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Mechanism and Machine Theory 44 (2009) 2127–2152

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

Modeling and performance evaluation of a ﬂexure-based XY parallel micromanipulator Yangmin Li *, Qingsong Xu Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira S.J., Taipa, Macao SAR, PR China

a r t i c l e

i n f o

Article history: Received 16 January 2009 Received in revised form 17 April 2009 Accepted 6 June 2009 Available online 21 July 2009

Keywords: Micro-positioning stages Parallel mechanisms Flexure hinges Lumped model Nonlinear modeling Finite element analysis

a b s t r a c t This paper presents the modeling and evaluation of a nearly uncoupled XY micromanipulator designed for micro-positioning uses. The manipulator is featured with monolithic parallel-kinematic architecture, ﬂexure hinge-based joints, and piezoelectric actuation. Its performances in terms of parasitic motion, cross-talk, lost motion, workspace, and resonant frequency have been evaluated via analytical approaches. Based on pseudorigid-body (PRB) simpliﬁcation and lumped model methods, the mathematical models for the kinematics and dynamics of the XY stage have been derived in closed-forms, which are veriﬁed by resorting to ﬁnite element analysis (FEA). Furthermore, a challenging full nonlinear kinematics model is established, which is based on the deformation of the entire manipulator since the above simpliﬁed models fail to predict its kinematic performances. The effectiveness of the nonlinear model is validated by both FEA and experimental studies on the prototype. Results show that the nonlinear model can predict the manipulator kinematics accurately, and the reason why simpliﬁed models fail is discovered. The established analytical models are helpful for both a reliable architecture optimization and performance improvement of the XY micromanipulator. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction As the increasing of activities around the research and development in micro- and nano-scales technology, micromanipulators with ultrahigh precision play more and more important roles in such applications as biological cell manipulation, optical ﬁbers alignment, micro component assembly, and scanning probe microscope (e.g., atomic force microscope (AFM)), etc. Due to a parallel mechanism usually owns a greater stiffness and lower inertia than a serial one [1–3], it enables the generation of higher stiffness and bandwidth of the servomechanism. Besides, ﬂexure hinges have dominant superiority over traditional mechanical joints in precision engineering since ﬂexure hinges endow a mechanism with such merits as no clearance and backlash, no friction and lubrication, vacuum compatibility and so on [4]. In this sense, the micromanipulator is preferred to be designed based on a parallel-kinematic structure composing of ﬂexure hinge-based joints [5–7], which is called a compliant parallel micromanipulator (CPM) in this paper. In the literature, a great number of CPMs have been designed and investigated for micro- or nano-positioning applications [8–11]. Particularly, in recent works dedicated to the investigations on XY micromanipulators, to name just a few, the concept design of an XY stage with linear motor actuation is presented in [12], which has a large workspace yet a coupled motion in the two axes. By employing a parallelogram mechanism, lever mechanisms and piezoelectric actuators (PZT), a ﬂexure XY stage with 30 lm displacement is designed in [13] for multiprobe nanorecording applications. The design * Corresponding author. Tel.: +853 83974464; fax: +853 28838314. E-mail address: [email protected] (Y. Li). 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2009.06.002

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methodology for constraint-based XY ﬂexure stages is proposed in [14,15] with the use of double parallelogram leaf ﬂexures to obtain uncoupled motion of the stage, where the mechanism occupies a complex architecture complicating the fabrication and fully modeling of the manipulator. Beside, the design of a totally decoupled XY CPM with both input and output decoupling properties is presented in [16]. Among previous works of the authors, a simple XY stage is proposed in [17] for the nanomanipulation application. Although a similar structure is then implemented on a silicon-on-insulator (SOI) wafer as a MEMS (micro-electro-mechanical system) stage in [18], the stage may undergoes vulnerabilities due to the stress stiffening and large parasitic motions. Hence, the CPM architecture has been recently improved and proposed in the preliminary works [19], where the architectural parameters of the stage are optimized to deliver minimal parasitic motions subject to performance constraints on actuation stiffness and resonant frequency, etc. Moreover, sine the improved XY CPM makes a compromise between the structure complexity and motion decoupling, it beneﬁts both the control scheme implementation and prototype fabrication in practice. The major objective of the current research is to establish efﬁcient models for the CPM so as to evaluate its performances in terms of kinematics and dynamics accurately, which will beneﬁt a more reliable optimal design of the CPM with respect to its performance speciﬁcations. As far as modeling methods for a CPM is concerned, the most frequently used technique is based upon various simpliﬁcation approaches. For example, the effective pseudo-rigid-body (PRB) [20] concept can simplify the design and modeling of compliant mechanisms greatly. In PRB model, a CPM is converted into an analogous rigid-body mechanism where each ﬂexure hinge is replaced by a one-degree-of-freedom (DOF) revolute joint with a torsional spring attached, while the remaining elements are treated as rigid bodies. Through this approach, the CPM can be modeled using the well-established rigid-body modeling techniques. For instance, based on the RPB model, a simple and efﬁcient modeling method has been proposed in [21] to derive the kinematics and dynamics models of a planar CPM in linear forms. Moreover, in order to establish a more accurate model of the CPM, the ﬂexure hinge can be treated as a multi-DOF joint instead, which is termed a lumped model of the CPM in the current research. It follows that, in addition to its rotational axis direction, the stiffness/compliance of a ﬂexure hinge in other directions are also desirable to obtain a lumped model [22]. Since the simpliﬁed models are sufﬁcient to predict most performances of CPMs, seldom works can be found on the full nonlinear modeling of the CPM. In this paper, it will be shown that although the dynamics property of the CPM can be adequately evaluated by resorting to PRB or lumped model, the kinematics performances can not be predicted by the simpliﬁed approaches properly, since only the compliances of ﬂexure hinges are taken into account. Instead, a full nonlinear modeling with the consideration of internal force stored in the CPM is attempted to derive the relationships between the input and output motions. Moreover, an insight into the nonlinear modeling results allows the deduction of a linear kinematics model of the CPM, which is validated by both the ﬁnite element analysis (FEA) and experimental investigations. In the rest of the paper, after a short description of the XY CPM architecture in Section 2, the PRB and lumped models are, respectively established in Sections 3 and 4 to evaluate the kinematics and dynamics characteristics of the CPM. The FEA simulation carried out in Section 5 via the ANSYS software package reveals that the two models can assess the dynamics characteristics of the CPM with small degrees of derivations, whereas neither of the two models can be used to predict the kinematics properties effectively. So, a nonlinear model is established in Section 6 to characterize the input and output relationship of the CPM motions. And then, the predications have been validated by experimental studies conducted in Section 7 where some discussions on the results and future works are also included. Finally, Section 8 concludes this research.

Fixing hole

Mobile platform

l2

y

l1

l4

x

m0

m4 PZT

m2

Limb 1

m5 Limb 2

m3

m2 m1

Fig. 1. Schematic diagram of the XY CPM.

a

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2. Architecture description of an XY CPM A schematic diagram of the designed XY CPM stage is illustrated in Fig. 1. As can be observed that, the stage employs ﬂexure hinges at all joints and consists of a mobile platform and two limbs with identical kinematic structure. Each limb consists of an outer parallelogram including four ﬂexure revolute (R) hinges (see Fig. 2) or two R–R legs, and a ﬂexure prismatic (P) joint in sequence. The P joint within each limb is ﬁxed at the base via three ﬁxing screws and actuated by a PZT interfacing with a coupling hinge as shown in Fig. 3. In addition, the P joint is composed of ﬁve R hinges, where an inner parallelogram is constructed to guide a linear output displacement of the P joint. The monolithic structure of the CPM beneﬁts a relatively easy manufacturing process. The major shortcoming of the ﬂexure hinge attributes to the center-shift and stress concentration phenomena, which lead to a degraded accuracy and fatigue risk of the CPM, respectively. Thus, the hinge with right-circular shape is adopted since it possesses the smallest magnitude of the center-shift value compared to other types, and the materials with higher ratio of yield strength to Young’s modulus (such as Ti alloy, Al alloy, stainless steal, etc.) can be selected to built the stage. Additionally, to overcome the short travel range of the PZT, a lever mechanism is adopted in the P joint (see Fig. 3) to amplify the output displacement of the PZT. To elaborate the working principle of the XY CPM, let us assume that limb 1 is driven by the PZT with a displacement q1 whereas the limb 2 remains un-driven. Under such case, the input q1 will be ampliﬁed by the lever in the P joint, which produces an ampliﬁed displacement d1 as shown in Fig. 3. Then, d1 is transmitted to the mobile platform through the two parallel R–R legs (along the longitudinal direction) in limb 1. The pure translation xp of the mobile platform along the x-axis is expected to be guaranteed by the two R–R legs which construct a parallelogram in limb 2. Due to a symmetric structure of the CPM, it can be deduced that the input displacement q2 will be ampliﬁed as d2 and then transferred to the output platform ðyp Þ along the y direction accordingly. However, due to the compliance of the ﬂexure mechanism, xp may be not equal to d1 . The difference between them is reﬂected as the lost motion of the structure, and the value of xp =q1 represents the ampliﬁcation ratio of the lever mechanism dedicating to the CPM. Meanwhile, the translation dy of the mobile platform along the y-axis direction (in case of q2 ¼ 0) is exhibited as the cross-talk error between the working axes. In the following discussions, the detailed performances of the CPM are evaluated based on the PRB, lumped, and nonlinear models, respectively.

y z

x

α r t

h

w

Fig. 2. The right-circular ﬂexure R hinge.

l2

Actuator coupling

Actuation direction Lever arm Lever

Fulcrum pivot

Passive leg

l1

q1

θ1

y x

y

z

x

e1 d1

Fig. 3. Actuation P joint parameters and its PRB model. For the P joint, the input displacement q1 is generated by a linear actuator and the output displacement d1 is provided by the lever mechanism.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152 R

k8

R

k4

R

R

k2

R

k9

k1 y

q1

R

k5

R

x

k3

d1 R

R

k6

k7 R

k7

q2 R

k3

R

k5

d2

R

k6

R

R

k9

R

k2

R

k4

R

k8

k1

Fig. 4. PRB model of the XY CPM.

3. PRB model simpliﬁcation approach By replacing each ﬂexure hinge with a one-DOF revolute joint with a torsional spring while considering the remainder elements as rigid bodies, the PRB model of the CPM is established as shown in Fig. 4. We can observe that the XY stage consists of eighteen ﬂexure hinges, which are designed to have the identical dimensions as elaborated in Fig. 2. Due to the symmetry of the stage, the nine hinges within limb 2 is numbered corresponding to those associated with limb 1. With reference to Fig. 4, we can observe that when the CPM is driven by a PZT inscribed in limb 1 with a displacement q1 , the parallelogram composing of revolute joints 2–5 will constrain a linear output d1 of the P joint. As a consequence, the ﬁve hinges (1–5) will rotate with the identical angle

h1 ¼

d1 l1

ð1Þ

The induced displacement d1 is then passed longitudinally to the mobile platform via the two R–R legs (6–7 and 8–9) within limb 1, which causes a rotation angle

b2 ¼

xp a

ð2Þ

to the four ﬂexure hinges of the two legs within in limb 2 accordingly as shown in Fig. 5. In view of the PRB model of the actuation P joint shown in Fig. 3, we can see that the ampliﬁcation ratio of the adopted lever mechanism is A ¼ l1 =l2 , which indicates that

d ¼ Aq

ð3Þ

where d ¼ ½d1 d2 T and q ¼ ½q1 q2 T denote the displacement vectors for the actuation P joints and linear actuators, respectively. Let R ki represent the rotational stiffness of the ith ﬂexure hinge around the z-axis, where the superscript R corresponds to the rotation. The stiffness equation of the right-circular ﬂexure hinge has been previously derived in [23], and an approximate stiffness model for the hinge can be expressed in terms of its parameters as: R

ki ¼ R k ¼

2Ewt5=2 9pr 1=2

ð4Þ

for i ¼ 1; 2; . . . ; 9. It has been shown in [24] that the accuracy of this equation is within 10% over a wide range of 0:05 6 t=r 6 0:5.

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t1

β1

R8

R9

δy

e1

xp

R6 R7

d1

t2

R9

R7

β2

R6

R8

Fig. 5. Parasitic motion illustration based on PRB model.

3.1. Parasitic motion analysis Let p ¼ ½xp yp T represent the displacement vector for the mobile platform. If the CPM is taken as a decoupled manipulator, once it is actuated by the ﬁrst PZT with an input displacement q1 only, the output motion of the mobile platform along the xaxis will cause a small parasitic motion dy in the other direction and hence a cross-talk error between the two working axes. With reference to Fig. 3, we can see that the parasitic motion caused by the parallelogram defection of the ﬁrst P joint can be calculated by

e1 ¼ l1

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 l1 d1

ð5Þ

At the same time, by referring to Fig. 5, the parasitic motion due to the parallelogram of the two legs in limb 2 can be derived in terms of the output motion xp :

t2 ¼ a

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 x2p

ð6Þ

In addition, the corresponding displacement in x direction due to the rotation of the two R–R legs in limb 1 can be derived as

t1 ¼ a

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 ðe1 þ t 2 Þ2

ð7Þ

Thus, the output motion x can be expressed by the following equation:

xp ¼ d1 t1 ¼ d1 a þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 a2 e1 þ a a2 x2p

ð8Þ

which allows the generation of the variable xp in terms of the input variable q1 . Then, the parasitic motion dy can be solved by

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dy ¼ a a2 x2p

ð9Þ

For an XY CPM with parameters shown in Table 1, with the input displacement q1 increasing from 50 to 1000 lm, the output and parasitic motions predicted by the above derivation are shown in Fig. 6. As can be observed that the ratios of output to input displacements are identical to 4, i.e., the ampliﬁcation ratio of the lever mechanism, which means a zero lost motion for both the lever mechanism and CPM system. Additionally, in case of q1 ¼ 100 lm, the output motion is xp ¼ 400 lm with a corresponding dy ¼ 0:53 lm, which indicates a cross-talk of CPRB ¼ 0:13% between the two axes. 3.2. Workspace determination The CPM workspace can be determined by considering the travel range of PZT and rotation limit of ﬂexure hinges at the same time. Assume that the PZT stroke is Q, i.e., qi 2 ½0; Q . In view of the zero lost motion for the CPM, the CPM workspace

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Table 1 Main parameters of the XY CPM. a

b

l1

Architectural parameters (mm) 152.0 98.5

152.0

l2

l4

38.0

195.0

r

t

h

2.5

1.0

10.0

w 12.7

Young’s modulus

Yield strength

Poisson’s ratio

Density

Material parameters 71.7 GPa

503 MPa

0.33

2:81 103 kg=m3

Output displacement x (μm) p

4000 3000 2000 1000 0 0

200

400

600

800

1000

800

1000

Parasitic displacement δy (μm) 0 −20 −40 −60 0

200

400 600 Input displacement q (μm) 1

Fig. 6. Output and parasitic motions versus input displacement based on PRB model.

can be theoretically calculated as V ¼ AQ AQ provided that the stresses caused by the ﬂexure hinge rotations remain within the yield strength range of the material. It can be deduced that for a notch hinge bearing a pure bending moment around its working direction, the maximum angular displacement amax arises when the maximum stress rmax , which occurs at the outermost surface of the thinnest portion of the hinge, reaches to the allowed stress ra which is less than the yield strength ry due to a predeﬁned safety factor na 2 ð1; 1Þ, i.e.,

rmax ¼ ra ¼ ry =na

ð10Þ

Referring to [25], the relationship between the maximum stress and maximum rotation of the ﬂexure hinge can be calculated as:

rmax ¼

Eð1 þ bÞ9=20 b2 f ðbÞ

amax

ð11Þ

where b ¼ 2rt is a dimensionless geometry factor with a valid range of 0 < b < 2:3, and f ðbÞ is a dimensionless compliance factor deﬁned by

f ðbÞ ¼

1

"

3 þ 4b þ 2b2

2b þ b2 ð1 þ bÞð2b þ b2 Þ

þ

6ð1 þ bÞ ð2b þ b2 Þ3=2

tan1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# 2þb b

ð12Þ

Once the PZT for driving the CPM is selected, its stroke Q is a constant. Substituting the rotation angles h1 and b2 with d1 ¼ Q into (11) allows the generation of the relationship:

infðl1 ; aÞ P

Eð1 þ bÞ9=20 na Q b2 f ðbÞry

which provides a guideline for the design of the CPM dimensions without the risk of inelastic deformation.

ð13Þ

Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

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3.3. Dynamic modeling In order to evaluate the dynamic property of the CPM, the Lagrange analysis is conducted below. Based on the simpliﬁed PRB model of the CPM, the kinematic energy equation can be derived as

T¼

! A2 A2 A2 1 1 4A2 m0 þ m1 þ m2 þ m3 þ m4 þ m5 q_ 21 þ q_ 22 6 2 2 2 3 3

ð14Þ

which is obtained with the consideration of relationships expressed by Eqs. (1)–(3). In addition, the equation for potential energy can be expressed in terms of qi :

V¼

! 2A2 R k 2 q1 þ q22 þ 2 2 a 2l2

5R k

ð15Þ

Then, the dynamic equation of the CPM can be generated through the Lagrange equation [26]:

d @T @T @V þ ¼ Fi dt @ q_ i @qi @qi

ð16Þ

where F i is the ith actuation force, for i ¼ 1 and 2. Substituting Eqs. (14) and (15) into Eq. (16) allows the derivation of dynamic equation describing a free motion of the CPM:

€ þ Kq ¼ 0 Mq

ð17Þ

where the two 2 2 diagonal matrices are

2 8 1 M ¼ diagfMg ¼ diag A2 m0 þ m1 þ m2 þ m5 þ m3 þ m4 3 3 3 ! ) ( 2 5 4A R þ 2 k K ¼ diagfKg ¼ diag 2 a l2

ð18Þ ð19Þ

which allows the calculation of the natural frequency based on PRB model:

1 f ¼ 2p

rﬃﬃﬃﬃﬃ K M

ð20Þ

For an XY CPM with parameters described in Table 1, the natural frequency assessed by PRB model can be calculated as f PRB ¼ 27:57 Hz. 4. Lumped model-based analysis The above PRB model approach with the consideration of only rotational stiffness provides a preliminary glance of kinematics and dynamics performances for the XY stage. A more detailed analysis along with the longitudinal stiffness taken into account can be employed to evaluate the XY performances in a more complete way, which also allows the generation of higher modes and mode shapes of the CPM. In this section, each ﬂexure hinge is treated as a two-DOF joint with both rotational and longitudinal stiffness to construct the lumped model of the XY stage. The mass and spring diagram is shown in Fig. 7, where ki denotes the longitudinal stiffness of the ﬂexure hinge. Whereas for the reason of clarity, the rotational stiffness R ki is omitted from the illustration. It has been derived that the longitudinal stiffness of the hinge along the y-axis (see Fig. 2) can be expressed as [23]:

2

, ki ¼ k ¼ Ew

6 2ð1 þ bÞ 4qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tan1 2b þ b2

3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ b p7 5 b 2

ð21Þ

with b ¼ 2rt for i ¼ 1 to 9 in two limbs of the CPM, which is accurate for 0:05 6 t=r 6 0:65 within 6% error [24]. 4.1. Dynamic analysis Referring to Fig. 7, in view of the linear displacements u1 and u2 (v 1 and v 2 ) associated with the actuation P joint within limb 1 (limb 2), the rotational angle can be expressed as

h1 ¼

u1 u2 d1 ¼ l2 l1

ð22Þ

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

k8

k4

u2

u1

k1

k5

k9

k2

y

q1

x

k3 d1

k6

u3

k7

k9

k7

q2 v1 d2

v3

k3 k5

k6

k2

k1

v2

k4

k8

Fig. 7. Lumped model of the XY CPM.

which allows the derivation of

d1 ¼ Aðu1 u2 Þ

ð23Þ

The platform displacement xp is different from d1 mainly due to the longitudinal stiffness k6 to k9 within limb 1. Additionally, taking into account the symmetry property, the following equation can be deduced:

u3 ¼

1 1 ðxp þ d1 Þ ¼ ½xp þ Aðu1 u2 Þ 2 2

ð24Þ

Similarly, we have

v3 ¼

1 y þ Aðv 1 v 2 Þ 2 p

ð25Þ

for the limb 2. Then, the kinetic energy equation for the XY CPM can be obtained in terms of q_ i ; u_ i ; v_ i and p_ as:

! 1 A2 A2 1 1 1 1 1 2 2 T ¼ m4 ðq_ 1 u_ 1 Þ þ m0 þ m5 x_ 2p m1 þ m2 þ m3 ðu_ 1 u_ 2 Þ2 þ ðm2 þ m3 Þu_ 22 þ m5 ½x_ þ Aðu_ 1 u_ 2 Þ þ 2 6 2 4 2 3 2 3 ! 2 2 1 A A 1 1 1 1 1 þ m4 ðq_ 2 v 1 Þ2 þ m0 þ m5 y_ 2p m1 þ m2 þ m3 ðv_ 1 v_ 2 Þ2 þ ðm2 þ m3 Þv_ 22 þ m5 ½y_ þ Aðv_ 1 v_ 2 Þ2 þ 2 6 2 4 2 3 2 3 ð26Þ which is derived by taking into account of Eqs. (2), (22), (23) and (24). Next, the potential energy equation can be derived as

V¼

! ! 1 5A2 R k 1 2 2R k 2 1 5A2 R k 2 2 þ 2k ðu u Þ þ þ x þ v Þ þ þ 2k ðv 1 v 2 Þ2 kðq1 u1 Þ2 þ ku kðq 1 2 1 2 2 2 2 2 2 a2 p 2 2l1 2l1 1 2 2R k þ kv 2 þ 2 y2p 2 a

ð27Þ

Again, substituting the kinetic and potential energies (T and V) into Lagrange’s equation as shown in Eq. (16) gives the equations of free motion in the form:

€ þ Kx ¼ 0 Mx where x ¼ ½q1 q2 u1 dix A.

ð28Þ

v1

u2

v2

xp yp T is the vector of generalized coordinates, and matrices M and K are presented in Appen-

Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

2135

Based on the theory of vibrations, the following modal equation describing free vibration of the system can be obtained:

ðK kj MÞUj ¼ 0

ð29Þ

for j ¼ 1 to 8. The condition of non-zero solutions for Eq. (29) can be derived by

jK kj Mj ¼ 0

ð30Þ

which allows the calculation of the eigenvalues, i.e., kj ¼ x2j with xj denoting the natural cyclic frequency of the system. Afterwards, the natural frequency can be computed as fj ¼ 21p xj . In addition, by substituting each eigenvalue kj into Eq. (29), the eigenvector Uj denoting the j-th mode shape of the system can be generated as well. And then, the modal matrix can be obtained as

U ¼ ½U1 U2 ; . . . ; U8

ð31Þ

As a case study, the modal analysis of the XY CPM with parameters described in Table 1 is carried out based on the above dynamic equation. The calculation results show that there are four pairs of natural frequencies due to the symmetry architecture of the CPM. And the lowest frequency can be predicted by the lumped model as f Lump ¼ 25:23 Hz. 4.2. Lost motion analysis Due to the compliances of ﬂexure hinges, the input displacement can not be totally transmitted to the output end of the lever mechanism and the output platform of the CPM accordingly. The lost motion of lever refers to the fractional loss of motion of the lever mechanism arising from the difference between the ideal and practical distortion of the lever arms. Likewise, the lost motion of the CPM can be deﬁned as the fractional loss of motion of the CPM system arising from the difference between the ideal and practical distortion of the whole ﬂexure system. Concerning the XY stage studied here, the lost motions come from the longitudinal stiffness of hinges associated with the actuator coupling ðK 1 ¼ k1 Þ, fulcrum pivot ðK 2 ¼ k2 Þ, and the ﬂexures ðK e Þ connecting the mobile platform to the ground. When the CPM is driven by the PZT within limb 1 only, the mass and spring lumped model is graphically shown in Fig. 8. Let K Pi and R K Pi denote the stiffness of the ith P joint along its longitudinal and rotational (working) directions, respectively. Similarly, assign K Li and R K Li be the two corresponding types of stiffness concerning the two legs associating with the ith limb. Then, referring to Fig. 8, the equivalent stiffness between the lever and the ground can be derived as:

K e ¼ R K P1 þ

1 K L1

1 þ R K1 þ K1P2

ð32Þ

L2

In view of the identical dimensions for the ﬂexure hinges, the combined longitudinal stiffness of the four hinges can be written as

K P2 ¼ K L1 ¼ k

ð33Þ

And the linear stiffness of the parallelogram consisting of four rotational stiffness along its working direction can be obtained as follows: R

K P1 ¼

4R k 2 l1

;

R

K L2 ¼

4R k a2

ð34Þ

In view of Eqs. (33) and (34), the equivalent stiffness in Eq. (32) can be simpliﬁed as:

Ke ¼

4R k 2 l1

þ

4R kk þ a2 k

ð35Þ

8R k

Then, a relationship between the input and output displacements can be written according to [25]:

d1 A ¼ q1 A2 K e þ A2 K e 1 þ 1 2 þ 1 A K1 K2

ð36Þ

l1 Ke

KP2 R

K L2

xp

K P1 d1 d1

u2 u1

Me K1

PZT

K L1

R

l2

Fig. 8. Lumped model for lost motion derivation.

q1 , F

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Moreover, the relation between the output displacements of the lever and CPM mobile platform can be calculated by:

xp 4R kk 4R kk 4R k þ a2 k ¼ ¼ R 8R k þ a2 k 4R k þ a2 k 8 k þ a2 k d1

ð37Þ

In view of Eqs. (36) and (37), the motion relationship between the CPM mobile platform and input can be described by:

xp Að4R k þ a2 kÞ i ¼h 2 2 2 q1 A KK1e þ A KK2e 1 þ A1 þ 1 ð8R k þ a2 kÞ

ð38Þ

which allows the derivation of an expression of lost motion for the CPM as: x

fcpm ¼

ðAÞ qp

1

A

¼1

h

A2 KK1e

þ

Ke K2

ð4R k þ a2 kÞ i 2 1 þ A1 þ A12 ð8R k þ a2 kÞ

ð39Þ

For the XY CPM as described in Table 1, the fractional loss of motion of the entire CPM system can be assessed to be LLump ¼ 0:20%. It is observed that the lost motion values are very small due to the ideal ratio between K e and K 1 ðK 2 Þ. 4.3. Parasitic motion analysis Let F c represent the force transmitted to limb 2, which brings a displacement xp of the CPM mobile platform accordingly. Based on the lumped model, the parasitic motion is illustrated in Fig. 9. In view of the linear stiffness of limb 2 in the x-axis, the force can be calculated by

K P2 R K L2 4R kk xp x ¼ p 4R k þ a 2 k K P2 þ R K L2

Fa ¼

ð40Þ

which is the output force of the lever mechanism actually. Meanwhile, the displacement v 2 due to the compliance of P joint 2 along its longitudinal direction can be calculated by:

v2 ¼

Fa Fa 4R k ¼ ¼ xp 4R k þ a2 k K P2 k

ð41Þ

Then, the linear displacement w2 of limb 2 along the x-axis can be derived as:

w2 ¼ xp v 2 ¼

a2 k xp 2 þa k

ð42Þ

4R k

δy

xp

w2

R7

R6

t2

R9

R8 v2

Fig. 9. Parasitic motion illustration based on lumped model.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Output displacement x (μm) p

4000 3000 2000 1000 0 0

200

400

600

800

1000

800

1000

Parasitic displacement δy (μm) 0 −20 −40 −60 0

200

400 600 Input displacement q1 (μm)

Fig. 10. Output motions versus input displacement based on lumped model.

Therefore, the parasitic motion can be calculated as follows.

91=2 8 > > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ = < 4 2 2 a k q1 dy ¼ ða a2 w22 Þ ¼ a þ a2 h i 2 > > 2 : A2 KK1e þ KK2e 1 þ A1 þ A12 ð8R k þ a2 kÞ2 ;

ð43Þ

which is expressed as a function of the input displacement q1 only. For the XY CPM as shown in Table 1, as the increasing of the input displacement q1 from 50 to 1000 lm, the output and parasitic motions predicted by the above derivation are shown in Fig. 10. It can be observed that the ratio of output to input displacements is 3.99 which can also be calculated from Eq. (38). In addition, when q1 ¼ 100 lm, the output motion is xp ¼ 399:22 lm with a corresponding dy ¼ 0:52 lm, which indicates a cross-talk of CLump ¼ 0:13% between the two axes. 5. Finite element analysis To verify the accuracy of the derived analytical equations based on the established PRB and lumped models, the FEA by resorting to the ANSYS software package is carried out. In ANSYS, the 20-node element SOLID 186 is used to create the mesh model using a suitable mesh size. And the six ﬁxing holes are constrained to simulate the real situation. In addition, the nonlinear statics analysis is adopted with the consideration of effects due to large deformation and stress stiffening. In the ﬁrst simulation, a displacement q1 ¼ 100 lm is applied on the input end of the ﬁrst P joint, while the input end of the other P joint in the actuation direction is constrained as q2 ¼ 0 lm. Under such case, the deformed shape is shown in Fig. 11. The results show that the output motion of the mobile platform in x direction is 320.4 lm, and the parasitic translation along the y-axis is 17.3 lm, which indicates a maximal cross-talk of 5.4% between the two axes that is larger than the value assessed by the PRB or lumped model. Besides, the output motion also reveals the lost motion of the stage is up to 19.9%, which is far larger than the one predicted by the lumped model. Then, the modal analysis is undertaken in the second simulation. The simulation results exhibit that the ﬁrst two natural frequencies are very close, and the ﬁrst value is 23.83 Hz. The results obtained by the PRB model, lumped model, and FEA simulation are elaborated in Table 2. It should be noted that the natural frequency of the designed CPM is relatively low for some applications such as the scanning stage in an AFM, which can be enhanced by further miniaturizing the CPM structure to reduce its mass or increasing the stiffness of the CPM. If we take the FEA result as the ‘‘true” value, by comparing the natural frequency predicted by the PRB and lumped models with the FEA result, we can see that the calculated frequencies are all higher with a derivation of 15.7% and 5.9%, respectively. Clearly, the lumped model is more accurate in dynamic analysis of the CPM. Nevertheless, although the dynamic property of the CPM can be predicted by the lumped model appropriately, the kinematics characteristics involving lost motion and parasitic motion can not be properly evaluated based on neither PRB nor lumped model, which is the motivation of nonlinear modeling performed in the subsequent section.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Fig. 11. Finite element model of the XY stage.

Table 2 Performances of an XY CPM assessed by PRB model, lumped model, and FEA. Ampliﬁcation ratio

Lost motion

Cross-talk

Frequency (Hz)

PRB Lumped FEA

4.00 3.99 3.20

0% 0.20% 19.90%

0.13% 0.13% 5.40%

27.57 25.53 23.83

Deviation/difference of PRB model to FEA results (%) Deviation/difference of lumped model to FEA results (%)

25.0 24.7

19.9 19.7

5.3 5.3

15.7 5.9

6. Nonlinear modeling For the sake of assessing the output and parasitic motions of the CPM accurately, the nonlinear modeling with the consideration of the elasticity of the whole CPM is the concentration of this section. The goal of nonlinear modeling is to establish the kinematic relations between the output motions (X; Y, and H) and input variables (q1 and q2 ). In the above evaluation of kinematics performances based on the PRB model, lumped model, and FEA simulation, it is assumed that the CPM is driven by PZT 1 with a displacement q1 ðq1 2 ½0; Q Þ while the actuation variable for PZT 2 remains q2 ¼ 0. This assumption beneﬁts the parasitic motion analysis for the CPM. In the nonlinear modeling of the CPM, this assumption is still adopted for the convenience of the modeling process. Furthermore, the corresponding input force F 1 exerted on limb 1 and internal forces and moments undergone by each ﬂexure element are taken into account in the nonlinear modeling. It will be shown that the relationships between the output and input motions of the CPM can be derived through such a case study. By inspecting the internal forces (conducted in the following discussions) within the entire CPM, we can deduce that in addition to the output displacement xp and parasitic translation dy, the mobile platform is prone to rotate clockwise with a parasitic angle of hp in the x–y plane so as to arrive at an equilibrium state. Fig. 12 illustrates the initial and deformed shape of the CPM. Now, the objective of the nonlinear modeling under such case is to seek for the equations predicting the output motions (xp ; dy, and hp ) once given the input displacement q1 . 6.1. Nonlinear modeling of output motions with input variable q1 6.1.1. Expression of output force F a of the P joint Let 2F a represent the output force of the P joint, which is transmitted to limb 2 and then brings a displacement xp of the CPM mobile platform accordingly. Due to the compliance of the actuation P joint 2 along the x direction, the induced linear displacement can be calculated by

v2 ¼

2F a 2F a ¼ K P2 k

ð44Þ

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

With reference to Fig. 12, we can derive an expression for the output motion xp of the platform:

xp ¼

2F a þ w2 cos hp þ b sin hp k

ð45Þ

where b is the length of the mobile platform, and w2 represents the linear displacement of the two legs within limb 2 which will be solved in the subsequent analysis. Since w2 is directly related to the R–R legs within limb 2, the free body diagram of one leg is depicted in Fig. 13a. At the equilibrium state, the equation of moments at the point O can be written as:

X

mO ¼ M R F a ða þ 2rÞ þ ML ¼ 0

ð46Þ

which gives the moment at point O as follows.

MR ¼ F a ða þ 2rÞ ML

ð47Þ

where ML is the internal moment constraining a zero angular deﬂection at the end point A. From the bending theory of materials mechanics, the bending equation of the ﬂexure element can be written as: 2

d y 2

dx

¼

MðxÞ EIðxÞ

ð48Þ

where E denotes the Young’s modulus of the material, MðxÞ is the moment exerted on a smaller length of dx at position x, and IðxÞ is the moment of inertia of the corresponding cross-section. It follows that the angular deﬂection with respect to the neutral axis can be derived as:

h¼

dy ¼ dx

Z

2

d y 2

dx

dx ¼

Z

MðxÞ dx EIðxÞ

ð49Þ

According to the shape of the cross-section, one leg of limb 2 is divided into segments I, II and III as shown in Fig. 13a, where the height of a cross-section at the position x can be, respectively expressed as:

h1 ðuÞ ¼ 2r þ t 2r cos u; h2 ðxÞ ¼ h;

for

p 2

6u6

p

ð50aÞ

2

for 2r 6 x 6 a

h3 ð/Þ ¼ 2r þ t 2r cos /;

ð50bÞ

for

p 2

6/6

p

ð50cÞ

2

where in segments I and III, the variable x is replaced by x ¼ r þ r sin u and x ¼ a þ r þ r sin /, respectively, for the convenience of calculation.

R8

β1

R9

d1

xp

t2

δy

e1 + u2

R6

θp b R7 w2 R7

R9

θp

R6

β2

R8

v2

Fig. 12. Parasitic motion illustration based on nonlinear model.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Rx = F1

Fx

ML

Fa

A

2r

Ry = Fc O

φ

III

l2 (V )

O

y MO

M1

y

ϕ

h I 2r + 2 II

F1

x 2r + a

2r + l1

x

x

O

Ry = Fa

Rx

MR

y

2r

ϕ

I

y

D F1

M1

III

φ

IV

A

Fc

O

MC

(b)

(a) Rx

Ry = F2 O

x

II

O

y

MO

2r

ϕ

I

y

y

VI h / 2

A

MA

x

B

MB

x

D F1

M1

II

y

ϕ

VII 2r

RBx

x

2r + l1

RBx = F1 B

RAx = F1 RAy

x

M B = M1

l2 (V )

(d) y Rx

M2

2r

Fx

(c)

y

Fy R9

II

I

III

y

(e) x

P

Fy1

Fy b

FE

ME E II

R7

R9

ML

Fy1

y

MR

R6

R7

Fa

x

θP

l4

M3

φ

O

M3

l3

2r

ϕ

Fa

(f)

Fa

A

2r + a

F2

Fa

x

M4 R = F y y

φ

III

A

Fy

M3

O

Fa

R8

MR

x I

Fy 2

l4

ML Fy 2

(g)

Fig. 13. Free body diagrams of: (a) one R–R leg in limb 2, (b) the lever, (c) the passive leg in lever, (d) the lever arm, (e) one R–R leg in limb 1, (f) the mobile platform, and (g) the connecting nod within limb 2.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

The corresponding moments in these three segments can be generated as:

M1 ðuÞ ¼ F a ðr sin u a rÞ þ M L ; M2 ðxÞ ¼ F a ðx a 2rÞ þ ML ; M3 ð/Þ ¼ F a ðr sin / rÞ þ M L ;

p

p

6u6 2 2 for 2r 6 x 6 a for

for

p 2

6/6

ð51aÞ ð51bÞ

p

ð51cÞ

2

Hence, according to Eq. (49), the angular deformations can be obtained as follows.

Z h1 ðuÞ ¼ Z h2 ðxÞ ¼ Z h3 ð/Þ ¼

M1 ðuÞ r cos u du þ C 1 EI1 ðuÞ M 2 ðxÞ dx þ C 2 EI2 ðxÞ M 3 ð/Þ r cos / d/ þ C 3 EI3 ð/Þ

ð52aÞ ð52bÞ ð52cÞ

In view of the zero angular deﬂection at the end of segment III, we have the equation:

hða þ 2rÞ ¼

Z

p

12½F a ðr sin u a rÞ þ M L

2

Ewð2r þ t 2r cos uÞ

p 2

þ

Z

p 2

p2

3

r cos u du þ

Z

a

2r

12½F a ðx a 2rÞ þ M L 3

Ewh

12½F a ðr sin / rÞ þ ML r cos / d/ ¼ 0 Ewð2r þ t 2r cos /Þ

dx ð53Þ

which allows the generation of an analytical expression for the internal moment ML by resorting to a symbolic computation software such as MATHEMATICA, MAPLE, MATLAB, etc. The result is a function of the variable F a , i.e.,

ML ¼

G Fa H

ð54Þ

where the expressions for G and H are shown in Appendix B, which are both constants in terms of the CPM dimensions. Substituting M L into Eq. (52) along with the consideration of the boundary conditions on angular deformations: h1 p2 ¼ 0; h2 ð2rÞ ¼ h1 p2 , and h3 p2 ¼ 0, we can obtain the three constants of C 1 to C 3 . Then, the linear deformations on the three segments can be written as:

Z y1 ðuÞ ¼ h1 ðuÞr cos u du þ D1 Z y2 ðxÞ ¼ h2 ðxÞdx þ D2 Z y3 ð/Þ ¼ h3 ð/Þr cos / d/ þ D3

ð55aÞ ð55bÞ ð55cÞ

In view of the boundary conditions on the linear deﬂections: y1 p2 ¼ 0; y2 ð2rÞ ¼ y1 p2 , and y3 p2 ¼ y2 ðaÞ, the other three parameters D1 to D3 can be calculated as well. Hence, the overall linear displacement w2 of the leg can be calculated by:

w2 ¼ y3

p 2

þ a sin hp

ð56Þ

which is a function of the unknown F a and hp . Substituting Eq. (56) into Eq. (45) yields

xp ¼

p 2F a þ y3 þ a sin hp cos hp þ b sin hp k 2

ð57Þ

which gives an expression of F a in terms of variables xp and hp , abbreviated as:

F a ¼ f ðxp ; hp Þ

ð58Þ

where f ðÞ denotes a function of variable ‘‘”. 6.1.2. Expression of output motion d1 of the lever Considering that F c stands for the output force of the lever mechanism, the free body diagram of the lever is drawn as shown in Fig. 13(b). The equation of moments at point O can be written as:

X

mO ¼ F 1 l2 Mo F c ðl1 þ 2rÞ M1 M c ¼ 0

which gives the moment at point O, i.e.,

ð59Þ

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Mo ¼ F 1 l2 M 1 M c F c ðl1 þ 2rÞ

ð60Þ

where M 1 and M c are the internal moments constraining a zero angular deﬂection at the input end D of the CPM and the end point A of the lever, respectively. Since the PZT can only produce the actuation force F 1 , the moment M1 is created by the ﬂat tip end of the PZT and interfacing coupling ﬂexure hinge R1. According to the force types and shapes of the cross-sections, one limb of the actuation P joint is divided into segments I– IV as shown in Fig. 13b. The height of cross-sections can be, respectively expressed as:

h1 ðuÞ ¼ 2r þ t 2r cos u;

for

p 2

6u6

p

ð61aÞ

2 h h2 ðxÞ ¼ h; for 2r 6 x 6 2r þ 2 h h3 ðxÞ ¼ h; for 2r þ 6 x 6 l1 2 p p h4 ð/Þ ¼ 2r þ t 2r cos /; for 6 / 6 2 2

ð61bÞ ð61cÞ ð61dÞ

where in segments I and IV, the variable x is replaced by x ¼ r þ r sin u and x ¼ l1 þ r þ r sin /, respectively, for the convenience of calculation. The corresponding moments in these four segments can be derived as:

M1 ðuÞ ¼ F 1 l2 M 1 M c F c ðl1 þ r r sin uÞ;

for

p 2

6u6

p

ð62aÞ

2

h M2 ðxÞ ¼ F 1 l2 M1 Mc F c ðl1 þ 2r xÞ; for 2r 6 x 6 2r þ 2 h M3 ðxÞ ¼ M c F c ðl1 þ 2r xÞ; for 2r þ 6 x 6 l1 2 M4 ð/Þ ¼ M c F c ðr r sin /Þ;

for

p 2

6/6

ð62bÞ ð62cÞ

p

ð62dÞ

2

Next, in view of Eq. (49), the angular deformations can be obtained as:

Z h1 ðuÞ ¼ Z h2 ðxÞ ¼ Z h3 ðxÞ ¼ Z h4 ð/Þ ¼

M1 ðuÞ r cos u du þ C 1 EI1 ðuÞ M 2 ðxÞ dx þ C 2 EI2 ðxÞ M 3 ðxÞ dx þ C 3 EI3 ðxÞ M 4 ð/Þ r cos / d/ þ C 4 EI4 ð/Þ

ð63aÞ ð63bÞ ð63cÞ ð63dÞ

Afterwards, taking into account the zero angular deﬂection at the end of segment IV, we have the equation:

hðl1 þ 2rÞ ¼

Z

p

12½F 1 l2 M1 Mc F c ðl1 þ r r sin uÞ

2

Ewð2r þ t 2r cos uÞ

p2

þ

Z

l1

12½M c þ F c ðl1 þ 2r xÞ

2rþ2h

3

Ewh

3

dx þ

r cos u du þ

Z

2rþ2h

12½F 1 l2 M 1 M c F c ðl1 þ 2r xÞ

2r

Z

p 2

p2

12½Mc þ F c ðr r sin /Þ Ewð2r þ t 2r cos /Þ3

3

Ewh r cos / d/ ¼ 0

dx ð64Þ

which allows the generation of an analytical expression for the internal moment M c in terms of the unknowns F 1 ; M 1 , and F c , i.e.,

Mc ¼ f ðF 1 ; M 1 ; F c Þ

ð65Þ

of boundary conditions on the angular deformations: Then, substituting Eq. (65)into Eq. (63) alongwith theconsideration h1 p2 ¼ 0; h2 ð2rÞ ¼ h1 p2 ; h3 2r þ 2h ¼ h2 2r þ 2h , and h4 p2 ¼ 0, we can obtain the four parameters of C 1 to C 4 . In succession, the linear deformations on the four sections can be written as:

Z y1 ðuÞ ¼ Z y2 ðxÞ ¼ Z y3 ðxÞ ¼ Z y4 ð/Þ ¼

h1 ðuÞr cos u du þ D1

ð66aÞ

h2 ðxÞdx þ D2

ð66bÞ

h3 ðxÞdx þ D3

ð66cÞ

h4 ð/Þr cos / d/ þ D4

ð66dÞ

Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

2143

In view of the boundary conditions on the linear deﬂections: y1 p2 ¼ 0; y2 ð2rÞ ¼ y1 p2 ; y3 2r þ 2h ¼ y2 2r þ 2h , and p y4 2 ¼ y3 ðl1 Þ, the other four parameters of D1 to D4 can also be calculated. Hence, the output motion d1 of the lever can be derived as:

d1 ¼ y4

p 2

¼ f ðF 1 ; M 1 ; F c Þ

ð67Þ

which is a function in terms of F 1 ; M 1 , and F c as well. Furthermore, the free body diagram of the passive leg in the lever is shown in Fig. 13c. Similar to the analysis of the R–R leg in Fig. 13a, a relationship between the internal force F 2 and moment M 2 can be derived as:

M2 ¼

G2 F2 H2

ð68Þ

where the expressions for G2 and H2 are identical to those in Eqs. (B.1) and (B.2) except that a is replaced by l1 instead. Likewise, the linear deﬂection at the end point A can be obtained as: c1 ¼ y3 p2 . Due to the displacement constraint of the parallelogram, we have d1 ¼ c1 , which allows the calculation of F 2 in terms of F 1 ; M 1 , and F c as well. In addition, another expression for the output force of the actuation P joint can be derived as:

2F a ¼ F c F 2 ¼ f ðF 1 ; M1 ; F c Þ

ð69Þ

which is transmitted to the mobile platform through the two parallel R–R legs within limb 1. 6.1.3. Expression of input displacement q1 In what follows, we will concentrate on the calculation of deﬂection u1 at the input end of the lever. As shown in Fig. 13d, the lever arm is denoted by segment V following segments I and II associated with the fulcrum pivot as illustrated in Fig. 13b. The segments VI and VII are related to the actuator coupling hinge. First, the angular deﬂection equation in Section 5 can be written as:

h5 ðxÞ ¼

Z

M 5 ðxÞ dx þ C 5 ¼ EI5 ðxÞ

Z

12F 1 ðl2 xÞ 12M1 3

Ewh

dx þ C 5

ð70Þ

Taking into consideration the boundary constraint: h5 ð0Þ ¼ h2 2r þ 2h , the parameter C 5 in Eq. (70) can be obtained. And then, the linear deﬂection on Section 5 and the corresponding boundary condition can be expressed by:

y5 ðxÞ ¼

Z

h5 ðxÞdx þ D5 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 h h h 2r þ y2 2r þ y5 ð0Þ ¼ 2r þ 2 2 2

ð71Þ ð72Þ

respectively, which enables the generation of the parameter D5 accordingly. Concerning the coupling hinge for the actuator, in segments VI and VII, the height of cross-sections can be respectively expressed as:

h6 ðxÞ ¼ h;

h 2

for 0 6 x 6

h7 ðuÞ ¼ 2r þ t 2r cos u;

ð73aÞ for

p 2

6u6

p 2

ð73bÞ

The corresponding moments in the two sections can be derived as:

M6 ðxÞ ¼ M 1 ;

for 0 6 x 6

M7 ðuÞ ¼ M 1 ;

for

p 2

h 2

ð74aÞ

6u6

p 2

ð74bÞ

Then, the angular deformations can be, respectively obtained as follows.

h6 ðxÞ ¼

Z

h7 ðuÞ ¼

12M1 3

Z Ewh

dx þ C 6 12M 1

Ewð2r þ t 2r cos uÞ3

ð75aÞ r cos u du þ C 7

ð75bÞ

In view of the zero angular deﬂection at the end of segment VII, i.e., the input end D of the CPM, we have

Z h Z p 2 2 12M h 12M 1 1 ¼ h5 ðl2 Þ þ h 2r þ dx þ r cos u d/ ¼ 0 3 3 2 0 p2 Ewð2r þ t 2r cos uÞ Ewh

ð76Þ

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

which allows the generation of an analytical expression for the internal moment M 1 in terms of the variables F 1 and F c , i.e.,

M1 ¼ f ðF 1 ; F c Þ

ð77Þ

Thus, the unknown M 1 in the above procedures has been eliminated. Moreover, in view of Eqs. (58), (69), and (77), the internal force F c can be expressed in terms of F 1 ; xp , and hp , i.e.,

F c ¼ f ðF 1 ; xp ; hp Þ

ð78Þ

Substituting Eqs. (77) and (78) into Eq. (71) allows the generation of linear deﬂection at the end point of segment V as: u1 ¼ y5 ðl2 Þ, which is also a function of the variables F 1 ; xp , and hp as well. Besides, by taking the actuator coupling hinge as a linear stiffness with the longitudinal stiffness of k1 , the input displacement can be derived as follows:

q1 ¼ u1 þ

F1 ¼ f ðF 1 ; xp ; hp Þ k1

ð79Þ

6.1.4. Expression of input force F 1 Referring to Fig. 12, we assume that the directions of two R–R legs in limb 1 remain along the x direction, i.e., b1 ¼ hp . This assumption is made based on the FEA simulation results. The free body diagram of one R–R leg in limb 1 is sketched in Fig. 13e. Following the process for dealing with one leg in limb 2, the relationship between the internal moment and force can be obtained as:

G Fy H

ð80Þ

F y ¼ f ðhp Þ

ð81Þ

M3 ¼

p which is similar to the expression in Eq. (54). In addition, the linear deﬂection at the end of the p leg can be derived as y3 2 , which is a function of the internal force F y only. In view of the above assumption, we have y3 2 ¼ a sin hp , which allows the derivation of the force F y in terms of parasitic angle hp :

Additionally, we can derive the output motion xp of the mobile platform through the two R–R legs in limb 1:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 xp ¼ d1 s1 b b ðb sin hp Þ2

ð82Þ

where s1 comes from the compliance of the two R–R legs along the axial direction and can be expressed as:

s1 ¼

2F a 2F a ¼ K L1 k

ð83Þ

Substituting Eqs. (67) and (83) into Eq. (82) along with the consideration of Eqs. (58), (77), and (78) gives an expression of the three variables F 1 ; xp , and hp . Hence, the actuation force F 1 can be solved as an expression in terms of unknowns xp and hp , i.e.,

F 1 ¼ f ðxp ; hp Þ

ð84Þ

6.1.5. Generation of output and parasitic motions (xp ; dy, and hp ) of the CPM The internal forces exerted on the mobile platform is elaborated in Fig. 13f. In view of the force equilibrium state, an equation for the internal forces can be written as:

F y2 ¼ F y1 þ 2F y

ð85Þ

Additionally, the moment equation at point P can be written as:

X

M p ¼ F y1

l4 l4 þ F y2 þ 2F y b þ 2M3 2F a b 2M L ¼ 0 2 2

ð86Þ

The combination of Eqs. (85) and (86) enables the generation of:

2 2 G ML þ F a b M3 F y b F y ¼ þ b ðF a F y Þ F y l4 l4 H 2 G ¼ þ b ðF a F y Þ þ F y l4 H

F y1 ¼

ð87Þ

F y2

ð88Þ

which are all expressed in terms of F a and F y , and then can be converted as the functions of xp and hp instead if the relations in Eqs. (58) and (81) are considered. Besides, the free body diagram of the connecting rod between the actuation P joint and R–R legs in limb 2 is shown in Fig. 13g. The rotation center of the connecting rod is located at its mid-point E interfacing with the actuation P joint. Due to the constraints of the parallelogram, the rotation angle for the rod is identical to hp .

Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

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With the rod slicing into two segments, we can obtain the moment equation in segment I as follows:

M1 ðxÞ ¼ M R F y2

l4 G 2 G l4 a 2r F a þ b ðF a F y Þ þ F y x ¼ x H l4 H 2 2

ð89Þ

where the relations in Eqs. (47) and (88) are considered. According to Eq. (49), the angular deformation can be written as

h1 ðxÞ ¼

Z

M 1 ðxÞ dx þ C 1 ¼ EI1 ðxÞ

Z

12M1 ðxÞ 3

Ewh

dx þ C 1

ð90Þ

which, with the consideration of zero angular deﬂection at the mid-point, i.e., h1 ð0Þ ¼ 0, gives an expression for the constant C1. Then, the linear deﬂection can be derived by:

y1 ðxÞ ¼

Z

h1 ðxÞdx þ D1

ð91Þ

where the parameter D1 can be solved by taking into account the boundary condition: y1 ð0Þ ¼ 0. Thus, the linear displacement at the end of the segment I can be express by y1 ðl4 =2Þ, which is a function in terms of xp and hp . Likewise, applying the similar process to segment II, the linear deﬂection at its end can be obtained as y2 ðl4 =2Þ. Hence, the rotation angle hp can be expressed as:

hp ¼ arcsin

y2 ðl4 =2Þ y1 ðl4 =2Þ l4

ð92Þ

which represents a relationship between the two unknowns xp and hp . Therefore, the parasitic rotation can be solved as a function of xp , abbreviated as:

hp ¼ f ðxp Þ

ð93Þ

Substituting Eqs. (93) and (84) into Eq. (79) gives an expression of output motion xp versus input displacement q1 as follows:

xp ¼ f ðq1 Þ

ð94Þ

Additionally, in view of Eq. (84), the stiffness of the CPM can be derived as: K p ¼ F 1 =q1 . Furthermore, the parasitic translation dy can be solved as:

dy ¼ t2 b sin hp ¼ l1

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F1 2 2 l1 d1 þ b sin hp k2

ð95Þ

which is also a function of the input variable q1 only. 6.1.6. Calculation of lost motion of the CPM For an XY CPM with parameters described in Table 1, the relations of the output displacement and parasitic motion versus the actuation variable q1 are plotted in Fig. 14. We can observe that the ratio of output to input motion for the CPM is about a constant of 3.26, which allows the generation of lost motion of the CPM system as follows: x

LNonlr ¼

A qp

1

A

¼ 18:5%

ð96Þ

Besides, the maximum translation in the y direction is up to 15.9 lm indicating a cross-talk of CNonlr ¼ 4:9% between the two axes. Comparing the results obtained by PRB or lumped model with those derived from the nonlinear or FEA model, we can deduce that the parasitic translation partially comes from the parasitic rotation. That is, the parasitic rotation is reﬂected in the cross-talk between the two working axes. 6.2. Predication of output motions with input variables q1 and q2 For the comparison of the results obtained by the nonlinear modeling and FEA simulation, the relations of the output displacement and parasitic motions versus the input variable of the XY CPM (see Table 1) are plotted in Fig. 14. It can be observed that the nonlinear modeling results agree well with the FEA simulation results. Moreover, both the nonlinear model and FEA results exhibit that the relationships between the output motions and input displacement are linear approximately. Thus, once given the input displacement values, the output motions can be predicted by an approach proposed as follows. Regarding to the parasitic rotation, it is observed that the rotation angle hp generated by Eq. (93) reaches to the maximum value around the clockwise direction in the case of q1 ¼ Q along with q2 ¼ 0. Due to the symmetry architecture of the CPM, it can be deduced that the effect of q2 is to drive the mobile platform rotate counterclockwise. Hence, the value of jhp j under the case of q1 ¼ Q and q2 ¼ 0 represents the maximum parasitic rotation of the CPM, i.e., Hmax ¼ jhp j.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Output displacement x (μm) p

400 Nonlinear FEA 200

0

0

20

40

60

80

100

80

100

80

100

Cross−axis translation δy (μm) 0 −10 −20 0

20

40

60

Parasitic rotation θp (mrad) 0 −0.1 −0.2 0

20

40

60

Input displacement q (μm) 1

Fig. 14. Output and parasitic motions obtained by nonlinear model and FEA simulation.

In view of the linear relations between the parasitic rotation and the input displacements of the CPM, the parasitic angle

H can be expressed in terms of q1 and q2 as:

Hðq1 ; q2 Þ ¼ c1 q1 þ c2 q2 þ c0

ð97Þ

where the three constants c0 ; c1 , and c2 can be solved by considering the extremum conditions:

q1 ¼ 0; q1 ¼ 0;

q2 ¼ 0; H ¼ 0 q2 ¼ Q; H ¼ Hmax

ð98aÞ ð98bÞ

q1 ¼ Q ;

q2 ¼ 0;

ð98cÞ

H ¼ Hmax

It follows that the parasitic rotation can be predicted by:

Hðq1 ; q2 Þ ¼

Hmax Q

ðq2 q1 Þ

ð99Þ

In the same way, we can deduce that in case of q1 ¼ Q and q2 ¼ 0, the output displacement xp represents the maximum translation ðX max ¼ AQÞ of the CPM in the x-axis direction, and the parasitic motion dy means the extremum translation along the y direction ðDmax Þ correspondingly. Let the output displacement X be expressed by:

Xðq1 ; q2 Þ ¼ n1 q1 þ n2 q2 þ n0

ð100Þ

The parameters n0 ; n1 , and n2 can be derived by considering the following sets of extremum values:

q1 ¼ 0; q1 ¼ 0;

q2 ¼ 0; X ¼ 0 q2 ¼ Q; X ¼ Dmax

ð101aÞ ð101bÞ

q1 ¼ Q ;

q2 ¼ 0;

ð101cÞ

X ¼ AQ

which enables the generation of the output motion in x-axis direction as:

Xðq1 ; q2 Þ ¼ Aq1

Dmax q Q 2

ð102Þ

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Likewise, the output motion Y of the CPM in the y direction can be predicted by the following expression.

Yðq1 ; q2 Þ ¼ Aq2

Dmax q Q 1

ð103Þ

Therefore, the kinematics of the CPM can be assembled into a matrix form:

2

3

2

A 6 7 6 Dmax 4Y 5¼6 4 Q H HQmax X

3 Dmax Q 7 q1 A 7 5 q 2

ð104Þ

Hmax Q

As the variation of the two input displacements q1 and q2 , the distributions of the small parasitic rotation obtained by the nonlinear predication and FEA simulation are depicted in Fig. 15. Since the small rotation value is in unit of micro-radian, it is neglectable in practice especially in the micro-positioning application.

7. Experimental validation The predicted input and output displacement relationships as shown in Eqs. (102) and (103) are veriﬁed by the experimental investigation conducted below. 7.1. Experiment setup and calibration

Parasitic rotation Θ (mrad)

The prototype of the XY CPM with the parameters in Table 1 is developed and the experiment equipments are graphically shown in Fig. 16. The stage is fabricated from the light material of Al 7075-T651 alloy. To make a tradeoff between the stroke and resolution of the PZT, one type of PZT, namely, PAZ015 produced by Thorlabs, Inc., is adopted to drive the XY stage. The PZT possesses a stroke of Q = 100 lm and a closed-loop resolution of 25 nm with an embedded strain gage sensory feedback. The embedded sensor enables a linear output motion of the PZT due to the elimination of the nonlinearity arising from hysteresis and drift, etc. Additionally, the two-axis controller BPC002 from the Thorlabs is used to drive the PZT with a voltage ranging from 0 to 75 V. As far as the displacement sensor is concerned, since the CPM possesses only a planar motion, a microscope vision system is used to detect the position of the CPM end-effector. As indicated in Fig. 16, the system consists of a light system, an inverted optical microscope, a CCD camera (from Watec Co., Ltd.), a PCI-based image acquisition board (from 10Moons Technology Development Co., Ltd.), and a PC (personal computer) (Intel Pentium 4 CPU 3.00 GHz, 512MB RAM) with image processing and control algorithms. The microscope allows a magniﬁcation ratio of 68–476 (calibrated on a 14-inch screen) and the light system enables a bright ﬁeld of view. In addition, the image acquisition board can capture images with the size of 720 pixel 576 pixel at the maximum acquisition rate of 15 frames per second. Moreover, an AFM probe (contact mode, from BudgetSensors, Innovative Solutions Bulgaria Ltd.) is adopted and mounted on the output platform of the CPM as an end-effector. The main parameters of the AFM probe are illustrated in Fig. 17, where the width of the probe cantilever is 50 lm.

FEA result

0.4 0.2

Nonlinear model 0 −0.2 −0.4 100 100 50

q (μm) 2

50 0

0

q1 (μm)

Fig. 15. Parasitic rotation distribution of the CPM.

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Y. Li, Q. Xu / Mechanism and Machine Theory 44 (2009) 2127–2152

Fig. 16. Photograph of the experimental setup.

Fig. 17. Parameters of the adopted AFM probe.

In the current research, the MATLAB with Image Acquisition Toolbox and Image Processing Toolbox is utilized to acquire and process the images. Once the image of the AFM probe is obtained, it is processed to identify the edge of the AFM cantilever, and the center of the tip in succession. According to the captured width of the cantilever, the image resolution can be calibrated as k = 0.78 lm/pixel. 7.2. Experimental results For the XY CPM as described in Table 1, in the case of q2 ¼ Q =2ðQ ¼ 100 lmÞ, the curves for the X and Y positions versus the input value q1 over the range of 0 to Q are plotted in Fig. 18. As the comparison, the FEA results are plotted therein as well. We can see that the predicted output motions through the nonlinear model are the largest among the three sets of results. Besides, the predicted X and Y motions agree well with the experiment results except for the deviations around 6.3% and 8.9%, respectively. With the two input variables q1 and q2 varying within the range of [0, Q], the CPM workspace range in terms of X and Y values can be predicted analytically based on Eqs. (102) and (103). In the case of Q ¼ 100 lm, the CPM workspace are sketched in Fig. 19, where the results obtained by the analytical model, FEA, and experiment are plotted for the reason of comparison. Within the overall workspace of the CPM obtained by the experiment, the maximum inscribed square shape workspace of 290 290 lm2 can be obtained as shown in Fig. 19.

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Output displacement X (μm) 300

Nonlinear FEA Experiment

200 100 0 0

20

40

60

80

100

80

100

Output displacement Y (μm) 170 160 150 140 130 0

20

40

60

Input displacement q (μm) 1

Fig. 18. Output displacements versus input displacement q1 .

350 300

Position Y (μm)

250 200

Nonlinear model FEA result Experiment result Maximum square

150 100 50 0 −50 −100

0

100

200

300

400

Position X (μm) Fig. 19. Workspace of the CPM.

Table 3 Kinematics performances of an XY CPM evaluated by nonlinear model, FEA, and experiment. Ampliﬁcation ratio

Lost motion

Cross-talk

Workspace (lm)

Nonlinear FEA Experiment

3.26 3.20 3.09

18.5% 19.9% 22.8%

4.9% 5.4% 6.2%

310 303 290

Deviation/difference of nonlinear model to FEA results (%) Deviation/difference of nonlinear model to experiments (%)

1.9 5.5

1.4 4.3

0.5 1.3

2.3 6.9

Moreover, the experimental results show that the ampliﬁcation ratio of the CPM is about 3.09 revealing a lost motion of 22.8%, and the cross-talk between the two axes is around 6.2%. For a clear comparison of the results, the performances of the CPM evaluated by the nonlinear modeling, FEA simulation, and experimental study are tabulated in Table 3. It can be observed that there are some derivations between these three types of results. Speciﬁcally, the derivation values of the

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nonlinear prediction are within 3% and 7% when compared to the FEA and experimental results, respectively, which indicates that the obtained analytical model is sufﬁcient for the assessment of the CPM performances. 7.3. Discussions Since the experimental results exhibit that the parasitic translation is up to 6.2% for the XY CPM, which is a relatively large value and can not be neglected for practical applications, the CPM can not be treated as a fully coupled XY manipulator. The cross-talk error between the two working axes can be compensated by a properly designed control strategy using the displacement sensory feedback which will be implemented in next step work of the research. In addition, if a larger workspace is required, a larger ampliﬁcation ratio can be designed for the lever or PZT with longer stroke can be adopted for such a purpose. Moreover, if the presented XY stage is implemented on a SOI wafer as the treatment in [18], a high resonant frequency with the magnitude of hundreds of Hertz can be easily achieved, which is more suitable for high-frequency applications. It is also observed that, due to a CPM consists of many passive joints and limbs, the nonlinear modeling is a non-trivial work although only the bending deformation of the limbs between ﬂexure hinges are taken into account. To establish a reliable relation between the output and input motions, a careful treatment of internal forces and moments associated with each ﬂexure element is required. However, as a simpliﬁcation approach, the use of PRB or lumped model considers the stiffness of ﬂexure hinges only, and hence fails to calculate the deformations of other components induced by the signiﬁcant internal effects. Nevertheless, the parasitic rotation of the output platform mainly comes from the compliances of components (such as connecting rods between the P joint and parallelogram limb) other than the ﬂexure hinges. Thus, the parasitic rotation and subsequent translation cannot be modeled without the consideration of internal forces and moments within the entire CPM. That is the reason why the PRB and lumped models can not predict the kinematics properties of the CPM properly. Moreover, the calculations of CPM performances with different approaches are implemented with MATLAB programs running on a PC as described in Section 7.1. For the computation of CPM performances in terms of ampliﬁcation ratio, lost motion, and cross-talk, the calculation time for the PRB, lumped, and nonlinear model approaches are 0.1, 0.5, and 38.3 s, respectively. Hence, except for a longer calculation time, the nonlinear modeling is a better method to characterize the kinematics or kinetostatics performances of the CPM. Even so, since the calculation time for the nonliner model is less than one minute, i.e., a relatively shorter time, it can still be employed for the architecture optimization of the CPM with respect to performance requirements. Besides, since the nonlinear modeling needs the calculation of each internal force in the CPM, it requires more efforts than both PRB and lumped model do in terms of analysis and computation process. From this point of view, the seeking for a simple approach that allows the generation of an accurate kinematics or kinetostatics model yet requires less computational work at the same time will be a meaningful future research direction.

8. Conclusion This paper is focused on the modeling and performance evaluation of a designed precision XY CPM. Preliminary investigations based on the simpliﬁed PRB and lumped models are carried out to assess the kinematics and dynamics performances of the CPM. It has been veriﬁed by the FEA simulation that the lumped model is sufﬁcient to evaluate the dynamics of the CPM with a derivation of less than 6%, while neither of the two models can be used to predict the CPM kinematics properly since they only consider the compliances due to the ﬂexure hinges. Thus, in view of elastic deformation of the whole CPM, a full nonlinear modeling is performed to obtain the relations between the input and output motions of the CPM with the consideration of internal force inﬂuences, and the results show that linear relations can be predicted. The experimental results based on a microscope vision system show that the established kinematic model can be used to assess the CPM kinematics performances with a deviation within 7%. Besides, the differences between the analytical model and FEA simulation results are less than 3%. However, a reliable FEA simulation is very time-consuming. Hence, the derived analytical models can be employed to evaluate the performances of the CPM as the variation of architecture parameters or to optimize the CPM parameters without extensive computation. The major contributions of this work involve the establishment of a proper dynamic model of the XY CPM through the simpliﬁcation approach and a linear kinematic model based on the nonlinear modeling method. The proposed methodology can be applied to the modeling of other types of CPMs as well. In addition, the conducted evaluation provides a guideline for both the performance improvement and positioning control of the CPM in the next step research.

Acknowledgements The authors appreciate the fund support from the research committee of University of Macau under Grant No.: UL016/08Y2/EME/LYM01/FST and Macao Science and Technology Development Fund under Grant No.: 016/2008/A1.

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Appendix A. Mass and stiffness matrices of lumped dynamics model

2

m4 6 0 6 6 6 m4 6 6 0 6 M¼6 6 0 6 6 0 6 6 4 0 2

0 k

0

0

m4

0

0

0

0

m4

0

m4

0

0

0

0

M33

0

M 35

0

A m5 2

m4 0

0 M53

M44 0

0 M 55

M 46 0

0 A2 m5

0

0

M64

0

M 66

0

0

A m5 2

0

A2 m5

0

M 77

0

0

A m5 2

0

A2 m5

k

6 k 0 6 0 6 6 k 0 K 33 6 6 6 0 k 0 K¼6 6 0 0 K 53 6 6 6 0 0 0 6 6 0 0 0 4 0 0 0

0

0

0

0

k

0

0

0

K 35

0

K 44 0

0 K 55

K 46 0

K 64

0

K 66

0

0

0

0

0

0

0

3

0

0

3

7 7 7 0 7 7 A m5 7 7 2 7 0 7 7 A2 m5 7 7 7 0 5 0

ðA:1Þ

M88

7 0 7 7 0 0 7 7 7 0 0 7 7 0 0 7 7 7 0 0 7 7 4 R k 0 7 5 a2 4 R 0 k a2 0

ðA:2Þ

where

2 1 1 M33 ¼ M44 ¼ A2 m1 þ A2 m2 þ m3 þ m4 þ A2 m5 3 3 2 2 2 1 1 2 M35 ¼ M53 ¼ M46 ¼ M64 ¼ A m1 þ A m2 þ m3 þ A2 m5 3 3 2 2 2 4 1 2 2 M55 ¼ M66 ¼ A m1 þ 1 þ A m2 þ m3 þ A m5 3 3 2 7 M77 ¼ M88 ¼ m0 þ m5 6

ðA:3aÞ ðA:3bÞ ðA:3cÞ ðA:3dÞ

and

K 33 ¼ K 44 ¼ K 55 ¼ K 66 ¼ k þ 4A2 k þ

2

K 35 ¼ K 53 ¼ K 46 ¼ K 64 ¼ 4A k þ

5 2

l1 5 2

l1

A2R k

ðA:4aÞ

! 2R

A k

ðA:4bÞ

Appendix B. Intermediate constants in nonlinear model

"

4r þ t

!#

ð8r 4 þ 8r 3 t þ 2r 2 t 2 þ 4ar 3 þ 4ar2 t þ art 2 Þ ð4rt þ t 2 Þ1=2 2 t þ ð4rt þ a2 Þ1=2 ða2 4r 2 Þð32r 3 þ 32r 2 t þ 10rt 2 þ t 3 Þ þ ð12r 3 þ 8r 2 t þ 2rt 2 þ 6ar 2 þ 4art þ at2 Þ 3 2rh " !# 4r þ t H ¼ 12 tan1 ð4r 3 þ 4r2 t þ rt 2 Þ ð4rt þ t 2 Þ1=2 2 t 3 2 2 3 2 2 þ ð4rt þ a2 Þ1=2 ða 2rÞð32r þ 32r t þ 10rt þ t Þ þ 2ð6r þ 4rt þ t Þ 3 2rh

G ¼ 6 tan

1

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ðB:2Þ

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