range is 20 mm), and the other parameters of the structure are designed in Table 2. The dimensions of the stage are 133 3 133 3 86 mm3. The material of the stage is aluminum alloy (7075-T6) with Young’s modulus E = 7.2 3 1010 Pa, yield strength sy = 5.05 3 108 Pa, and density r = 2810 kg/m3. Substituting the above geometrical dimensions and material property into equations (16) and (18), we obtain the central point displacement (in millimeter and milliradian) on the moving platform under four actuation forces 2

xsc

3

2

1:183 3 101 ð f1  f3 Þ + 5:657 3 105 ð f2  f4 Þ

3

7 6 7 6 6 ysc 7 6 5:657 3 105 ð f  f Þ + 1:183 3 101 ð f  f Þ 7 1 3 2 4 7 6 7 6 7 6 7 6 7 6 7 6 7 6 zsc 7 6 3 104 ð f1 + f2 + f3 + f4 Þ 7 6. . .7 6. . . . . . 2:586 . . . . . . . . . . . . . . . . . . . . . . . . . . . dsc = 6 7 = 6 7 6 u 7 6 2:150 3 104 ð f  f Þ  3:482 3 103 ð f  f Þ 7 6 x7 6 1 3 2 4 7 7 6 7 6 7 6 7 6 6 uy 7 6 3:482 3 103 ð f1  f3 Þ  2:150 3 104 ð f2  f4 Þ 7 5 4 5 4 uz

1:431 3 101 ð f1 + f2 + f3 + f4 Þ

ð31Þ Hence, in order to reach the travel range (Sa = 10 mm), the sum of actuation forces for X- or Yaxial motion should not be less than 87 N, and 43.5 N for each actuator.

Motion precision analysis According to equation (31), the cross-axis coupling relationship between X and Y axes is

dy ð fx Þ

= 0:00005657 ’ 0:048% C= 0:1183 dx ð fx Þ

ð32Þ

The maximum cross-axis coupling error between the two axes is about 4.8 mm, which indicates that the stage can be viewed as totally decoupled. Besides, the cross-

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Figure 6. Monolithic manufacturability: (a) milling machine and (b) wire-EDM.

axis coupling relationship can be depressed by adopting decoupling control. It is obvious that the first and third actuators provide force for the X-axial motion and the second and fourth actuators are for the Y-axial motion. The ideal driving condition is that the magnitudes of actuators’ forces for the same axis motion are equal to each other but the directions are opposite in the respective leg’s local coordinate, that is, f1 = 2f3, f2 = 2f4. Therefore, both the center displacement of the moving platform in Z-axis and the parasitic rotation angle about Z-axis in the ideal condition are equal to zero. The parasitic rotation angle about X-axis (Y-axis) is proportional to the actuation force or the Y-axis (Xaxis) primary motion, which is the main cause for the Z-axial positioning error. Given a certain parasitic rotation angle, the Z-axial positioning error is proportional to the concerned point position on the moving platform. According to equation (32), the maximum parasitic rotation angle is about 0.3 mrad. Assuming that the maximum dimension of the moving platform is 40 3 40 mm2, the maximum position errors in Z-axis on the moving platform caused by parasitic rotation angle are about 12 mm, which could not be neglected.

Monolithic manufacturability Monolithic fabrication for SPXYS complaint mechanisms is more difficult than that for PPXYS compliant mechanisms using water jet, laser cutting, wire electrodischarge machine (wire-EDM) and so on9,29 because of the complex structures of the spatial compliant mechanisms. For example, water jet may not be used in

spatial compliant mechanisms because the inner structures of spatial compliant mechanisms are normally hollow and discontinuous; laser cutting is not suitable for spatial compliant mechanism since the cutting thickness of spatial compliant is too large for laser cutting; although wire-EDM is not constrained by the large cutting thickness and inner discontinuity of the work piece, the auxiliary hole in wire-EDM for spatial complaint mechanism is hard to drill. Despite all of this, by taking the monolithic fabrication into consideration in the design phase, Hao has managed to monolithically manufacture two spatial parallel XYZ prototypes by milling machine.29,30 Hence, in addition to sorting to better manufacturing approaches for spatial compliant mechanisms, one can make it possible to monolithically fabricate the spatial compliant mechanism by considerate design. In consideration of machining difficulty and the requirement of high precision, it is finally decided that wire-EDM is still adopted as the machining method for the flexure features in this article. Accordingly, the SPLCM is designed as a cut-through type which is adapted to the characteristics of the wire-EDM machining procedure. Since there is bulk metal to be removed in the computer-aided design (CAD) model of the SPLCM as shown in Figure 3, milling machining could be used to remove the bulk metal before the wire-EDM machining procedure, which would increase the machining efficiency. In order to guarantee the machining precision, the corresponding fixtures and supporting blocks31 need to be designed and manufactured first. Therefore, there are two main machining processes in the

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Figure 7. FEA deformation: (a) 10-mm deformation in Y-axis and (b) 10-mm deformations in both X and Y axes.

monolithic fabrication for the SPLCM: milling machine and wire-EDM. Figure 6 shows the simple simulation of the machining procedures in Solidworks: first, remove the bulk metal by milling machine and then fabricate the leaf spring by wire-EDM.

FEA comparison The performance characteristics, such as travel range, cross-axis coupling, lost motion, and actuator isolation, of the SPXYS are simulated in ANSYS. The reference point is the center of the bottom surface of the stage.

Static analysis comparison Figure 7 illustrates the large range of motion for two cases under static elastic domain: single axial driving force and double axial driving force. The driving force is 45 N for each actuator. The maximum displacements in two cases are both more than 10 mm. And the area ratio of the stage is about 2.26% which is more than that of any stage in existing research works. Since the proposed SPXYS is symmetric about the two translational axes, only X- or Y-axis is selected as an example in the following comparisons. Figure 8 shows that the X-axis compliance which is obtained according to equation (18) under the condition of zero Y-actuation is slightly larger than that obtained using the FEA model, with a difference of 4.87%. From Figure 9, we can see that the cross-axis coupling effect (the motion along Y-axis in response to the X-axial actuation and vice versa) of FEA model is

Figure 8. Primary motion in X-axis.

much smaller than that of the theoretical model. This cross-axis coupling effect results in a slight reduction in the primary stiffness. The FEA results reveal that the maximal cross-axis coupling error in the Y-direction is 1.3 mm when the X-axial deformation reaches 10 mm. In addition, the FEA model shows that the cross-axis coupling effect varies nonlinearly with the associated input force. Figure 10 shows that the parasitic rotation angle about X (Y) axis obtained by theoretical model is much larger than that of FEA model because the nonlinearity factors are not taken into account in the theoretical model. The maximum parasitic rotation angle of FEA model is less than 0.132 mrad and the Z-axis maximum positioning error is about 5.28 mm, less than half of the theoretical result.

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Figure 9. Cross-axis coupling in Y-axis under X-axial actuation.

Figure 11. Motion lost percentage.

Figure 10. Parasitic rotation angle.

Figure 12. Actuator isolation.

Lost motion indicates the transmission performance from the actuator to the end-effector which is the key indicator that whether the sensor could be placed at the end of actuators forming a half-closed loop, expressed as the percentage of displacement difference to the actuator’s displacement. Figure 11 shows the comparison of the lost motion in the X-direction. It is observed that the lost motions obtained by FEA model and theoretical model are about 1.1% and 2%, respectively. According to the lost motion of FEA model, the maximum displacement difference between the actuator and the end-effector is about 0.11 mm, which is not accepted in precision fields and needs to be calibrated if taking feedback sensors measuring the actuator displacement. Figure 12 shows the actuator isolation9 expressed in the form of input coupling displacement of FEA and theoretical model: the input coupling displacement of

FEA model is larger than that of theoretical model. The maximum input coupling displacement of FEA model is about 0.225 mm which means the mover of Y-axial actuator shifts 0.225 mm along its non-working direction. In case of the mover contacting with the stator of the actuator, the clearance between the mover and the stator should be larger than the maximum input coupling displacement. The deviation between the theoretical model and finite element model (FEM) in terms of compliance (the reciprocal of stiffness) may be caused by the improper assumption about the stress–strain state.32 The theoretical model is based on the planar stress assumption which may be not proper because the leaf springs in the SPLCM are usually under spatial loads. According to Hao’s analysis in Hao et al.32 and the specific parameters of the leaf spring, the difference between the forces under no planar assumption and

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Figure 13. First three mode shapes of the stage: (a) first mode shape, (b) second mode shape, and (c) third mode shape.

planar stress assumption may be about 6%–8%, which is close to the compliance error between the theoretical model and FEM. The relative large deviation between the theoretical model and FEM in terms of cross-axis coupling, parasitic rotation angle, lost motion, and actuator isolation may be caused by the lack of nonlinear factor in the stiffness modeling, which is partially confirmed by the obvious nonlinear results of the cross-axis coupling and parasitic rotation angle obtained by FEM. Hence, in order to precisely predict the characteristics of the SPXYS, the proper stress–strain assumption, as well as nonlinear factor such as elastokinematic effects, should be taken into account in the future research.

Table 3. Resonant frequencies of the stage. Mode

Theoretical (Hz)

FEA (Hz)

Error (%)

1 2 3

34.2 34.2 62.0

34.471 34.300 54.356

0.79 0.29 14.06

Modal analysis comparison The comparison of FEA results by ANSYS software and theoretical results of the first three natural frequencies are shown in Table 3. Figure 13 shows the first three mode shapes. The first and second mode shapes are translation along two

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Figure 14. Out-of-plane payload analysis: (a) the curve of Z-axis deformation of the moving platform, (b) the curve of maximum stress, (c) the maximum stress, and (d) the deformation of the moving platform under 24-kg out-of-plane payloads and the driving force of 90 N in each translational axis.

working axes, respectively. The translational modes of theoretical results are confirmed by FEA within the error of 1%. The third mode shape is a rotation about the Z-axis. As shown in the table, the error between the theoretical and FEA result is 14.06%, which may be caused by the estimation error of the equivalent rotation moment Jz in equation (26). The third resonant frequency obtained by FEA is relatively close to the second resonant frequency which should be carefully taken into account in the positioning control.

Out-of-plane payload analysis The out-of-plane payload of the stage is assessed by the nonlinear buckling analysis in ANSYS. The nonlinear buckling analysis is conducted under the constant driving forces of 90 N in the two translational axes and a

vertical force ranging from 0 to 800 N applied on the moving platform in the Z-axis direction. The critical buckling load is about 565 N as the Z-axial deformation dramatically increases, as shown in Figure 14(a). However, the critical buckling load is obtained without taking the maximum allowed stress into consideration. Figure 14(b) shows the maximum stress curve under different out-of-plane payloads. It can be concluded from the maximum stress curve that the SPLCM would yield first at the out-of-plane of 345 N before buckling as the increment of out-of-plane payload. Therefore, the maximum out-of-plane payload should be decided by the yield strength rather than the critical buckling load. If the maximum allowed stress is decided at 404 MPa, that is, the safe factor is set as 1.25, the out-

16 of-plane payload cannot exceed 235.2 N (24 kg) as shown in Figure 14(c). At this point, the deformations in the two translational axes shown in Figure 14(d) are still larger than 10 mm. Hence, the SPXYS is capable of sustaining a relative large out-of-plane payload of about 24 kg without decreasing the travel range.

Conclusion A novel monolithic SPXYS with large travel, compact size, and high out-of-plane payload capacity has been proposed. The theoretical model has been derived, and FEA model is simulated in ANSYS. It can be concluded from both the theoretical and FEA results in the example case that the SPXYS has a large range of motion up to 20 3 20 mm2 with a relatively compact structure only 133 3 133 mm2, which makes the area ratio of the stage about 2.26% higher than that of any existing stage reported in the literature. The stage exhibits a relatively high precision including well-constraint cross-axis decoupling displacement less than 1.5 mm, the maximum parasitic rotation angle about the X-axis/ Y-axis about 0.132 mrad, and acceptable lost motion below 1.1% of the primary motion. The stage also shows relative good dynamic characteristics with the resonant frequencies of about 34 Hz along both X-axis and Y-axis, and a capability of sustaining a large outof-plane payload of about 24 kg without decreasing the travel range, which makes it superior in space-limited precision applications. Further investigations of the SPXYS would be carried out including nonlinear modeling, parametric sensitivity, and experiment verification. Acknowledgements The authors would like to thank the reviewers for their excellent comments and suggestions.

Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (grant nos 51135009, 61405256, and 51475467).

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