J Intell Robot Syst (2009) 55:377–402 DOI 10.1007/s10846-008-9300-z

Design and Optimization of an XYZ Parallel Micromanipulator with Flexure Hinges Yangmin Li · Qingsong Xu

Received: 30 November 2007 / Accepted: 24 November 2008 / Published online: 11 December 2008 © Springer Science + Business Media B.V. 2008

Abstract In this paper, a nearly decoupled XYZ translational compliant parallel micromanipulator (CPM) is designed for micro/nano scale manipulation with features of piezo-driven actuators and flexure hinges. The CPM structure improvement is made to enlarge the workspace and eliminate or reduce the stress stiffening, buckling phenomenon, and parasitic motions of the original XYZ CPM, which leads to a new CPM with a more compact structure. The CPM kinematics, parasitic motions, and workspace are determined analytically, and the mathematical models describing statics and dynamics of the CPM are established to evaluate its related performances, which are verified by the finite element analysis (FEA) undertaken in ANSYS environment. Based on the analytic models, the CPM dimensions have been optimized by resorting to the particle swarm optimization (PSO) approach, which produces a CPM having minimum parasitic motions and satisfying other performance specifications as validated by the FEA simulations. Keywords Parallel manipulator · Precision machine · Flexure hinge · Robotic modeling · Optimum design · Particle swarm optimization (PSO)

1 Introduction In the fields of micro/nano scale manipulation, micromanipulators with ultra-high precision are urgently required to perform such tasks as bio-cell manipulation,

Categories (2), (3). Y. Li (B) · Q. Xu Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa, Macao SAR, People’s Republic of China e-mail: [email protected] Q. Xu e-mail: [email protected]

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optical fibers alignment, micro device assembly, and operation under scanning probe microscopes, etc. It is well known that parallel kinematic mechanisms possess inherent advantages over conventional serial manipulators in terms of high rigidity, high load carrying capacity, and high accuracy, etc [1–3]. However, traditional parallel manipulators suffer from errors due to clearance and backlash, hysteresis, and frictions in the mechanical joints. Therefore, it is a major challenge to obtain ultrahigh precision using conventional joints. On the other hand, flexure hinge-based compliant mechanisms, which achieve the motion from deflections of flexure elements, can be employed into parallel manipulators for the applications of requiring very high precision [4, 5] thanks to their excellent characteristics of vacuum compatibility, no backlash property, no nonlinear friction, simple structure and facilitated manufacture, and so on. These attributes have endowed compliant parallel manipulators (CPMs) with potential ability in micro or nano scales precision operations. In the literature, many CPMs have been designed for such purposes [6–9]. However, most of the existing manipulators can provide only a planar motion, or a spatial combined motion of translation and rotation. In such applications as optical alignment and cell injection operation, an XYZ-stage with three translational DOF is preferred. For example, two positioning CPMs are proposed in [10] and [11], however, the three translation motions are coupled, which complicate the control of the micromanipulator and cost excessive fees and energies for the actuators from the economy point of view. For this reason, several decoupled translational CPMs have been proposed in [12, 13] recently. Nevertheless, these CPMs possess a complex structure with a large volume which may not be suitable if the CPM residing in a limited space is required. In view of this point, we designed a novel micromanipulator in our previous work [14], which was called a 3-PRC (three-prismatic-revolute-cylindrical) CPM as shown in Fig. 1, the CPM provided three translational DOF in space, while possessed a very simple structure. But we noticed that the previous 3-PRC CPM exhibited several disadvantages in the aspects of stiffening, buckling, and parasitic motions after further investigations, which could prevent its practical application in micro/nano

Fig. 1 The original 3-PRC CPM

Mobile platform

y z

x

C joint

P joint R joint P joint

Fixed base

PZT

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manipulation fields. To resolve these problems, the objective of the current research is to improve the original 3-PRC CPM structure in order to eliminate or reduce its vulnerabilities and to optimize its dimensions so as to obtain a new CPM with optimal performance. It will be shown that even if the CPM architecture can be improved intuitively by an inspection of its structure, the CPM performance can be predicted by resorting to the finite element analysis (FEA) via software package ANSYS since it is not a straightforward work to evaluate the CPM performance analytically. Besides, it is observed that the structure of improved 3-PRC CPM looks like the Delta Cube proposed in [15]. However, an insightful comparison reveals that the presented CPM has a different type of actuation, i.e., piezo-driven with a displacement amplification mechanism to enlarge the workspace, and the three actuation directions intersect at one common point to reduce parasitic motion. In a sense, the designed CPM is an improved version of the Delta Cube. The novelty of the current research lies in the design improvement and dimension optimization of the 3-PRC CPM with significantly enhanced performances. The remainder of the paper is organized in the following way. The architecture of the original 3-PRC CPM is briefly described in Section 2, and the design modification procedures are presented in Section 3 where a performance preview of the improved CPM is provided. Then in Section 4, the DOF of the CPM is derived and the kinematics models are established. Besides, the complicated parasitic motions are evaluated and the workspace is determined as well due to elastic limits of flexure hinges. Afterwards, the analytical approaches for the assessment of statics and dynamics of the CPM are carried out in Section 5 and 6, respectively, along with the established models validated by FEA simulations. Furthermore, the optimal design with the consideration of parasitic motions, stiffness, and natural frequency requirements, etc., are conducted in Section 7 by means of the particle swarm optimization (PSO) method along with the CPM performances validated through the FEA. Finally, some concluding remarks are summarized in Section 8. 2 Architecture Description of a 3-PRC CPM The original 3-PRC CPM composed of flexure hinges is illustrated in Fig. 1. The CPM consists of a mobile platform, a fixed base, and three limbs with identical kinematic structure. Each limb connects the fixed base to the mobile platform by one flexure prismatic (P) hinge, two flexure revolute (R) hinges, and another passive flexure P joint in sequence, where the passive P joint is a parallelogram structure involving four flexure R hinges. The first P joint within each limb is equivalent to a leaf spring, which is fixed at the base and actuated by a linear actuator. Thus, each limb is a PRRP kinematic linkage indeed. It is observed that the axes of the last R and P joints are parallel to each other, and the combined effect of these two joints is analogy to a cylindrical (C) joint. Hence, the mobile platform can be viewed to be attached to the base by three identical PRC limbs. For a 3-PRC parallel manipulator with conventional mechanical joints, it has been shown that such an overconstraint mechanism can act as a translational manipulator with some certain geometric conditions satisfied [16]: briefly, the axes of the R and C joints within the same limb are parallel to each other. Due to the equivalent kinematic architecture, the proposed CPM possesses three translational DOF. Moreover, in order to generate a cuboid shape workspace of the manipulator, the

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three actuated P joints are arranged in an orthogonal manner. With a proper endeffector mounted on the mobile platform or a mobile platform located under a specified microscope, the CPM is expected to find its way into three-dimensional micro/ nano scale positioning manipulation. The flexure hinge with right-circular notch type is adopted since it possesses the smallest center-shift compared to other types. As far as the linear actuator is Fig. 2 Flowchart of the CPM design

Performance Requirement

Architecture Design/Selection

Actuator/Sensor

Material/ Flexure Hinge Structure Improvement • Parasitic Motion Suppression • Stiffness Enhancement • Stress Stiffening Elimination • Displacement Amplifier • Buckling Elimination • Compact Structure Dimension Optimization Performance Evaluation

Satisfied? Yes Prototype Fabrication Experimental Test

No

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concerned, the piezoelectric actuator (PZT) is selected to drive the CPM due to its major advantages of large blocking force, high stiffness, fast response, and compact size. Whereas compared with other types of linear actuators, the main drawback of PZT lies in its small travel stroke. If the stroke of the adopted PZT can not meet the application requirements, a proper amplification mechanism will benefit the design. Moreover, a flowchart of the CPM design is illustrated in Fig. 2, where the FEA simulation instead of the experimental test can be adopted to evaluate the CPM performances in the design stage so as to speed up the design process and reduce the manufacture cost. Additionally, the analytical models for the CPM are indispensable to implement an efficient dimension optimization. In accordance with the design flowchart, the structure improvement of the original 3-PRC CPM is carried out in the following section.

3 Structure Improvement of a 3-PRC CPM 3.1 Design Modification Procedures The original 3-PRC CPM as represented by one limb in Fig. 3a is modified in terms of six steps as conducted below. Step 1: Due to the three actuation axes do not intersect at one point, parasitic moments and parasitic motions in succession will be generated when the CPM is driven by the actuators. For instance, the moment Mx exists if the input force F y is exerted which produces a parasitic rotation θx as indicated in Fig. 3a. Therefore, the actuation axes of the three limbs are assembled to intersect at one common point as shown in Fig. 3b, so as to eliminate or reduce parasitic motions of the CPM. Step 2: Intuitively, one leg is too slender to support the mobile platform. Hence, a parallel RR leg is added to each limb to enhance the structure stiffness of the CPM as described by Fig. 3c. Step 3: For the original actuated flexure P joint, the deflection mainly comes from the two flexure leaves. Once actuated by PZT, the two leaves suffer from additional axial loads in the direction vertical to the translation. The exerted axial loads are the sources of stress stiffening phenomenon in the flexure P joint. It can be shown that stiffening phenomenon increases the actuation stiffness of the flexure P joint, which may reduce the output stroke of PZT and the workspace of CPM correspondingly. Hence, in order to reduce the effect of stress stiffening, the actuation P joint is modified as shown in Fig. 3d, where the axial loads are avoided. Step 4: Add the displacement amplification mechanism to enlarge the stroke of PZT actuators. The main drawback of PZT is its limited stroke, which leads to a small workspace of the CPM. In order to enlarge the CPM workspace, the stroke of the PZT is amplified by using a lever amplification mechanism as illustrated in Fig. 3e. Step 5: Buckling may occur in the flexure hinges once their axial loads are compressive and sufficient large. An observation of one limb in Fig. 3a reveals that when the PZT pushes the leaf-based P joint, the six flexure hinges in the limb all bear the axial compressive load. So, buckling phenomenon may

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y

y

z

z

θx

Fy

(a) O riginal limb.

(b) Step 1.

(c) Step 2.

Fy

Fy

(d) Step 3.

(e) Step 4.

(f) Step 5.

(g) Step 6.

Fig. 3 Modification procedures of a 3-PRC CPM (a–g)

arise as long as the load is large enough, which may lead to instability of the CPM and also restricts the CPM workspace. In order to reduce buckling in the CPM, the concept of reverse actuation is introduced. By inverting the actuation direction, the flexure hinges constructing the R and C joints in one PRC limb are all suffer from axial tensile loads instead. And the notch hinges in the actuation P joint do not bear only axial compressive loads either. The only hinge under the compression state is the coupling flexure hinge between the PZT and the actuation P joint. If the coupling hinge is replaced by a metallic ball with the point-contact manner instead, then the

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Fig. 4 One limb of an improved 3-PRC CPM

y

z

O ( P)

Mobile platform

C joint x

b

R joint P joint

a

l

PZT

R joint

P joint

buckling will be fully excluded from the current CPM. Otherwise, we can check the elastic buckling load of this single hinge by using the guidelines elaborated in [17]. Moreover, in order to make full use of the limited space to build up a compact CPM, the PZT is inserted into the amplification P joint as depicted in Fig. 3f. Step 6: In order to generate a more compact CPM, the limb architecture is further improved as described in Fig. 3g, where the passive P joint is inserted between the two passive R joints with a PRPR linkage generated instead. Since the effect of the last P and R joints is also equivalent to a C joint as depicted in Fig. 4, the resulting CPM can still be viewed as a 3-PRC CPM, whose virtual prototype is shown in Fig. 5. 3.2 Performance Preview for the New CPM In order to have a preview for the merits of the improved new 3-PRC CPM, both the original and the current CPM are analyzed via the nonlinear statics analysis in ANSYS software. For the sake of comparison, both the original and improved CPM are created with the same dimensions as described in Table 1, where the notch hinge parameters (t, r and w) are depicted in Fig. 6 and other parameters refer to the distances between the centers of relevant notch hinges. In addition, the

384 Fig. 5 A virtual prototype of the improved 3-PRC CPM

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Limb 1

Mobile platform

Limb 3

PZT

Fixed hole

Fixed block

Limb 2

same large displacement (4 mm) is assigned as input of the CPM to have a more apparent comparison of the two CPM performances. The corresponding loads can be generated after the solution, and the output motion of the CPM can be monitored as well. The relationship between the input and output displacements for the two CPMs is plotted in Fig. 7. It is found that with the same input displacements of linear actuators, the ratio of output displacements between the modified and original CPM is about 1.99. Thus, the CPM workspace volume has been enlarged by 1.993 ≈ 7.9 times. Moreover, the force-displacement relationships of the two CPMs are shown in Fig. 8, where the curve for the original CPM exhibits that the slope, i.e., the actuation stiffness, grows from 437.2 N/mm to 620.6 N/mm as the increasing of the input displacement due to the existence of stress stiffening. Whereas the actuation

Table 1 Main parameters of the 3-PRC CPM

Parameter Architectural parameters (mm) a b l l1 l2 l3 r t w CPM material parameters Young’s modulus Yield strength Poisson’s ratio Density

Value 129 66 103 48 24 92 5.0 0.5 16 113.8 GPa 880 Mpa 0.342 4.43 g/cm3

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Fig. 6 Parameters of the modified flexure P joint

l1 m2

θ2 l3

y x

z

y

Actuation

m1 w t

z

m3 l2

e2

r m2

θ2

q2 =

d2 2

d2 Right-cicular notch hinge

stiffness of the modified CPM almost remains a constant of 136.0 N/mm, which means that the stiffening phenomenon has been eliminated. In addition, with the second P joints of the CPM driven, the parasitic motions versus output displacements for the original and modified CPM are elaborated in Figs. 9 and 10, respectively. It is seen that the parasitic motions of the modified CPM are all smaller than those of the original CPM. Although the adopted amplification P joint possesses a parasitic translation along the z-axis as shown in Fig. 6, the parasitic motion of the CPM in this direction (uz = 7.3 μm) is very small and less than 10% motion of the original CPM (uz = 78.7 μm). Besides, the parasitic displacement ux has been significantly reduced from 559.2 μm to 2.7 μm, and the parasitic rotations around the x and z axes due to the non-intersection of the original three actuation axes are reduced to 0.0087 and 0.0145 degree which are only 46% and 8% of those in the original CPM, respectively.

Fig. 7 Input–output displacement relationship of the two CPMs

8

Output displacement (mm)

7

Original CPM Modified CPM

6 5 4 3 2 1 0 0

1

2 3 Input displacement (mm)

4

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Fig. 8 Force-displacement relationships of the two CPMs

2500 Original CPM Modified CPM

Force (N)

2000

1500

1000

500

0 0

1

2 Deflection (mm)

3

4

Regarding the critical load (Pcr ) that can induce elastic buckling to the modified CPM, it can be roughly assessed with the consideration of the equation [17]: Pcr =

π 2 EImin 2 lcr

(1)

where Imin = wt3 /12 is the minimum moment of inertia for the coupling notch hinge, and lcr = 2 × 2r denotes the critical length of the hinge which is taken as a fixedfree type of column. The critical load predicted by (1) is 468 N which corresponds to a constant rectangular cross section, one-sensitivity-axis flexure hinge under axial compression conditions. For the right-circular notch hinge, the cross section is not constant, and the average moment of inertia is larger than Imin . As a consequence, the critical load for elastic buckling of the modified 3-PRC CPM is much larger than

Fig. 9 Relationship between the output motion and parasitic motion of the original 3-PRC TPM

1.2

Parasitic motion

1

x translation (mm) z translation (mm) x rotation (degree) z rotation (degree)

0.8 0.6 0.4 0.2 0 0

1 2 3 Output displacement (mm)

4

J Intell Robot Syst (2009) 55:377–402 Fig. 10 Relationship between the output motion and parasitic motion of the improved 3-PRC TPM

387

0.25

Parasitic motion

0.2

x translation (mm) z translation (mm) x rotation (degree) z rotation (degree)

0.15 0.1 0.05 0 0.05 0

2 4 6 Output displacement (mm)

8

468 N. Taking into account the actuation stiffness value (136 N/mm) of the CPM, the critical load 468 N can only be reached by the actuation with a displacement of 3.44 mm. It follows that the actuation value leading to elastic buckling of the CPM is larger than 3.44 mm, which is safe enough for the CPM driven by the PZT actuator whose stroke is usually less than 1 mm. The preview of the CPM performance reveals that the modified CPM possesses a larger workspace and less parasitic displacements than the original one along with the unfavorable stiffening and buckling phenomena eliminated. The CPM performances in terms of kinematics, statics and dynamics are evaluated in details for a multiobjective architecture optimization conducted in the following discussions.

4 Kinematics, Parasitic Motion, and Workspace Assessment In the early design stage of a 3-PRC CPM, the main objective is to establish a simple yet accurate enough model to assess the CPM performance by resorting to computer simulations. Since the rotation center shifted value of the flexure hinge with a right-circular shape is smaller than the hinges with other shapes, we assume that the rotation center is not changed during the operation of the CPM. Then, the pseudo-rigid-body (PRB) model [18] is utilized to facilitate the design and evaluation of CPM. 4.1 Mobility Determination With the mechanism topology identified and each flexure hinge replaced by a revolute joint and a torsional spring, the PRB model of the CPM can be easily developed. And limb 2 is shown in Fig. 11. As is well-known, a planar parallelogram (Pa) joint has only one translational DOF. So, each Pa joint of the CPM can be considered as a 1-DOF P joint pair with two links connected to each side of it. Hence, each limb of the modified CPM is equivalent to a PRPR structure as illustrated in Fig. 11. Furthermore, since the effect

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Fig. 11 PRB model of one limb of the modified CPM

y

z

O ( P)

x

R P

l P R

a

of the last P and R joints is equivalent to a C joint, it still can be seen as a 3-PRC parallel mechanism, which possesses three translational DOF as can be shown by resorting to the screw theory (see [16] for more details). 4.2 Kinematics Modeling Assign d = [d1 d2 d3 ]T be the vector of three actuated P joint variables, and q = [q1 q2 q3 ]T be the vector for the displacements of three PZT. Since the adopted lever mechanism possesses an amplification ratio of A p = ll12 = 2 as illustrated in Fig. 6, we have q = 12 d. Referring to the limb parameters as indicated in Fig. 6, we can observe that the difference between parameters a and l is only a small offset. For simplicity, we replace the value of a with l in the kinematics analysis. In addition, a fixed reference frame O is assigned as the initial position of the center point P for the mobile platform. Thus, the position of the mobile platform with respect to the reference frame can be described by a position vector p = [x y z]T of the reference point P. With reference to the vectors representation shown in Fig. 12, a vector-loop equation can be written for the i-th chain: lli0 = Li − di d i0

(2)

with the notation of Li = p + bi − ai − di d i0 , where l is the leg length and li0 is the −−→ unit vector along the leg direction Ci Bi ; di represents the linear displacement of the i-th actuated P joint with d i0 denoting the unit vector along the actuated direction; ai and bi represent the position vector of point Ai and Bi at the home position, respectively. In view of (2), a necessary calculation leads to the inverse displacement solutions:  T T di = d i0 Li + (d i0 Li )2 − LiT Li + l 2 (3)

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Fig. 12 Vectors representation for the CPM

z

y

y

O

O

p

p P

x

P bi

Bi

ai

Bi

l li0 Ai

Ai

d i d i0 Ci

which can be expanded into the following forms:  d1 = −x − l + l 2 − y2 − z2  d2 = −y − l + l 2 − z2 − x2  d3 = −z − l + l 2 − x2 − y2

Ci

(4a) (4b) (4c)

Assume that the CPM is away from singularities (which is true for the CPM with micro motion), three velocity equations can be obtained by differentiating (2) with respect to time, which are then assembled in the matrix form: ˙ d˙ = J p

(5)

˙ = [x˙ y˙ z˙ ]T denotes the vector of linear velocities for the mobile platform, where p and the Jacobian matrix is ⎡ ⎤ 1 Jy1 Jz1 ⎢ ⎥ J = − ⎣ Jx2 1 Jz2 ⎦ (6) x y 1 J3 J3 with J1 = x + l − d1 , J2 = y + l − d2 , and J3 = z + l − d3 .

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Considering that the units for the mobile position (x, y, and z) and actuated joint variable (di ) of the CPM are micrometers, whereas the leg length l is in unit of millimeter, we can deduce that x, y, z, di  l. And then, the Jacobian matrix can be approximately written as J ≈ −I

(7)

which indicates that d1 ≈ −x

d2 ≈ −y

d3 ≈ −z

(8)

The above kinematics analysis exhibit that the designed CPM possesses a nearly decoupled translational motion. 4.3 Parasitic Motion Analysis The explicit expression for the CPM parasitic deflections is necessary for the sake of alleviating them. In view of the symmetric architecture of the three limbs of the CPM, we can consider the case that the second limb solely is driven by a PZT with an input displacement d2 while the other two limbs remain zero displacement input. Then, one parasitic displacement (e1 ) along the −x direction is caused by limb 1, and two parasitic displacements (e2 , e3 ) along the −z direction can be induced by limbs 2 and 3, respectively. Referring to Figs. 6 and 13, we can observe that the parasitic displacements subject to the three limbs can be calculated by:

 e1 = l 1 − cos(β2 ) (9)

 (10) e2 = l1 1 − cos(θ2 )

 (11) e3 = a 1 − cos(γ2 ) where β2 = sin−1 (d2 /l) ≈ d2 /l

(12)

θ2 = sin−1 (d2 /l1 ) ≈ d2 /l1

(13)

−1

γ2 = sin (d2 /a) ≈ d2 /a

Fig. 13 Parasitic displacements due to limbs 1 and 3

(14)

y

y z

x

a

l

β2 Limb 1

e3

e1

γ2 Limb 3

d2

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Taking into account the directions of the parasitic motion, we can see that both e2 and e3 are along the negative z-axis. According to the magnitude relationship between e2 and e3 , three cases in terms of e2 > e3 , e2 = e3 , and e2 < e3 may occur. For the sake of eliminating the parasitic motion of the CPM in z direction, e2 and e3 should be along the opposite direction with the same magnitude (e2 = e3 ). That is, if e3 is along −z and e2 is in z-axis direction with the same size, then the two opposite motions compensate for the CPM parasitic motion in the z direction. Therefore, we adopt the manner as shown in Fig. 14 to mount the three limbs of the CPM at the fixing blocks. Moreover, in view of (10), (11), (13), and (14), we can deduce that the relationship of e2 = e3 leads to l1 = a

(15)

Another factor that may cause parasitic motions of the CPM is the adopted flexure hinges themselves. Although the employed right-circular notch hinge has better accuracy than other types, the deviation of rotation center and the compliances in other working directions still exist. Referring to the flexure hinge indicated in Fig. 6, we can see that the rotation around the x-axis (with stiffness Kθx −Mx ) is the working direction of flexure hinge. However, the hinges within legs also bear loads along the z-axis (with stiffness Kuz −Fz ) and moments around the y-axis (with stiffness Kθ y −My ) directions during the operation. In order to make the flexure hinge more sensitive to the rotation about the working direction and more insensitive to the passive directions, the following two stiffness ratios should be made as small as possible:

Fig. 14 CAD model of the final CPM

ε1 =

Kθx −Mx Kuz −Fz

(16)

ε2 =

Kθx −Mx Kθ y −My

(17)

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Based upon the approximate stiffness models in [19], the two stiffness ratios can be expanded in terms of the hinge parameters:

 2+π t t2 2− (18) ε1 = 9 π r

 2+π t 12t2 2− (19) ε2 = 9w 2 π r The above equations are valid in the ranges of 0<

t ≤ 0.2 2r

(20)

so as to keep the deviations with respect to the true values within 10% [20]. 4.4 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 ∈ [0, Q], then in view of (8) and amplification ratio of the lever mechanism, the CPM workspace can be theoretically calculated as V = (2Q)3 provided that the stresses caused by the flexure hinge rotations remain within the yield strength of the material. For a notch hinge bearing a pure bending moment around its working direction, the maximum angular displacement αmax arises when the maximum stress σmax , which occurs at the outermost surface of the thinnest portion of the hinge, reaches to the allowed stress σa that is less than the yield strength σ y due to an assigned safety factor na ∈ (1, ∞), i.e., σmax = σa = σ y /na

(21)

The relationship between the maximum stress and maximum rotation of the flexure hinge can be calculated as [20]: σmax =

E(1 + β)9/20 αmax β 2 f (β)

(22)

where β = 2rt is the dimensionless geometry factor with a valid range of 0 < β < 2.3, and f (β) is a dimensionless compliance factor derived in [20] as:    1 6(1 + β) 2+β 3 + 4β + 2β 2 −1 f (β) = + (23) tan 2β + β 2 (1 + β)(2β + β 2 ) (2β + β 2 )3/2 β Once the PZT for driving the CPM is selected, its stroke Q is a constant. Substituting the rotation angles described by (12), (13), and (14) with d2 = Q into (22) allows the generation of the relationships: inf(l, l1 , a) ≥

E(1 + β)9/20 na Q β 2 f (β)σ y

(24)

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

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5 Statics Analysis 5.1 Static Forces Generation Differing from the statics of a conventional parallel manipulator, the statics of a CPM solves the actuation forces which are expected to balance both the external forces applied on the mobile platform and the internal forces arising from the deformations of flexure elements constructing the CPM. Assume that an external force F = [Fx F y Fz ]T is exerted on the mobile platform of the CPM, which brings a displacement u = [ux u y uz ]T of the CPM along the three translational directions at an equilibrium state. Let the output displacements of the three PZT be denoted by the vector q. In view of the kineto-statics duality property of the manipulator, the external force F can be balanced by the actuation force: f ex = J −T F

(25)

As far as the internal forces are concerned, referring to Fig. 6, we consider the CPM displacement u y along the −y direction as an example. From (8), we derive that d2 = −u y . 5.1.1 Internal Forces in Limb 1 Within the limb 1, the deformation u y is mainly suffered by the passive P joint. It is observed that all of the four notch hinges rotate with the same angle β2 . Assign K2y be the stiffness of the passive P joint in y direction, then the induced potential energy in the P joint due to elastic deflections can be computed by P1 =

1 1 K1y d22 = 4 × kβ22 2 2

(26)

where k is the approximate rotation stiffness of the right-circular flexure hinge around the working direction as expressed by k=

2Ewt2.5 9πr0.5

(27)

and the rotation angle β2 (see Fig. 15) is described by (12). Then, in view of (26), the internal elastic force due to limb 2 becomes: f1y = K1y d2

(28)

where K1y =

8Ewt2.5 9πr0.5 l 2

(29)

5.1.2 Internal Forces in Limb 2 and Limb 3 In limb 2, the deformation u y is mainly supported by the actuation P joint which includes five notch hinges as illustrated in Fig. 6. While in limb 3, the deformation u y is mainly carried by the two passive R-R legs involving four flexure hinges.

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Fig. 15 Convergence processes of the PSO optimization

ps=40, Inertia wt=0.4, Common PSO, 5D, cost=0.061652 0.16

gBest

0.14

0.12

0.1

0.08

0.06 0

500

1000

1500

Epoch

By the potential energy expression, the stiffness K2y and K3y of limbs 2 and 3 in the y direction can be respectively calculated as: K2y =

10Ewt2.5 , 9πr0.5 l22

K3y =

8Ewt2.5 9πr0.5 a2

(30)

which allows the generation of the internal elastic forces subjected to limb 2 and limb 3 as follows. f2y = K2y d2 ,

f3y = K3y d2

(31)

Therefore, the internal forces of the whole CPM corresponding to the displacement u y of the mobile platform can be computed by: f y = f1y + f2y + f3y = Kq q2

(32)

where Kq =

4Ewt2.5 9πr0.5



4 5 4 + 2 + 2 2 l a l1

 (33)

which describes the relationship between the actuation force and displacement of the CPM, i.e., the actuator stiffness. 5.1.3 Statics of the Whole CPM In the same way, the internal forces owing to displacements ux and uz can also be calculated, which can be balanced by the actuation forces: f in = K q q where the 3×3 diagonal stiffness matrix K q = diag{Kq }.

(34)

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With the consideration of (25) and (34) at the same time, we can see that the following balance forces need to be created by actuators under statics condition: f = J −T F + K q q

(35)

which provides a guideline for the selection of actuators in terms of the output force. 5.2 Simulation Results and Discussions Concerning a 3-PRC CPM with parameters described in Table 1, assume that a force F = [0 8.1 0]T N is applied on the mobile platform which requires a displacement q = [0 0.2 0]T mm of the linear actuators to arrive at an equilibrium state. The actuation force can be calculated via (35) as f cal = [0 27.55 0]T N. The FEA is performed with ANSYS to validate the derived statics equation. In the simulation, the force F and input displacement q are assigned on the mobile platform and the second actuation P joint of the CPM, respectively. After the simulation, the actuation force can be determined as f sim = [0 30.74 0]T N. We can observe that calculated actuation force f cal is very closed to the one f sim generated by the FEA approach. And the deviation of the calculated value with respect to the simulation result is 10.4%. The deviation mainly comes from the assumption of the PRB model concept, which can be reduced by adopting a fully nonlinear modeling of the CPM instead. It should be noted that, in the above statics analysis for the 3-PRC CPM, only the external forces applied on the mobile platform are considered, which induce internal forces due to the elastic deflection of flexure hinges. The external forces and induced internal forces are balanced by the forces created by the three PZT. Moreover, the mobile platform of the spatial 3-PRC CPM may suffer from external moments in addition to the forces. The external loads should be balanced by the three actuation force created by the PZT and constrained wrenches exerted by the flexure elements. The wrenches can introduce deformations of flexure hinges in directions other than their working axes, which may even enlarge the parasitic rotation of the mobile platform. The said deformations can be calculated by resorting to a full stiffness modeling of the CPM as proposed in the literature [21, 22].

6 Dynamics Evaluation 6.1 Dynamics Modeling of the 3-PRC CPM In the current research, the Lagrange’s equation is adopted for the dynamics modeling of a 3-PRC CPM. The variables q = [q1 q2 q3 ]T are chosen as the generalized coordinates. Additionally, we assume that the kinetic energies are induced by the rigid rods connecting the flexure hinges and potential energies introduced by the elastic deformations of flexure hinges and gravity of the CPM. By expressing the kinetic (T) and elastic potential (U) energies of the entire CPM in terms of the selected coordinates and their derivatives, the Lagrangian function (L = T − U) for the CPM can be generated. For the reason of compactness, the derivation processes are omitted here, more details can be found in relevant works

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[23] of the authors. The Lagrangian equation of motion can be derived based on the generalized coordinates q according to ∂L d ∂L · − = Fi dt ∂ q˙ i ∂qi

(i = 1, 2, 3)

(36)

where qi denotes the i-th generalized coordinate and Fi is the i-th actuation force. The generated dynamic equations take on the following form: M q¨ + Kq + G = F

(37)

where M = diag{M} is the mass matrix, K = diag{K} is the stiffness matrix, G = [0 G 0]T and F = [F1 F2 F3 ]T denote the gravity force vector and actuation force vector, respectively, with the following notations: 8 40 M = 4m0 + 4m1 + m2 + m3 + m4 3 3   16 16 2Ewt2.5 5 + 2 + 2 K= 9πr0.5 l22 l a G = 2(m0 + 3m1 + 6m2 + 3m3 + 6m4 )g

(38) (39) (40)

where m0 is the mass of the mobile platform, m4 denotes the mass of one R-R leg, and other masses are indicated in Fig. 6. Therefore, the actuation force Fi for the i-th limb can be derived in view of (37). It is noticeable that Lagrange multipliers do not appear in the established dynamic equation since no constraint equations are adopted in the dynamic modeling. In addition, the consideration of gravity effects is necessary once the CPM motion along the y direction is concerned, and other forces due to the interacting between the end-effector and the environment should be taken into account as well for practical applications. 6.2 Modal Analysis and Simulation Validations The modal analysis is necessary for the design of the manipulators as far as the control frequency is concerned. Commonly, to avoid exciting the structural oscillation and resonance of the CPM system, the natural frequency of the mechanism ( fmech ) in a servo control system should be no less than two times higher than the frequency of the driving system ( fctrl ). It follows that a higher natural frequency of the mechanism allows a higher control frequency of the driving system. Based on the theory of vibrations, the dynamic equation of undamped free vibration of the CPM system can be expressed as: M q¨ + Kq = 0

(41)

Besides, the modal equation describing free vibration of the system can be obtained as follows: (K − λi M) i = 0

(42)

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where λi and i are the eigenvalue and eigenvector associating with the i-th mode shape of the system. The condition of non-zero solutions for (42) can be derived by |K − λi M| = 0

(43)

which allows the calculation of the eigenvalue, i.e., λi = ωi2 with ωi denoting the natural cyclic frequency of the system. Then, the natural frequency can be computed 1 as fi = 2π ωi . For instance, with the kinematic and physical parameters elaborated in Table 1, the natural frequency of the 3-PRC CPM can be calculated by

1 K fmech = = 12.6 Hz (44) 2π M Moreover, the modal analysis of the 3-PRC CPM with the parameters described in Table 1 is performed with ANSYS. Both the natural frequency and the corresponding mode shapes are derived by the simulation, which exhibits that the first three vibration shapes are translations along the three axes of the frame with almost the same natural frequency. The first natural frequency (12.4 Hz) obtained by ANSYS agrees well with the value (12.6 Hz) calculated from the dynamic model, which partly verifies the validity of the performed dynamic modeling for a 3-PRC CPM.

7 Multi-Objective Dimension Optimization A review of the above analyses reveals that the performances of the CPM in terms of kinematics, statics and dynamics are all dependent on its architectural parameters. Consequently, in order to fabricate a CPM for practical application, it is a key step to determine its dimensions by taking into account its performances simultaneously. To increase the natural frequency of the CPM, a lighter material, i.e., Al 7075 alloy (density = 2.81×103 kg/m3 , Young’s modulus = 71.7 GPa, yield strength = 503 Mpa, Poisson’s ratio = 0.33) is adopted, and the CPM mobile platform weight is reduced by removing unnecessary masses. In addition, to make a tradeoff between the stroke and resolution of the PZT, one type of PZT, namely, P-178.50 produced by the Physik Instrumente (PI), Inc., is selected to drive the 3-PRC CPM. The PZT actuator possesses a stroke of Q = 80 μm with a resolution of 1.6 nm. Additionally, the true values for the CPM stiffness and natural frequency are taken to be those generated from the FEA. Considering the differences between the calculated values and FEA results for the stiffness and natural frequency are within ±15%, a compensation factor η = 0.85 is adopted in the optimization process to compensate for the derived analytical models. 7.1 Optimization Statement With the aim of enhancing the accuracy property of the CPM, the optimal design problem can be stated as follows: • •

Variables to be optimized: a, l1 , t, r and w. Minimize: Parasitic motions (e1 , e2 , e3 , ε1 , ε2 )

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•

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Subject to: 1. 2. 3. 4. 5. 6.

Actuation stiffness value Kq /η ≤ K PZ T Natural frequency 2 fctrl ≤ fmech η ≤ fmax /15 Parasitic motion elimination ensured by (15) Free of plastic deflections guaranteed by (24) with a safety factor na = 1.25 Accuracy valid range ensured by (20) Parameter ranges: 60 mm ≤ a ≤ 160 mm, 60 mm ≤ l1 ≤ 160 mm, 0.3 mm ≤ t ≤ 2 mm, 2 mm ≤ r ≤ 6 mm, and 10 mm ≤ w ≤ 16 mm

In this optimization, five parameters (n = [a, l1 , t, r, w]T ) need to be optimized since the parameter l can be determined with an offset from the parameter a and other parameters have little effects on the CPM performances. For the sake of minimizing the parasitic motions, the objective function for minimization is taken as: f (n) = e21 + e22 + e23 + ε12 + ε22          2Q 2 2Q 2 2Q 2 = l 2 1 − cos + l12 1 − cos + a2 1 − cos l l1 a 2 2

2+π t 2+π t t4 144t4 2− 2− + + (45) 4 81 π r 81w π r As far as the constraint conditions are concerned, the actuation stiffness of the CPM should not exceed the stiffness of the adopted PZT, i.e., K PZ T = 17.6 N/μm. The natural frequency should be no less than two times of the frequency of the control system which is taken to be fctrl = 18 Hz in this case, and be no greater than one fifteenth of the maximum sampling rate fmax = 1000 Hz of the numerical control so as to avoid exciting resonance of the system. Meanwhile, the CPM should be designed with the elimination of some parasitic motions and plastic failure phenomena for the safety reason. Besides that, since the CPM will be manufactured by the wire EDM (electrical discharge machining) process, the thinnest portion of the notch hinge should be no less than 0.3 mm corresponding the maximum tolerance of ±0.01 mm. Additionally, the minimum value of the distance a is restricted by the length of the adopted PZT (68 mm) with the addition of a proper assembling space, and the upper bounds for design variables are all limited so as to generate a compact manipulator. 7.2 PSO Optimization and Results Compared to the genetic algorithm (GA), the PSO has no evolutionary operators such as crossover and mutation. Thus, from the viewpoint of programming, the advantages for PSO are ease to implement and few parameters to adjust. In the current five-dimensional optimization problem, a particle can be described by Xi = (xi1 , xi2 , xi3 , xi4 , xi5 ) with the particle velocity Vi = (vi1 , vi2 , vi3 , vi4 , vi5 ), which corresponds to a set of the CPM design variables (a, l1 , t, r, w). In the n-th generation, there are N particles popn = (X1 , X2 , · · · , X N ), where the population size N is set to be 40 for the current problem. The inertia weight w determines the impact of previous velocities on the current velocity, and the initial and final values are selected as 0.9 and 0.4, respectively, where 1000 epoches are allowed to take from the initial

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Table 2 Optimized dimensions of a 3-PRC CPM

Value (mm)

Parameter a

l1

t

r

w

120.00

120.00

1.19

2.97

10.00

value to the final one linearly. In addition, the local and global acceleration constants are assigned as c1 = 2.0 and c2 = 2.0, respectively. And the swarm is manipulated by the equations [24]:



  Vi (k + 1) = wVi (k) + c1 r1 Pi (k) − Xi (k) + c2 r2 Pg (k) − Xi (k) (46) Xi (k + 1) = Xi (k) + Vi (k + 1)

(47)

where r1 and r2 are random numbers uniformly distributed between 0 and 1, and the particle index i = 1, 2, · · · , N. As far as the termination criterion is concerned, three items are set. One criterion is the maximum number of iterations (3000) for the optimization procedure, another one is the minimum global error gradient (1.0E-6) which is the error between two neighboring particles with the best fitness values, and the third one is the maximum number of iterations without error change, that is chosen as 500. The optimization is implemented with MATLAB via a PSO toolbox developed by [25], and the PSO is initialized with random start values within the search space. Ten independent runs are carried out on a personal computer, and one convergence process is plotted in Fig. 15. Although the PSO has the stochastic property, i.e., it makes random choices, each run converges to almost the same value for the objective function due to the reason that the selected termination criteria are sufficient. The optimized CPM dimensions are described in Table 2. 7.3 Performance Validation with FEA The FEA is accomplished on a virtual prototype of the optimized CPM with the above parameters. As shown in Fig. 16, the 20-node element SOLID186 is adopted Fig. 16 FEM of the optimized 3-PRC CPM

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Fig. 17 Relationship between the output motion and parasitic motion of the improved 3-PRC TPM

0.02 0.018 0.016 Parasitic motion

0.014

x translation (mm) z translation (mm) x rotation (degree) y rotation (degree) z rotation (degree)

0.012 0.01 0.008 0.006 0.004 0.002 0 0

0.02

0.04

0.06 0.08 0.1 0.12 Output displacement (mm)

0.14

0.16

to create the finite element models of the three limbs, while the mobile platform is meshed using the 10-node element SOLID187 to enhance the calculation efficiency. And the six fixing holes are constrained in the simulation. The results show that the actuator stiffness of the CPM is 128.6 N/mm, which is far less than that provided by the adopted PZT actuator. Moreover, the CPM possesses a high natural frequency of 35.0 Hz, and a workspace range of 141×141×141 μm3 with a resolution of 2.8 nm. Besides, the parasitic motions are shown in Fig. 17, which indicates that the maximum values are ux = 10.9 μm, uz = 0.6 μm, θx = 0.002 degree, θ y = 0.001 degree, and θz = 0.019 degree, respectively. We can observe that when the CPM translates along the y direction, its parasitic translation in the z direction has been greatly 0.6 reduced to 141 = 0.4%. While the parasitic translation along the x-axis is 10.9 = 7.7%, 141 which is relatively large and can be compensated by control the PZT actuating in the x direction. Additionally, the parasitic rotations around the three axes are all not significant and can be neglected. So, the 3-PRC CPM can be seen as a partially decoupled XYZ micromanipulator. And the FEM results also validate the effectiveness of the conducted dimension optimization for the 3-PRC CPM.

8 Conclusion The structure improvement and dimension optimization for a 3-PRC CPM are carried out in details in this paper. The previous version of CPM has been modified in considerations of limb architecture assembly, actuation joint type, actuation direction, and displacement amplification mechanism adoption, etc., which results in a new CPM with improved performances. The FEA results show that the workspace volume of the improved CPM is enlarged by 7.9 times along with the parasitic motions reduced tremendously, and both stiffening and buckling phenomena have been eliminated. Hence, the new 3-PRC CPM is more suitable for the application in micro/nano scale manipulation. Furthermore, the mobility of this new CPM is determined by resorting to the screw theory, while the kinematics are solved based on PRB model approach. The CPM workspace subjected to elastic limits of flexure hinges are determined and

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the parasitic motions are formulated analytically. Meanwhile, both statics and dynamics models of the CPM are derived and then verified through the FEA. Based upon the derived models, the optimal dimension design with the goal of minimizing the CPM parasitic motions subjected to the stiffness and natural frequency requirements under other parameter constraints are conducted by means of the particle swarm optimization. The FEA results show that the optimized CPM possesses a partially decoupled translation with other performances suited well with the requirements. The main contribution of this research is the design improvement of a new CPM with the generation of its mathematical models and optimal dimensions considering kinematics, statics, and dynamics performances. The results presented in this paper provide a sound base for the hardware development of a 3-PRC CPM for such applications as biological cell injection, micro components assembly and so on, which are planned in our future investigations. In addition, other approaches to enhance the CPM precision and experimental studies to verify the CPM accuracy will be attempted as well. Acknowledgements The authors appreciate the fund support from the research committee of University of Macau under Grant No. RG-UL/07-08S/Y1/LYM/FST and Macao Science and Technology Development Fund under Grant No. 069/2005/A.

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