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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014

Design of a Double Triangular Parallel Mechanism for Precision Positioning and Large Force Generation Hyunpyo Shin and Jun-Hee Moon

Abstract—This paper presents the design of a double triangular parallel mechanism for precision positioning and large force generation. In recent years, with the acceleration of miniaturization in mobile appliances, the demand for precision aligning and bonding has been increasing. Such processes require both high precision and large force generation, which are difficult to obtain simultaneously. This study aimed at constructing a precision stage that can perform submicrometer resolution alignment, several-hundred micrometer stroke, and several-hundred Newton force generation. Piezoelectric actuators were tactically placed to constitute a parallel mechanism with a double triangular configuration. In addition, flexure hinges were carefully designed and optimized. The three actuators in the inner triangle function as an in-plane positioner, whereas the three actuators in the outer triangle as an out-of-plane positioner. To facilitate rapid control of the developed stage, two mapping matrices were derived from the inverse and forward kinematics and the corresponding iterative numerical calculations. Finite-element analyses and experimental results proved that the developed stage with the double triangular configuration sufficiently met the requirements of precision positioning and large force generation. Index Terms—Design optimization, flexure mechanism, kinematic analysis, parallel mechanism, precision stage.

I. INTRODUCTION RECISION positioning stages are designed for micro-/ nano-precision positioning and aligning of an object in a plane or space. They have facilitated precision manufacturing, assembly, and measurement. For example, precision positioning stages have been critical in laser welding; in lithography for semiconductors, and thin-film transistor liquid crystal displays [1], [2]; and in probe and object positioning of atomic force microscopes and scanning probe microscopes [3]. To ensure superior resolution and accuracy, precision positioning stages use actuators, sensors, and linkages that differ from those of conventional positioners. Especially, lead zirconate titanate (PZT) actuators, thermo–mechanical actuators, pressure-elongated cylinder, and magnetostrictive materials are used as actuators for precision positioning. PZT is widely used due to its high energy conversion efficiency, theoretically infinitesimal resolution, high stiffness, and fast response. As

P

Manuscript received September 8, 2012; revised January 22, 2013; accepted April 23, 2013. Date of publication May 22, 2013; date of current version April 11, 2014. Recommended by Technical Editor G. Schitter H. Shin is with the School of Robot and Automation Engineering, Dongyang Mirae University, Seoul 152-714, Korea (e-mail: [email protected]). J.-H. Moon is with the Department of Mechanical Design, Yuhan University, Bucheon 422-749, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2013.2261822

for sensors, capacitive proximity sensors, laser interferometers, and linear variable differential transformers are used because of their high resolution. For kinematic linkages, flexure hinges are predominantly used for force and displacement transmission without wear and backlash [4], [5]. Precision positioning stages are representatively used for the stepper in semiconductor processes and as devices for various photolithography processes. Most of them are designed for planar alignments such as x-, y-translations or x-, y-translations with z-rotation, depending on the process. However, advancements in micromanufacturing, biotechnology, optical measurement, and wafer aligning processes require spatial movement and alignment using more degrees-offreedoms (DOFs) [6], [7]. In addition, a large force generation is essential for micro-/nano-processes such as nanoimprinting and wafer bonding for smart device components. Thus, six-DOF precision positioning stages with high resolution and large force generation are needed. Various actuators and kinematic structures have been devised for six-DOF motion stages and applied. Gao and Swei [8] combined the in-plane (x-, y-translations, z-rotation) and out-ofplane (x-, y-rotations, z-translation) motions in a serial mechanism. Moreover, a PZT actuator consisting of a monolithic leaf spring and a preload mechanism has been designed. A six-DOF magnetically levitated nanoprecision positioning system with a long travel range was presented in [9]; it has high resolution and a relatively long stroke. However, the maximum payload is limited to tens of Newton. Choi and Gweon [10] applied a Halbach linear active magnetic bearing (HLAMB) for six-DOF high precision stage in gravity compensation and control of the out-of-plane motion. In addition, a small-scale nanopositioner, the μHexFlex, comprised of a six-axis compliant mechanism and three pairs of twoaxis thermo–mechanical microactuators is presented in [11]; this positioner is small with high natural frequency. However, it is also not appropriate to large force generation applications because of low structural stiffness. A high-precision positioner using a superimposed concentrated field permanent magnet is presented in [12]; it has a high translational resolution of 20 nm, but cannot generate large forces. The wafer stage of a six-DOF compliant mechanism for a single-step nanoimprint lithography is proposed in [13]. The proposed wafer stage consists of two kinds of flexure-based mechanisms: an inner mechanism and an outer mechanism. The outer mechanism, which is for out-of-plane motion, has three leaf-type flexures between the connection frame and the outer frame. The inner mechanism for the in-plane motion consists of a moving plate at the center, a connection frame that surrounds the moving plate and three

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SHIN AND MOON: DESIGN OF A DOUBLE TRIANGULAR PARALLEL MECHANISM

pairs of inner flexures. A piezoelectrically actuated six-DOF stage for micropositioning is developed in [14] by using orthogonal actuators and lever linkages. As for the control of a precision positioning stage, Janaideh et al. [15] formulated a generalized Prandtl–Ishlinskii model to compensate for hysteresis nonlinearities of smart actuators. Kenton and Leang [16] evaluated four approaches (PID feedback, discrete-time repetitive control, and so on) to control the lateral motion of a nanopositioning stage for precision tracking scanning probe microscopy. To estimate the stiffness of a flexure hinge, the fundamental mathematical equations of flexure hinges were suggested by Paros and Weisbord [17]. In this study, design of a novel double triangular parallel mechanism (DTPM) and a vertical actuation unit (VAU) are presented to obtain both high-resolution and large-force generation. Four performance objectives of the precision stage design were set as follows. The first objective is to achieve high resolution of submicron motions. For this objective, the precision stage is designed by using flexure hinges, which have no backlash, and PZT actuators, which have theoretically infinitesimal resolution and low thermal expansion under the Curie temperature [18], [19]. The second is to achieve large-force generation. The DTPM is a parallel mechanism in which two independent parallel mechanisms of triangular shape are combined into one to generate sixDOF motion. Compared to the serial mechanism, which positions an end-effector by using a series of stacked or nested stages comprised of a serial kinematic chain, a parallel mechanism has a fixed base and a movable end-effector, which are directly connected to multiple independent kinematic chains. Flexure hingebased parallel mechanisms are frequently adopted for precision positioning since they have relatively higher stiffness than serial mechanisms with flexure hinges [4]. Moreover, the actuators for the horizontal and VAUs construct two kinds of equilateral triangles to share payload equally. Hinge design optimization contributes to large force generation of the precision stage because it finds the hinge thicknesses that minimize normal and bending stresses in the hinges. Since structural parameters such as link lengths were decided to have minimum values in the early design step, we optimized only the hinge thicknesses. The third is to enlarge workspace with a restricted actuator stroke. To enlarge the out-of-plane motion workspace, lever linkage is applied to VAU. By placing horizontal actuation units (HAUs) and constructing the inner triangle close to the center of the precision stage, the z-rotation workspace among the in-plane motion was enlarged. The fourth is to simplify kinematics. Two equilateral triangular placements of actuation units can organize inverse and forward kinematics in a simple mathematical form. Moreover, by applying the kinematic analyses results and defining constant mapping matrices, real-time feedback control can be realized. This paper is organized as follows. DTPM and VAU are explained in Section II. Design optimization of the flexure hinges is presented in Section III with the formulation of the performance index. Inverse and forward kinematic analyses followed by the design of the mapping matrices are presented in Section IV. The control system design and the experimental re-

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Fig. 1. Positions and directions of the actuators and sensors in the developed double triangular parallel mechanism.

Fig. 2.

Exploded view displaying three mechanical parts.

sults for performance verification are presented in Section V. Finally, some concluding remarks are provided in Section VI. II. KINEMATIC STRUCTURE OF THE SIX-DOF PRECISION POSITIONING STAGE In this section, the kinematic structures of the DTPM and VAU are discussed. The positions and directions of the actuators and sensors in the developed stage is shown in Fig. 1. Bi and Si represent actuator and sensor, respectively. B1 , B2 , and B3 actuators produce in-plane motion (x-, y-translations and z-rotation). B4 , B5 , and B6 actuators produce out-of-plane motion via lever linkages (x-, y-rotations and z-translation). With regard to sensing, S1 , S2 , and S3 sensors detect in-plane motion (x-translation and z-rotation are detected by S1 and S2 sensors, and y-translation is detected by S3 alone). S4 , S5 , and S6 sensors detect out-of-plane motion (x-, y-rotations and z-translation). On the whole, the precision stage is composed of a fixed part, and an actuation part, and a moving part as shown in Fig. 2. The actuation part has two kinds of components: HAU and

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014

Fig. 4. Stroke amplification mechanism and overlapping structure of the vertical actuation unit.

Fig. 3.

Hinges (circles) and generation of motions (arrows).

VAU. Each actuation unit has a PZT actuator and several flexure hinges. Among many types of hinges (circular, corner-filleted, parabolic, hyperbolic, elliptic, etc) that are utilized to deliver precision motion [20], the circular type hinge is adopted due to the ease of machining and the simplicity of corresponding geometry which facilitate clear expression for robotics. The HAU has only 2-D hinges and the VAU has both 1-D and 2-D hinges as annotated in Fig. 3. A. Double Triangular Parallel Mechanism The inner triangular configuration consists of actuation units for in-plane motion. Similarly, the outer triangular configuration is composed of actuation units for out-of-plane motion. All the actuators contribute six-DOF motion by connecting every action point of force of actuation units to the moving part (endeffector). The distances between actuators and the stage center are equal. The in-plane motion includes x-, y-translations, and z-rotation while the out-of-plane motion includes x-, y-rotations, and z-translation. As shown in Fig. 3, the inner triangle generates the in-plane motion and the outer triangle generates the out-of-plane motion. Especially, the VAU of the outer triangle configuration determines the height of the stage. The horizontally placed PZT actuator and lever linkage of the VAU reduce the stage height, resulting in increasing the horizontal stiffness of the stage. The advantages of the DTPM are as follows. Uniform vertical load distribution to minimize the maximum stress in each hinge is easily realized in this triangular configuration. Moreover, the triangular configuration facilitates rotational motions. All the actuation units related to in-plane and out-of-plane motions are located on the same plane. B. Vertical Actuation Unit The DTPM is composed of three HAUs and three VAUs. The VAU is carefully designed for large force generation and minimum height. In Fig. 4, the side view of the VAU is displayed.

The VAU is designed to have several technological features. Most of all, to amplify the stroke of PZT actuator and to increase the stiffness of whole stage by reducing the height of actuation unit, the VAU is designed to have a lever linkage structure. Generally, a PZT actuator generates displacement by as much as 0.1% of its length. To realize a stroke longer than 300 μm, we must select a PZT actuator of 300 mm or more in length. If we place the actuator vertically on the base of the stage equipped with hinges and mechanical components for mounting, the height of the stage becomes much longer than the length of actuator. However, in this precision stage, the actuator is placed horizontally by applying a lever linkage structure and rotating the displacement direction from horizontal to vertical at the motion amplification ratio, so the height of VAU is minimized and motion range is amplified. Moreover, all the hinges are orthogonally aligned in horizontal and vertical directions to reduce parasitic motion. To minimize the width and length as well as the height of the stage, the VAU should be designed to have minimum total length, while including hinge blocks. Through this study, the VAU is designed with a kinematic structure in which the actuator and hinge blocks almost fully overlap. A pole hinge block, which has two 2-D hinges and a hole, is located at the middle of the PZT actuator. The front hinge and back hinge blocks are designed for maximum overlap. Most of the hinge blocks are overlapped by the actuator, except for the space of mounting. To quantify the overlapping ratio, Ro is defined as the ratio of the length of the actuator La to the total length of the VAU, Lt , as in Ro =

La × 100. Lt

(1)

In our case, the overlapping ratio of the VAU was 88.8%. This means that most of the mechanical parts and the actuator of the VAU overlapped each other and consequently, the VAU occupies minimum apace. This overlapping design is expected to be very useful for designing a compact positioning stage. III. DESIGN OPTIMIZATION OF HINGES A. Mathematical Formulation and Optimization Results During the operation of a precision stage, actuation forces and external load are delivered to the hinges, which then experience

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normal and bending stresses. If the total stress exceeds the yield strength of the hinge, the hinge will deform permanently. Similarly, if the total stress exceeds the ultimate strength, the hinge will fracture. Equations(2) and (3) represent the normal and bending stresses, respectively. F is the applied external force, A is the cross- sectional area of the neck of the hinge, M is the moment at the hinge, Kθ is the rotational stiffness, I is the moment of inertia, θ is the rotation angle of hinge, and y is the position of maximum stress. In (3), ri is the radius of the hinge circle of a 2-D hinge, E is the Young’s modulus, t is the hinge thickness, and h is the width of a 1-D hinge F A Kθ θ t M σb = y= I I 2

σn =

where I =

I=

π 4 t , 64 i

(2)

7/2

Kθ =

1 hi t3 , 12 i

Ei t i

1/2

,

i = 0, 3

20ri

5/2

Kθ =

2Ei hi ti

1/2

,

i = 1, 2.

(3)

Fig. 5. Optimal hinge thicknesses that minimize stresses (a) Thickness of t0 . (b) Thickness of t1 . (c) Thickness of t2 . (d) Thickness of t3 .

9πri

Total stresses in 2-D and 1-D hinges are represented in (4) and (5), respectively. The right sides of (4) and (5) are composed of normal and bending stress terms ti Fi 8 E θi,m ax , i = 0, 3 (4) σi = π 2 + 5π ri 4 ti ti Fi 12 σi = E + θi,m ax , i = 1, 2. (5) ti hi 9π ri

VAU. In addition, (c) is related to the 1-D lever hinge of VAU and (d) the 2-D hinges of the pole hinge block of VAU. All the values determined by this optimization were applied to the precision stage manufacturing except for t2 , which was increased from 0.3 to 1 mm because of difficulties in manufacturing.

The hinge thickness appearing in both of the stress terms is considered a more dominant design parameter than the radius of the hinge circle [20] and the stress concentration factors at the necks of the hinges are all less than 1.2 when the hinge thicknessto-radius ratio does not exceed 0.5 [21]. We selected a radius such that the ratio does not exceed 0.5. Hinge thickness can be changed more easily than the hinge radius without restriction of space; therefore, the hinge thickness was set as the design variable. To decrease the normal stress in the hinge, the cross-sectional area of the neck of hinge should be widened. However, to decrease the bending stress, the cross-sectional area should be narrowed because the bending stress is proportional to the square root of the hinge thickness. To achieve stress minimization, the total stress is set as a performance index as defined

In ascertaining the force generation and the maximum stress in the optimized hinges of a precision stage, static finite-element analysis (FEA) is performed with respect to the HAU and VAU by using ANSYS Workbench with the modeling data of SolidEdge. In Figs. 6 and 7, stress analysis results with magnified views of the hinges and deformation analysis results are displayed. The FEA result of the HAU is displayed in Fig. 6(a). A lateral force of 100 N was applied as a simulation condition. Through static FEA, the maximum stress of 56.6 MN/m2 was obtained both in the front and back hinges. It did not exceed the yield stress of stainless steel SUS420 (1360 MN/m2 ), which was adopted as the material of the hinges. For the VAU displayed in Fig. 6(b), a vertical force of 250 N was applied as a simulation condition, which is bigger than one third of the sum of the payload (500 N) and the stage moving unit weight (200 N). The payload is determined by the wafer bonding process. The front upper 1-D hinge, which is magnified in Fig. 6(b), is under maximum stress of 152.4 MN/m2 . The FEA results for the whole precision stage are presented in Fig. 7. The actuation part, moving part, and fixed part are included in the FEA. Target force generation of 500 N and the weight of moving unit of the precision stage were applied as analysis conditions. The maximum stress in the front upper hinge of the VAU was similar to that of the unit level analysis case. This result shows that the VAUs undertook most of the

min σ = σn ,m ax + σb,m ax t

subject to σ ≤ σY .

(6)

In (6), σn ,m ax , σb,m ax , and σY represent maximum normal and bending stresses, and yield strength, respectively. The optimization result is displayed in Fig. 5. In the figure, (a) is related to the 2-D hinge of HAU and (b) the 1-D hinges, which are included in the front and back hinge blocks of the

B. Static Finite-Element Analysis Results

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Fig. 8. Actuator arrangement and notations of the precision stage. (a) Top view. (b) Side view. Fig. 6. Static FEA of the horizontal and vertical actuation units. (a) Horizontal actuation unit. (b) Vertical actuation unit.

Fig. 9.

Fig. 7.

Static FEA results of the whole precision stage.

vertical force. It also shows the independence of roles of the inplane and out-of-plane motion parts with respect to the carried load. By the FEA results, the hinge optimization process is considered to be helpful in designing positioning stages with large force generation. IV. KINEMATIC ANALYSES AND DERIVATION OF MAPPING MATRICES A. Inverse Kinematics Inverse kinematics is investigated to find actuator displacements for certain positions and orientations of the end-effector (in this stage: top center of the moving part). Top and side views of the actuator arrangement are displayed in Fig. 8. Bi designates the actuator. pi represents the front hinge position for HAU and the upper pole hinge position for VAU. qi represents the back hinge position for HAU and the

Schematic diagram of the vertical actuation unit.

lower pole hinge position for VAU, respectively (pi is moving and qi is fixed in HAU and moving in VAU). l is the distance between the front hinge and back hinge positions, and L1 and L2 are distances from the stage center to the center lines of HAUs and VAUs, respectively. LH is the distance between the front hinge positions of two HAUs. m is the distance between two hinges of the pole hinge block and zL is the height difference between the pi hinge positions of in-plane and out-of-plane. In Figs. 3 and 8, a coordinate system is introduced for the kinematic analysis. The origin is located at the center of the actuator configuration. x-axis is parallel to B1 actuator. The actuating directions of the VAUs (denoted by arrows in Fig. 3) are parallel to z-axis. Thus,y-axis is determined by the righthand rule for the Cartesian coordinate system. Fig. 9 is a kinematic equivalent of the VAU shown in Fig. 4. The VAU consists of one PZT actuator, three 1-D hinges, and two 2-D hinges. The PZT actuator can be regarded as a prismatic joint, 1-D hinge as a revolute joint, and 2-D hinge as a spherical joint, respectively, as shown in Fig. 9. In the figure, S, U , R, and P represent spherical, universal, revolute, and prismatic joints, respectively. The pole hinge block has two 2-D hinges which are connected to the moving part and the front hinge block,

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1 2 2 2 (zp − zq ,0 ) − (zp − zq ,0 ) − pi − qi,0 + m . ai = AR (11) In (9)–(11), AR, qi,0 , and qi represent the amplification ratio, and the initial and final positions of the lower hinge of the pole hinge block, respectively. zq ,0 and zp mean the z-axis coordinate values of qi,0 and pi . r1 and r2 are the distances from the center lever hinge to the centers of the front hinge and back hinge. ai is the actuator displacement, and Ai is the displacement amplified by the lever hinge. m is the distance between pi and qi . Equation (11) is derived from (9) and (10), then the displacements a4 , a5 , and a6 are determined by (11). These results will be applied to design the mapping matrices for control. Fig. 10.

B. Forward Kinematics

Schematic diagram of the whole precision stage.

respectively. The pole hinge block moves in z-axis direction through the front hinge block, which perpendicularly transmits the displacement generated by the PZT actuator. Therefore, the VAU can be simplified into an equivalent system on the righthand side of Fig. 9. For succeeding kinematic analysis, we made kinematic equivalents of the actuation units constituting double triangular configuration as shown in Fig. 10. Two triangular lines are constructed by connecting the lines of action of actuation units corresponding to the in-plane motion and the out-of-plane motions, respectively. The inverse kinematics starts from (7). pi = T + (Rx + Ry + Rz − 2I)pi,0 , pi ∈ 3×1 , i = 1 ∼ 6.

(7)

In (7), pi,0 is a vector representing the initial position of pi . T is the translation vector. Rx , Ry , and Rz , are the rotation matrices related to the x-, y-, and z-axes, respectively. I is the identity matrix. When the stage goes through translation and rotation, pi,0 becomes pi . This formulation of linear form is applicable because the workspace of the flexure-based precision stage is very small. ai = pi − qi − l, i = 1, 2, 3 .

(8)

Displacements a1 , a2 , and a3 of HAUs are obtained directly by (8). However, the displacements of VAUs cannot be obtained immediately. The position of the upper pole hinge of VAU pi can be obtained by (9). From the position of upper pole hinge pi , the corresponding lower pole hinge position qi should be determined. Consecutively, the lever hinge rotation, front hinge position, and PZT actuator displacements are determined from the lower pole hinge positions. From the schematic diagram shown in Fig. 9, we derived (10) and (11) to determine the actuator displacements (9) pi − qi = m, i = 4, 5, 6 r2 qi = qi,0 + [ 0 0 Ai ], Ai = AR × ai , AR = (10) r1

Forward kinematics is analyzed to find the final position and orientation of the end-effector for given displacements of actuators. The analysis results are used for ascertaining the workspace of the precision stage and for designing the real-time controller. Unlike serial mechanisms, parallel mechanisms have forward kinematics that cannot be uniquely determined, which means that a numerical method needs to be applied to find the solution. We defined the 18 constraints in terms of distance, because the precision stage is controlled using displacement information. Distance relationships between the six actuation units producing the in-plane and out-of-plane motions are depicted in Fig. 11. The upper triangle (Plane #1) is configured by connecting three front hinge centers related to the out-of-plane motion. In the same way, the lower triangle (Plane #2) is configured by connecting three upper pole hinge centers related to the in-plane motion. We applied the following simple distance constraints to the forward kinematics because all the actuation units are symmetrically placed in equilateral triangular forms. The constraints are represented as follows. Constraint #1: Distances between the positions of the front hinges of HAUs are constant. pi − pj 2 − L2H = 0, i, j = 1, 2, 3 (i = j).

(12)

Constraint #2: Distances between the positions of the front hinges of HAUs and the upper pole hinges of VAUs are constant as L3 or L4 . pi − pj 2 − L23 = 0, i = 1, 2, 3 , j = 4, 5, 6

(j = i + 3) (13)

pi − pj − 2

L24

= 0, i = 1, 2, 3

j = i + 3.

(14)

Constraint #3: Distances between the positions of the front and back hinges of HAUs are equal to the summation of the initial length l and the displacement of the actuator ai (i = 1, 2, 3) pi − qi 2 − (l + ai )2 = 0, i = 1, 2, 3.

(15)

Constraint #4: Distances between the positions of the upper pole hinges and the initial positions of the lower pole hinges of VAUs are equal to the total distance m and the displacements

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Fig. 11.

Geometrical relationships between the plane #1 and plane #2.

Fig. 12.

Various views of the in-plane workspace (a) x-y view (b) x-θz view (c) y-θz view.

Fig. 13.

Various views of the out-of-plane workspace (d) θx -θy view (f) θy -z view (e) θx -z view.

of the actuators ai (i = 4, 5, 6), which is multiplied by the amplification ratio AR, pi − qi,0 2 − (m + AR · ai )2 = 0, i = 4, 5, 6.

(16)

These constraints are consolidated into (17) and (18). g(xn ) of (17) is the constraint function including (12)–(16) g(xn ) = 0

of the processes (x-, y-translations: ±50 μm, z-translation: ±150 μm, x-, y-rotations: ±244 μrad, z-rotation: ±733 μrad) at the wafer level. Especially, the workspace requirements emphasize z-translation and x-, y-rotations. With the help of the stroke amplification mechanism of the VAU, the requirements are met.

(17) −1

xn+1 = xn − ∇g(xn )

· g(xn ), xn ∈

18×1

.

(18)

Equation (18) implies the Newton–Raphson method applied in forward kinematics to find the final position and orientation of the moving part by numerical convergence. xn represents 18 coordinate values of the neck centers of six hinges (p1 , p2 , p3 , p4 , p5 , p6 ) connected to the moving part (see Fig. 8). The workspaces of the in-plane and out-of plane motions obtained from the forward kinematics analysis are shown in Figs. 12 and 13. The results meet the workspace requirements

C. Mapping Matrices Formulation For the real-time controller design, the two mapping matrices, Ra and Rs are derived in this section. Although the actuator displacements and the position and orientation of the end-effector can be perfectly determined by the inverse and forward kinematics, the corresponding calculation time is too long for real time application. Because the ratio of the motion range to the size of the precision stage is below 0.3%, we carefully extracted the linear mapping matrices, which can be applied to positioning

SHIN AND MOON: DESIGN OF A DOUBLE TRIANGULAR PARALLEL MECHANISM

Fig. 14.

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Schematic diagram of the closed-loop control.

control across the whole working space, from the inverse and forward kinematics through the following procedure. The control loop is shown in Fig. 14. When a reference command x∗ is given, Ra produces actuator displacements a. The inverse of Rs computes the real movement of the precision stage x from the sensor signal s. Because x is too small, a and s can be linearly approximated as shown in (19) and (20), where i and j represent the numbers of row and column, respectively,

∂ai ∗ a = Ra x , Ra = ∈ 6×6 ∂xj

ai

(19) when xj 1 xj

∂si s, R = x = R−1 ∈ 6×6 s s ∂xj

si

(20) when xj 1. xj Finally, the Ra and Rs are represented as ⎡ −1 0 0 0 0.042 ⎢ ⎢ 0.500 0.866 0 0.036 −0.021 ⎢ ⎢ 0.500 −0.866 0 −0.036 −0.021 ⎢ Ra = ⎢ ⎢ 0 0 1 0.118 0 ⎢ ⎢ 0 1 −0.059 0.102 ⎣ 0 0 ⎡

1 ⎢ ⎢1 ⎢ ⎢0 ⎢ Rs = ⎢ ⎢0 ⎢ ⎢ ⎣0 0

0

1

−0.059

0

0

−0.042

0

0

0

−0.042

−0.042

0

0

1

0.038

0.060

0

1

0.038

−0.060

0

1

−0.060

0

⎤

⎥ 0.073 ⎥ ⎥ 0.073 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦

−0.102

0

−1 −0.001

0.073

0 −0.112

(21) ⎤

⎥ 0.112 ⎥ ⎥ 0 ⎥ ⎥ ⎥. 0 ⎥ ⎥ ⎥ 0 ⎦ 0 (22)

V. CONTROLLER DESIGN AND EXPERIMENTAL RESULTS A. Controller Design The developed precision stage is manufactured through precision grinding, pin-hole fit, and bolting to secure high precision. Fig. 15 presents the hardware system configuration. To enlarge the out-of-plane workspace, PZT actuators for the out-of-plane motion were selected to have longer strokes than those for the in-plane motion (for the in-plane motion: maximum stroke of

Fig. 15.

Hardware system configuration.

150 μm, maximum load of 5000 N, and stiffness of 17 N/μm; for the out-of-plane motion: maximum stroke of 200 μm, maximum load of 15000 N, and stiffness of 35 N/μm). For sensing, capacitive gap sensors were selected for the high sensing resolution (sensing range of 500 μm and resolution of 10 nm). For signal processing, a real-time controller, which is interfaced to a host PC, processes the analog signal from −10 to 10 V. The real-time controller have 16-bit resolution and operates with the LabVIEW real-time module. PI-control is applied to the stage positioning. The proportional constants are 0.3, 0.4 and 0.2 for x-, y-, and z-translations; and 0.03, 0.025, and 0.2 for x-, y-, and z-rotations, respectively. The integral time constants are 0.00 005, 0.00 005, and 0.0 002 for x-, y-, and z-translations; and 0.00 005 for x-, y-, and z-rotations.

B. Experimental Results Test results of multistep responses across almost full travel ranges are shown in Fig. 16. In the figures, submicron multistep responses are displayed at the right upper side. According to the requirements of the wafer bonding process, the required resolution of translation and rotation were 0.1 μm and 1 μrad, respectively. The figures show that the precision stage meets the resolution specifications. As shown in Fig. 16, the x-, y-translational strokes of the precision stage were more than ±50 μm and that the z-translational stroke was more than ±150 μm. In addition, the x-, y-rotational strokes were more than ±733 μrad (±0.042◦ ) and that the z-rotational stroke was more than ±244 μrad (±0.014◦ ). The noise in the submicrometer step responses shown in the right upper corners of insets in Fig. 16, originated from electromagnetic noise induced by electric power and unsystematic white noise. The developed stage positions the end-effector under the weight of 500 N as shown in Fig. 17. Even though the transient responses become worse as the payload increases, the final location is still exact. The multistep responses show that the flexure hinges, which are the weakest parts of the developed stage, overcome the heavy weight and the PZT actuators generate large force successfully.

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Fig. 16. Experimental results for the multistep translations and rotations. (a) Multistep translations (left: x-axis, 20 μm; middle: y-axis, 20 μm; right: z-axis, 60 μm). (b) Multistep rotations (left: x-axis, 300 μrad; middle: y-axis, 300 μrad; right: z-axis, 100 μrad).

horizontal arrangement of the actuators by the lever linkage minimized the stage height and the overlapped structure in the VAU reduced the width and length of the precision stage. The developed precision stage is useful for the precision alignment of optical devices and nanoimprinting as well as to high payload precision positioning processes such as wafer level alignment and bonding. REFERENCES

Fig. 17. Experimental result for the multistep response in case of applying external force of 500 N (z-translation).

VI. CONCLUSION In this study, a six-DOF precision positioning stage with large force generation of 500 N as well as submicron resolution of 0.1 μm in translation and 1 μrad in rotation was designed. Flexure hinges and PZT actuators with the design of VAU, in which all the hinges were orthogonally aligned in horizontal and vertical directions, gave high resolution. The DTPM, which is smaller with higher stiffness than serial mechanisms, were devised for large force generation. Moreover, the hinge design optimization was performed for large force generation of the precision stage because it found the optimal hinge thicknesses that minimized normal and bending stresses. To enlarge workspace, the lever linkage was designed, and the actuation units were placed closer to the center of the precision stage. The mapping matrices with the results of inverse and forward kinematics were defined for real-time feedback control. The

[1] R. Yang, M. Jouaneh, and R. Schweizer, “Design and characterization of a low-profile micropositioning stage,” Precis. Eng., vol. 18, no. 1, pp. 20– 29, 1996. [2] K. Sugihara, I. Mori, T. Tojo, C. Ito, M. Tabata, and T. Shinozaki, “Piezoelectrically driven XYθ table for submicron lithography systems,” Rev. Sci. Instrum., vol. 60, no. 9, pp. 3024–3029, 1989. [3] A. R. Smith, S. Gwo, and C. K. Shih, “A new high-resolution twodimensional micropositioning device for scanning probe microscopy applications,” Rev. Sci. Instrum., vol. 65, no. 10, pp. 3216–3219, 1994. [4] Q. Yao, J. Dong, and P. M. Ferreira, “Design, analysis, fabrication and testing of a parallel-kinematic micropositioning XY stage,” Int. J. Mach. Tools Manuf., vol. 47, no. 6, pp. 946–961, 2007. [5] H. S. Kim, Y. M. Cho, and J. H. Moon, “Active vibration control using a novel three-DOF precision micro-stage,” Smart Mater. Struct., vol. 19, no. 5, 055001, 2010. [6] M. Y. Chen, S. C. Huang, S. K. Hung, and L. C. Fu, “Design and Implementation of a New Six-DOF maglev positioner with a fluid bearing,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 3, pp. 449–458, Jun. 2011. [7] S. Polit and J. Dong, “Development of a high-bandwidth XY nanopositioning stage for high-rate micro-nanomanufacturing,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 724–733, Aug. 2011. [8] P. Gao and S.-M. Swei, “A six-degree-of-freedom micro-manipulator based on piezoelectric translators,” Nanotechnology, vol. 10, pp. 447– 452, 1999. [9] S. Verma, W.-J. Kim, and J. Gu, “Six-Axis nanopositioning device with precision magnetic levitation technology,” IEEE/ASME Trans. Mechatronics, vol. 9, no. 2, pp. 384–391, Jan. 2004. [10] Y. M. Choi and D. G. Gweon, “A high-precision dual-servo stage using halbach linear active magnetic bearings,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 5, pp. 925–931, Oct. 2011.

SHIN AND MOON: DESIGN OF A DOUBLE TRIANGULAR PARALLEL MECHANISM

[11] S.-C. Chen and M. L. Culpepper, “Design of a six-axis micro-scale nanopositioner—μHexFlex,” Precision Eng., vol. 30, pp. 314–324, 2006. [12] T. Hu and W.-J. Kim, “Extended range six DOF high-precision positioner for wafer processing,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 6, pp. 682–689, Dec. 2006. [13] K.-B. Choi and J. J. Lee, “Passive compliant wafer stage for single-step nano-imprint lithography,” Rev. Sci. Instrum., vol. 76, 075106, 2005. [14] J.-H. Moon, H. J. Pahk, and B.-G. Lee, “Design, modeling, and testing of a novel 6-DOF micropositioning stage with low profile and low parasitic motion,” Int. J. Adv. Manuf. Technol., vol. 55, no. 1–4, pp. 163–176, 2011. [15] M. A. Janaideh, S. Rakheja, and C. Y. Su, “An analytical generalized Prandtl–Ishlinskii model inversion for hysteresis compensation in micropositioning control,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 734–744, Aug. 2011. [16] B. J. Kenton and K. K. Leang, “Design and control of a three-axis serialkinematic high-bandwidth nanopositioner,” IEEE/ASME Trans. Mechatronics, vol. 17, no. 2, pp. 356–369, Apr. 2012. [17] J. M. Paros and L. Weisbord, “How to design flexure hinges,” Mach. Des., vol. 37, no. 8, pp. l5l–156, 1965. [18] Piezomechanik GmbH (2012, Aug. 17). Catalog webpage. [Online]. Available: http://www.piezomechanik.com. Accessed [19] M. Kamlah, “Ferroelectric and ferroelastic piezoceramics—modeling of electromechanical hysteresis phenomena,” Continuum Mech. Thermodyn., vol. 13, no. 4, pp. 219–268, 2001. [20] S. T. Smith, Flexures: Elements of Mechanical Mechanisms. Boca Raton, FL, USA: CRC Press, 2000. [21] D. W. Pilkey and D. F. Pilkey, Peterson’s Stress Concentration Factors, 3rd ed. New York, NY, USA: Wiley, 2008.

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Hyunpyo Shin received the B.S. degree in biosystem engineering and the M.S. and Ph.D. degrees in mechanical and aerospace engineering from Seoul National University (SNU), Seoul, Korea, in 2002, 2004, and 2009, respectively. He was a Postdoctoral Researcher at the Institute of Advanced Machinery and Design (IAMD) at SNU from 2009 to 2010. He was a Senior Researcher at Samsung Electro-Mechanics, Suwon, Korea, from 2011 to 2013. He is currently an Assistant Professor in the School of Robot and Automation Engineering, Dongyang Mirae University, Seoul, Korea. His current research interests include the design of redundantly actuated parallel mechanisms, precision positioning stages, stiffness enhancement algorithms for robots, 6-DOF motion simulators, and multifunctional machine tools. Jun-Hee Moon received the B.S., M.S., and Ph.D. degrees in mechanical design and production engineering from Seoul National University (SNU), Seoul, Korea, in 1992, 1994, and 2002, respectively. From 2000 to 2004, he was a Research Engineer with SNU Precision Company, Ltd. From 2005 to 2008, he was a Senior Researcher at the Micro Thermal System Research Center at SNU. From 2009 to 2010, he was an Assistant Professor in the Department of Mechatronics, Daelim University College, Anyang, Korea. Since 2011, he has been a Professor in the Department of Mechanical Design, Yuhan University, Bucheon, Korea. His research interests include the analysis and control of precision systems using air pressure, piezoelectric materials, and mechanical linkages.

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014

Design of a Double Triangular Parallel Mechanism for Precision Positioning and Large Force Generation Hyunpyo Shin and Jun-Hee Moon

Abstract—This paper presents the design of a double triangular parallel mechanism for precision positioning and large force generation. In recent years, with the acceleration of miniaturization in mobile appliances, the demand for precision aligning and bonding has been increasing. Such processes require both high precision and large force generation, which are difficult to obtain simultaneously. This study aimed at constructing a precision stage that can perform submicrometer resolution alignment, several-hundred micrometer stroke, and several-hundred Newton force generation. Piezoelectric actuators were tactically placed to constitute a parallel mechanism with a double triangular configuration. In addition, flexure hinges were carefully designed and optimized. The three actuators in the inner triangle function as an in-plane positioner, whereas the three actuators in the outer triangle as an out-of-plane positioner. To facilitate rapid control of the developed stage, two mapping matrices were derived from the inverse and forward kinematics and the corresponding iterative numerical calculations. Finite-element analyses and experimental results proved that the developed stage with the double triangular configuration sufficiently met the requirements of precision positioning and large force generation. Index Terms—Design optimization, flexure mechanism, kinematic analysis, parallel mechanism, precision stage.

I. INTRODUCTION RECISION positioning stages are designed for micro-/ nano-precision positioning and aligning of an object in a plane or space. They have facilitated precision manufacturing, assembly, and measurement. For example, precision positioning stages have been critical in laser welding; in lithography for semiconductors, and thin-film transistor liquid crystal displays [1], [2]; and in probe and object positioning of atomic force microscopes and scanning probe microscopes [3]. To ensure superior resolution and accuracy, precision positioning stages use actuators, sensors, and linkages that differ from those of conventional positioners. Especially, lead zirconate titanate (PZT) actuators, thermo–mechanical actuators, pressure-elongated cylinder, and magnetostrictive materials are used as actuators for precision positioning. PZT is widely used due to its high energy conversion efficiency, theoretically infinitesimal resolution, high stiffness, and fast response. As

P

Manuscript received September 8, 2012; revised January 22, 2013; accepted April 23, 2013. Date of publication May 22, 2013; date of current version April 11, 2014. Recommended by Technical Editor G. Schitter H. Shin is with the School of Robot and Automation Engineering, Dongyang Mirae University, Seoul 152-714, Korea (e-mail: [email protected]). J.-H. Moon is with the Department of Mechanical Design, Yuhan University, Bucheon 422-749, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2013.2261822

for sensors, capacitive proximity sensors, laser interferometers, and linear variable differential transformers are used because of their high resolution. For kinematic linkages, flexure hinges are predominantly used for force and displacement transmission without wear and backlash [4], [5]. Precision positioning stages are representatively used for the stepper in semiconductor processes and as devices for various photolithography processes. Most of them are designed for planar alignments such as x-, y-translations or x-, y-translations with z-rotation, depending on the process. However, advancements in micromanufacturing, biotechnology, optical measurement, and wafer aligning processes require spatial movement and alignment using more degrees-offreedoms (DOFs) [6], [7]. In addition, a large force generation is essential for micro-/nano-processes such as nanoimprinting and wafer bonding for smart device components. Thus, six-DOF precision positioning stages with high resolution and large force generation are needed. Various actuators and kinematic structures have been devised for six-DOF motion stages and applied. Gao and Swei [8] combined the in-plane (x-, y-translations, z-rotation) and out-ofplane (x-, y-rotations, z-translation) motions in a serial mechanism. Moreover, a PZT actuator consisting of a monolithic leaf spring and a preload mechanism has been designed. A six-DOF magnetically levitated nanoprecision positioning system with a long travel range was presented in [9]; it has high resolution and a relatively long stroke. However, the maximum payload is limited to tens of Newton. Choi and Gweon [10] applied a Halbach linear active magnetic bearing (HLAMB) for six-DOF high precision stage in gravity compensation and control of the out-of-plane motion. In addition, a small-scale nanopositioner, the μHexFlex, comprised of a six-axis compliant mechanism and three pairs of twoaxis thermo–mechanical microactuators is presented in [11]; this positioner is small with high natural frequency. However, it is also not appropriate to large force generation applications because of low structural stiffness. A high-precision positioner using a superimposed concentrated field permanent magnet is presented in [12]; it has a high translational resolution of 20 nm, but cannot generate large forces. The wafer stage of a six-DOF compliant mechanism for a single-step nanoimprint lithography is proposed in [13]. The proposed wafer stage consists of two kinds of flexure-based mechanisms: an inner mechanism and an outer mechanism. The outer mechanism, which is for out-of-plane motion, has three leaf-type flexures between the connection frame and the outer frame. The inner mechanism for the in-plane motion consists of a moving plate at the center, a connection frame that surrounds the moving plate and three

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SHIN AND MOON: DESIGN OF A DOUBLE TRIANGULAR PARALLEL MECHANISM

pairs of inner flexures. A piezoelectrically actuated six-DOF stage for micropositioning is developed in [14] by using orthogonal actuators and lever linkages. As for the control of a precision positioning stage, Janaideh et al. [15] formulated a generalized Prandtl–Ishlinskii model to compensate for hysteresis nonlinearities of smart actuators. Kenton and Leang [16] evaluated four approaches (PID feedback, discrete-time repetitive control, and so on) to control the lateral motion of a nanopositioning stage for precision tracking scanning probe microscopy. To estimate the stiffness of a flexure hinge, the fundamental mathematical equations of flexure hinges were suggested by Paros and Weisbord [17]. In this study, design of a novel double triangular parallel mechanism (DTPM) and a vertical actuation unit (VAU) are presented to obtain both high-resolution and large-force generation. Four performance objectives of the precision stage design were set as follows. The first objective is to achieve high resolution of submicron motions. For this objective, the precision stage is designed by using flexure hinges, which have no backlash, and PZT actuators, which have theoretically infinitesimal resolution and low thermal expansion under the Curie temperature [18], [19]. The second is to achieve large-force generation. The DTPM is a parallel mechanism in which two independent parallel mechanisms of triangular shape are combined into one to generate sixDOF motion. Compared to the serial mechanism, which positions an end-effector by using a series of stacked or nested stages comprised of a serial kinematic chain, a parallel mechanism has a fixed base and a movable end-effector, which are directly connected to multiple independent kinematic chains. Flexure hingebased parallel mechanisms are frequently adopted for precision positioning since they have relatively higher stiffness than serial mechanisms with flexure hinges [4]. Moreover, the actuators for the horizontal and VAUs construct two kinds of equilateral triangles to share payload equally. Hinge design optimization contributes to large force generation of the precision stage because it finds the hinge thicknesses that minimize normal and bending stresses in the hinges. Since structural parameters such as link lengths were decided to have minimum values in the early design step, we optimized only the hinge thicknesses. The third is to enlarge workspace with a restricted actuator stroke. To enlarge the out-of-plane motion workspace, lever linkage is applied to VAU. By placing horizontal actuation units (HAUs) and constructing the inner triangle close to the center of the precision stage, the z-rotation workspace among the in-plane motion was enlarged. The fourth is to simplify kinematics. Two equilateral triangular placements of actuation units can organize inverse and forward kinematics in a simple mathematical form. Moreover, by applying the kinematic analyses results and defining constant mapping matrices, real-time feedback control can be realized. This paper is organized as follows. DTPM and VAU are explained in Section II. Design optimization of the flexure hinges is presented in Section III with the formulation of the performance index. Inverse and forward kinematic analyses followed by the design of the mapping matrices are presented in Section IV. The control system design and the experimental re-

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Fig. 1. Positions and directions of the actuators and sensors in the developed double triangular parallel mechanism.

Fig. 2.

Exploded view displaying three mechanical parts.

sults for performance verification are presented in Section V. Finally, some concluding remarks are provided in Section VI. II. KINEMATIC STRUCTURE OF THE SIX-DOF PRECISION POSITIONING STAGE In this section, the kinematic structures of the DTPM and VAU are discussed. The positions and directions of the actuators and sensors in the developed stage is shown in Fig. 1. Bi and Si represent actuator and sensor, respectively. B1 , B2 , and B3 actuators produce in-plane motion (x-, y-translations and z-rotation). B4 , B5 , and B6 actuators produce out-of-plane motion via lever linkages (x-, y-rotations and z-translation). With regard to sensing, S1 , S2 , and S3 sensors detect in-plane motion (x-translation and z-rotation are detected by S1 and S2 sensors, and y-translation is detected by S3 alone). S4 , S5 , and S6 sensors detect out-of-plane motion (x-, y-rotations and z-translation). On the whole, the precision stage is composed of a fixed part, and an actuation part, and a moving part as shown in Fig. 2. The actuation part has two kinds of components: HAU and

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Fig. 4. Stroke amplification mechanism and overlapping structure of the vertical actuation unit.

Fig. 3.

Hinges (circles) and generation of motions (arrows).

VAU. Each actuation unit has a PZT actuator and several flexure hinges. Among many types of hinges (circular, corner-filleted, parabolic, hyperbolic, elliptic, etc) that are utilized to deliver precision motion [20], the circular type hinge is adopted due to the ease of machining and the simplicity of corresponding geometry which facilitate clear expression for robotics. The HAU has only 2-D hinges and the VAU has both 1-D and 2-D hinges as annotated in Fig. 3. A. Double Triangular Parallel Mechanism The inner triangular configuration consists of actuation units for in-plane motion. Similarly, the outer triangular configuration is composed of actuation units for out-of-plane motion. All the actuators contribute six-DOF motion by connecting every action point of force of actuation units to the moving part (endeffector). The distances between actuators and the stage center are equal. The in-plane motion includes x-, y-translations, and z-rotation while the out-of-plane motion includes x-, y-rotations, and z-translation. As shown in Fig. 3, the inner triangle generates the in-plane motion and the outer triangle generates the out-of-plane motion. Especially, the VAU of the outer triangle configuration determines the height of the stage. The horizontally placed PZT actuator and lever linkage of the VAU reduce the stage height, resulting in increasing the horizontal stiffness of the stage. The advantages of the DTPM are as follows. Uniform vertical load distribution to minimize the maximum stress in each hinge is easily realized in this triangular configuration. Moreover, the triangular configuration facilitates rotational motions. All the actuation units related to in-plane and out-of-plane motions are located on the same plane. B. Vertical Actuation Unit The DTPM is composed of three HAUs and three VAUs. The VAU is carefully designed for large force generation and minimum height. In Fig. 4, the side view of the VAU is displayed.

The VAU is designed to have several technological features. Most of all, to amplify the stroke of PZT actuator and to increase the stiffness of whole stage by reducing the height of actuation unit, the VAU is designed to have a lever linkage structure. Generally, a PZT actuator generates displacement by as much as 0.1% of its length. To realize a stroke longer than 300 μm, we must select a PZT actuator of 300 mm or more in length. If we place the actuator vertically on the base of the stage equipped with hinges and mechanical components for mounting, the height of the stage becomes much longer than the length of actuator. However, in this precision stage, the actuator is placed horizontally by applying a lever linkage structure and rotating the displacement direction from horizontal to vertical at the motion amplification ratio, so the height of VAU is minimized and motion range is amplified. Moreover, all the hinges are orthogonally aligned in horizontal and vertical directions to reduce parasitic motion. To minimize the width and length as well as the height of the stage, the VAU should be designed to have minimum total length, while including hinge blocks. Through this study, the VAU is designed with a kinematic structure in which the actuator and hinge blocks almost fully overlap. A pole hinge block, which has two 2-D hinges and a hole, is located at the middle of the PZT actuator. The front hinge and back hinge blocks are designed for maximum overlap. Most of the hinge blocks are overlapped by the actuator, except for the space of mounting. To quantify the overlapping ratio, Ro is defined as the ratio of the length of the actuator La to the total length of the VAU, Lt , as in Ro =

La × 100. Lt

(1)

In our case, the overlapping ratio of the VAU was 88.8%. This means that most of the mechanical parts and the actuator of the VAU overlapped each other and consequently, the VAU occupies minimum apace. This overlapping design is expected to be very useful for designing a compact positioning stage. III. DESIGN OPTIMIZATION OF HINGES A. Mathematical Formulation and Optimization Results During the operation of a precision stage, actuation forces and external load are delivered to the hinges, which then experience

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normal and bending stresses. If the total stress exceeds the yield strength of the hinge, the hinge will deform permanently. Similarly, if the total stress exceeds the ultimate strength, the hinge will fracture. Equations(2) and (3) represent the normal and bending stresses, respectively. F is the applied external force, A is the cross- sectional area of the neck of the hinge, M is the moment at the hinge, Kθ is the rotational stiffness, I is the moment of inertia, θ is the rotation angle of hinge, and y is the position of maximum stress. In (3), ri is the radius of the hinge circle of a 2-D hinge, E is the Young’s modulus, t is the hinge thickness, and h is the width of a 1-D hinge F A Kθ θ t M σb = y= I I 2

σn =

where I =

I=

π 4 t , 64 i

(2)

7/2

Kθ =

1 hi t3 , 12 i

Ei t i

1/2

,

i = 0, 3

20ri

5/2

Kθ =

2Ei hi ti

1/2

,

i = 1, 2.

(3)

Fig. 5. Optimal hinge thicknesses that minimize stresses (a) Thickness of t0 . (b) Thickness of t1 . (c) Thickness of t2 . (d) Thickness of t3 .

9πri

Total stresses in 2-D and 1-D hinges are represented in (4) and (5), respectively. The right sides of (4) and (5) are composed of normal and bending stress terms ti Fi 8 E θi,m ax , i = 0, 3 (4) σi = π 2 + 5π ri 4 ti ti Fi 12 σi = E + θi,m ax , i = 1, 2. (5) ti hi 9π ri

VAU. In addition, (c) is related to the 1-D lever hinge of VAU and (d) the 2-D hinges of the pole hinge block of VAU. All the values determined by this optimization were applied to the precision stage manufacturing except for t2 , which was increased from 0.3 to 1 mm because of difficulties in manufacturing.

The hinge thickness appearing in both of the stress terms is considered a more dominant design parameter than the radius of the hinge circle [20] and the stress concentration factors at the necks of the hinges are all less than 1.2 when the hinge thicknessto-radius ratio does not exceed 0.5 [21]. We selected a radius such that the ratio does not exceed 0.5. Hinge thickness can be changed more easily than the hinge radius without restriction of space; therefore, the hinge thickness was set as the design variable. To decrease the normal stress in the hinge, the cross-sectional area of the neck of hinge should be widened. However, to decrease the bending stress, the cross-sectional area should be narrowed because the bending stress is proportional to the square root of the hinge thickness. To achieve stress minimization, the total stress is set as a performance index as defined

In ascertaining the force generation and the maximum stress in the optimized hinges of a precision stage, static finite-element analysis (FEA) is performed with respect to the HAU and VAU by using ANSYS Workbench with the modeling data of SolidEdge. In Figs. 6 and 7, stress analysis results with magnified views of the hinges and deformation analysis results are displayed. The FEA result of the HAU is displayed in Fig. 6(a). A lateral force of 100 N was applied as a simulation condition. Through static FEA, the maximum stress of 56.6 MN/m2 was obtained both in the front and back hinges. It did not exceed the yield stress of stainless steel SUS420 (1360 MN/m2 ), which was adopted as the material of the hinges. For the VAU displayed in Fig. 6(b), a vertical force of 250 N was applied as a simulation condition, which is bigger than one third of the sum of the payload (500 N) and the stage moving unit weight (200 N). The payload is determined by the wafer bonding process. The front upper 1-D hinge, which is magnified in Fig. 6(b), is under maximum stress of 152.4 MN/m2 . The FEA results for the whole precision stage are presented in Fig. 7. The actuation part, moving part, and fixed part are included in the FEA. Target force generation of 500 N and the weight of moving unit of the precision stage were applied as analysis conditions. The maximum stress in the front upper hinge of the VAU was similar to that of the unit level analysis case. This result shows that the VAUs undertook most of the

min σ = σn ,m ax + σb,m ax t

subject to σ ≤ σY .

(6)

In (6), σn ,m ax , σb,m ax , and σY represent maximum normal and bending stresses, and yield strength, respectively. The optimization result is displayed in Fig. 5. In the figure, (a) is related to the 2-D hinge of HAU and (b) the 1-D hinges, which are included in the front and back hinge blocks of the

B. Static Finite-Element Analysis Results

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Fig. 8. Actuator arrangement and notations of the precision stage. (a) Top view. (b) Side view. Fig. 6. Static FEA of the horizontal and vertical actuation units. (a) Horizontal actuation unit. (b) Vertical actuation unit.

Fig. 9.

Fig. 7.

Static FEA results of the whole precision stage.

vertical force. It also shows the independence of roles of the inplane and out-of-plane motion parts with respect to the carried load. By the FEA results, the hinge optimization process is considered to be helpful in designing positioning stages with large force generation. IV. KINEMATIC ANALYSES AND DERIVATION OF MAPPING MATRICES A. Inverse Kinematics Inverse kinematics is investigated to find actuator displacements for certain positions and orientations of the end-effector (in this stage: top center of the moving part). Top and side views of the actuator arrangement are displayed in Fig. 8. Bi designates the actuator. pi represents the front hinge position for HAU and the upper pole hinge position for VAU. qi represents the back hinge position for HAU and the

Schematic diagram of the vertical actuation unit.

lower pole hinge position for VAU, respectively (pi is moving and qi is fixed in HAU and moving in VAU). l is the distance between the front hinge and back hinge positions, and L1 and L2 are distances from the stage center to the center lines of HAUs and VAUs, respectively. LH is the distance between the front hinge positions of two HAUs. m is the distance between two hinges of the pole hinge block and zL is the height difference between the pi hinge positions of in-plane and out-of-plane. In Figs. 3 and 8, a coordinate system is introduced for the kinematic analysis. The origin is located at the center of the actuator configuration. x-axis is parallel to B1 actuator. The actuating directions of the VAUs (denoted by arrows in Fig. 3) are parallel to z-axis. Thus,y-axis is determined by the righthand rule for the Cartesian coordinate system. Fig. 9 is a kinematic equivalent of the VAU shown in Fig. 4. The VAU consists of one PZT actuator, three 1-D hinges, and two 2-D hinges. The PZT actuator can be regarded as a prismatic joint, 1-D hinge as a revolute joint, and 2-D hinge as a spherical joint, respectively, as shown in Fig. 9. In the figure, S, U , R, and P represent spherical, universal, revolute, and prismatic joints, respectively. The pole hinge block has two 2-D hinges which are connected to the moving part and the front hinge block,

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1 2 2 2 (zp − zq ,0 ) − (zp − zq ,0 ) − pi − qi,0 + m . ai = AR (11) In (9)–(11), AR, qi,0 , and qi represent the amplification ratio, and the initial and final positions of the lower hinge of the pole hinge block, respectively. zq ,0 and zp mean the z-axis coordinate values of qi,0 and pi . r1 and r2 are the distances from the center lever hinge to the centers of the front hinge and back hinge. ai is the actuator displacement, and Ai is the displacement amplified by the lever hinge. m is the distance between pi and qi . Equation (11) is derived from (9) and (10), then the displacements a4 , a5 , and a6 are determined by (11). These results will be applied to design the mapping matrices for control. Fig. 10.

B. Forward Kinematics

Schematic diagram of the whole precision stage.

respectively. The pole hinge block moves in z-axis direction through the front hinge block, which perpendicularly transmits the displacement generated by the PZT actuator. Therefore, the VAU can be simplified into an equivalent system on the righthand side of Fig. 9. For succeeding kinematic analysis, we made kinematic equivalents of the actuation units constituting double triangular configuration as shown in Fig. 10. Two triangular lines are constructed by connecting the lines of action of actuation units corresponding to the in-plane motion and the out-of-plane motions, respectively. The inverse kinematics starts from (7). pi = T + (Rx + Ry + Rz − 2I)pi,0 , pi ∈ 3×1 , i = 1 ∼ 6.

(7)

In (7), pi,0 is a vector representing the initial position of pi . T is the translation vector. Rx , Ry , and Rz , are the rotation matrices related to the x-, y-, and z-axes, respectively. I is the identity matrix. When the stage goes through translation and rotation, pi,0 becomes pi . This formulation of linear form is applicable because the workspace of the flexure-based precision stage is very small. ai = pi − qi − l, i = 1, 2, 3 .

(8)

Displacements a1 , a2 , and a3 of HAUs are obtained directly by (8). However, the displacements of VAUs cannot be obtained immediately. The position of the upper pole hinge of VAU pi can be obtained by (9). From the position of upper pole hinge pi , the corresponding lower pole hinge position qi should be determined. Consecutively, the lever hinge rotation, front hinge position, and PZT actuator displacements are determined from the lower pole hinge positions. From the schematic diagram shown in Fig. 9, we derived (10) and (11) to determine the actuator displacements (9) pi − qi = m, i = 4, 5, 6 r2 qi = qi,0 + [ 0 0 Ai ], Ai = AR × ai , AR = (10) r1

Forward kinematics is analyzed to find the final position and orientation of the end-effector for given displacements of actuators. The analysis results are used for ascertaining the workspace of the precision stage and for designing the real-time controller. Unlike serial mechanisms, parallel mechanisms have forward kinematics that cannot be uniquely determined, which means that a numerical method needs to be applied to find the solution. We defined the 18 constraints in terms of distance, because the precision stage is controlled using displacement information. Distance relationships between the six actuation units producing the in-plane and out-of-plane motions are depicted in Fig. 11. The upper triangle (Plane #1) is configured by connecting three front hinge centers related to the out-of-plane motion. In the same way, the lower triangle (Plane #2) is configured by connecting three upper pole hinge centers related to the in-plane motion. We applied the following simple distance constraints to the forward kinematics because all the actuation units are symmetrically placed in equilateral triangular forms. The constraints are represented as follows. Constraint #1: Distances between the positions of the front hinges of HAUs are constant. pi − pj 2 − L2H = 0, i, j = 1, 2, 3 (i = j).

(12)

Constraint #2: Distances between the positions of the front hinges of HAUs and the upper pole hinges of VAUs are constant as L3 or L4 . pi − pj 2 − L23 = 0, i = 1, 2, 3 , j = 4, 5, 6

(j = i + 3) (13)

pi − pj − 2

L24

= 0, i = 1, 2, 3

j = i + 3.

(14)

Constraint #3: Distances between the positions of the front and back hinges of HAUs are equal to the summation of the initial length l and the displacement of the actuator ai (i = 1, 2, 3) pi − qi 2 − (l + ai )2 = 0, i = 1, 2, 3.

(15)

Constraint #4: Distances between the positions of the upper pole hinges and the initial positions of the lower pole hinges of VAUs are equal to the total distance m and the displacements

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 3, JUNE 2014

Fig. 11.

Geometrical relationships between the plane #1 and plane #2.

Fig. 12.

Various views of the in-plane workspace (a) x-y view (b) x-θz view (c) y-θz view.

Fig. 13.

Various views of the out-of-plane workspace (d) θx -θy view (f) θy -z view (e) θx -z view.

of the actuators ai (i = 4, 5, 6), which is multiplied by the amplification ratio AR, pi − qi,0 2 − (m + AR · ai )2 = 0, i = 4, 5, 6.

(16)

These constraints are consolidated into (17) and (18). g(xn ) of (17) is the constraint function including (12)–(16) g(xn ) = 0

of the processes (x-, y-translations: ±50 μm, z-translation: ±150 μm, x-, y-rotations: ±244 μrad, z-rotation: ±733 μrad) at the wafer level. Especially, the workspace requirements emphasize z-translation and x-, y-rotations. With the help of the stroke amplification mechanism of the VAU, the requirements are met.

(17) −1

xn+1 = xn − ∇g(xn )

· g(xn ), xn ∈

18×1

.

(18)

Equation (18) implies the Newton–Raphson method applied in forward kinematics to find the final position and orientation of the moving part by numerical convergence. xn represents 18 coordinate values of the neck centers of six hinges (p1 , p2 , p3 , p4 , p5 , p6 ) connected to the moving part (see Fig. 8). The workspaces of the in-plane and out-of plane motions obtained from the forward kinematics analysis are shown in Figs. 12 and 13. The results meet the workspace requirements

C. Mapping Matrices Formulation For the real-time controller design, the two mapping matrices, Ra and Rs are derived in this section. Although the actuator displacements and the position and orientation of the end-effector can be perfectly determined by the inverse and forward kinematics, the corresponding calculation time is too long for real time application. Because the ratio of the motion range to the size of the precision stage is below 0.3%, we carefully extracted the linear mapping matrices, which can be applied to positioning

SHIN AND MOON: DESIGN OF A DOUBLE TRIANGULAR PARALLEL MECHANISM

Fig. 14.

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Schematic diagram of the closed-loop control.

control across the whole working space, from the inverse and forward kinematics through the following procedure. The control loop is shown in Fig. 14. When a reference command x∗ is given, Ra produces actuator displacements a. The inverse of Rs computes the real movement of the precision stage x from the sensor signal s. Because x is too small, a and s can be linearly approximated as shown in (19) and (20), where i and j represent the numbers of row and column, respectively,

∂ai ∗ a = Ra x , Ra = ∈ 6×6 ∂xj

ai

(19) when xj 1 xj

∂si s, R = x = R−1 ∈ 6×6 s s ∂xj

si

(20) when xj 1. xj Finally, the Ra and Rs are represented as ⎡ −1 0 0 0 0.042 ⎢ ⎢ 0.500 0.866 0 0.036 −0.021 ⎢ ⎢ 0.500 −0.866 0 −0.036 −0.021 ⎢ Ra = ⎢ ⎢ 0 0 1 0.118 0 ⎢ ⎢ 0 1 −0.059 0.102 ⎣ 0 0 ⎡

1 ⎢ ⎢1 ⎢ ⎢0 ⎢ Rs = ⎢ ⎢0 ⎢ ⎢ ⎣0 0

0

1

−0.059

0

0

−0.042

0

0

0

−0.042

−0.042

0

0

1

0.038

0.060

0

1

0.038

−0.060

0

1

−0.060

0

⎤

⎥ 0.073 ⎥ ⎥ 0.073 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦

−0.102

0

−1 −0.001

0.073

0 −0.112

(21) ⎤

⎥ 0.112 ⎥ ⎥ 0 ⎥ ⎥ ⎥. 0 ⎥ ⎥ ⎥ 0 ⎦ 0 (22)

V. CONTROLLER DESIGN AND EXPERIMENTAL RESULTS A. Controller Design The developed precision stage is manufactured through precision grinding, pin-hole fit, and bolting to secure high precision. Fig. 15 presents the hardware system configuration. To enlarge the out-of-plane workspace, PZT actuators for the out-of-plane motion were selected to have longer strokes than those for the in-plane motion (for the in-plane motion: maximum stroke of

Fig. 15.

Hardware system configuration.

150 μm, maximum load of 5000 N, and stiffness of 17 N/μm; for the out-of-plane motion: maximum stroke of 200 μm, maximum load of 15000 N, and stiffness of 35 N/μm). For sensing, capacitive gap sensors were selected for the high sensing resolution (sensing range of 500 μm and resolution of 10 nm). For signal processing, a real-time controller, which is interfaced to a host PC, processes the analog signal from −10 to 10 V. The real-time controller have 16-bit resolution and operates with the LabVIEW real-time module. PI-control is applied to the stage positioning. The proportional constants are 0.3, 0.4 and 0.2 for x-, y-, and z-translations; and 0.03, 0.025, and 0.2 for x-, y-, and z-rotations, respectively. The integral time constants are 0.00 005, 0.00 005, and 0.0 002 for x-, y-, and z-translations; and 0.00 005 for x-, y-, and z-rotations.

B. Experimental Results Test results of multistep responses across almost full travel ranges are shown in Fig. 16. In the figures, submicron multistep responses are displayed at the right upper side. According to the requirements of the wafer bonding process, the required resolution of translation and rotation were 0.1 μm and 1 μrad, respectively. The figures show that the precision stage meets the resolution specifications. As shown in Fig. 16, the x-, y-translational strokes of the precision stage were more than ±50 μm and that the z-translational stroke was more than ±150 μm. In addition, the x-, y-rotational strokes were more than ±733 μrad (±0.042◦ ) and that the z-rotational stroke was more than ±244 μrad (±0.014◦ ). The noise in the submicrometer step responses shown in the right upper corners of insets in Fig. 16, originated from electromagnetic noise induced by electric power and unsystematic white noise. The developed stage positions the end-effector under the weight of 500 N as shown in Fig. 17. Even though the transient responses become worse as the payload increases, the final location is still exact. The multistep responses show that the flexure hinges, which are the weakest parts of the developed stage, overcome the heavy weight and the PZT actuators generate large force successfully.

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Fig. 16. Experimental results for the multistep translations and rotations. (a) Multistep translations (left: x-axis, 20 μm; middle: y-axis, 20 μm; right: z-axis, 60 μm). (b) Multistep rotations (left: x-axis, 300 μrad; middle: y-axis, 300 μrad; right: z-axis, 100 μrad).

horizontal arrangement of the actuators by the lever linkage minimized the stage height and the overlapped structure in the VAU reduced the width and length of the precision stage. The developed precision stage is useful for the precision alignment of optical devices and nanoimprinting as well as to high payload precision positioning processes such as wafer level alignment and bonding. REFERENCES

Fig. 17. Experimental result for the multistep response in case of applying external force of 500 N (z-translation).

VI. CONCLUSION In this study, a six-DOF precision positioning stage with large force generation of 500 N as well as submicron resolution of 0.1 μm in translation and 1 μrad in rotation was designed. Flexure hinges and PZT actuators with the design of VAU, in which all the hinges were orthogonally aligned in horizontal and vertical directions, gave high resolution. The DTPM, which is smaller with higher stiffness than serial mechanisms, were devised for large force generation. Moreover, the hinge design optimization was performed for large force generation of the precision stage because it found the optimal hinge thicknesses that minimized normal and bending stresses. To enlarge workspace, the lever linkage was designed, and the actuation units were placed closer to the center of the precision stage. The mapping matrices with the results of inverse and forward kinematics were defined for real-time feedback control. The

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SHIN AND MOON: DESIGN OF A DOUBLE TRIANGULAR PARALLEL MECHANISM

[11] S.-C. Chen and M. L. Culpepper, “Design of a six-axis micro-scale nanopositioner—μHexFlex,” Precision Eng., vol. 30, pp. 314–324, 2006. [12] T. Hu and W.-J. Kim, “Extended range six DOF high-precision positioner for wafer processing,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 6, pp. 682–689, Dec. 2006. [13] K.-B. Choi and J. J. Lee, “Passive compliant wafer stage for single-step nano-imprint lithography,” Rev. Sci. Instrum., vol. 76, 075106, 2005. [14] J.-H. Moon, H. J. Pahk, and B.-G. Lee, “Design, modeling, and testing of a novel 6-DOF micropositioning stage with low profile and low parasitic motion,” Int. J. Adv. Manuf. Technol., vol. 55, no. 1–4, pp. 163–176, 2011. [15] M. A. Janaideh, S. Rakheja, and C. Y. Su, “An analytical generalized Prandtl–Ishlinskii model inversion for hysteresis compensation in micropositioning control,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 734–744, Aug. 2011. [16] B. J. Kenton and K. K. Leang, “Design and control of a three-axis serialkinematic high-bandwidth nanopositioner,” IEEE/ASME Trans. Mechatronics, vol. 17, no. 2, pp. 356–369, Apr. 2012. [17] J. M. Paros and L. Weisbord, “How to design flexure hinges,” Mach. Des., vol. 37, no. 8, pp. l5l–156, 1965. [18] Piezomechanik GmbH (2012, Aug. 17). Catalog webpage. [Online]. Available: http://www.piezomechanik.com. Accessed [19] M. Kamlah, “Ferroelectric and ferroelastic piezoceramics—modeling of electromechanical hysteresis phenomena,” Continuum Mech. Thermodyn., vol. 13, no. 4, pp. 219–268, 2001. [20] S. T. Smith, Flexures: Elements of Mechanical Mechanisms. Boca Raton, FL, USA: CRC Press, 2000. [21] D. W. Pilkey and D. F. Pilkey, Peterson’s Stress Concentration Factors, 3rd ed. New York, NY, USA: Wiley, 2008.

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Hyunpyo Shin received the B.S. degree in biosystem engineering and the M.S. and Ph.D. degrees in mechanical and aerospace engineering from Seoul National University (SNU), Seoul, Korea, in 2002, 2004, and 2009, respectively. He was a Postdoctoral Researcher at the Institute of Advanced Machinery and Design (IAMD) at SNU from 2009 to 2010. He was a Senior Researcher at Samsung Electro-Mechanics, Suwon, Korea, from 2011 to 2013. He is currently an Assistant Professor in the School of Robot and Automation Engineering, Dongyang Mirae University, Seoul, Korea. His current research interests include the design of redundantly actuated parallel mechanisms, precision positioning stages, stiffness enhancement algorithms for robots, 6-DOF motion simulators, and multifunctional machine tools. Jun-Hee Moon received the B.S., M.S., and Ph.D. degrees in mechanical design and production engineering from Seoul National University (SNU), Seoul, Korea, in 1992, 1994, and 2002, respectively. From 2000 to 2004, he was a Research Engineer with SNU Precision Company, Ltd. From 2005 to 2008, he was a Senior Researcher at the Micro Thermal System Research Center at SNU. From 2009 to 2010, he was an Assistant Professor in the Department of Mechatronics, Daelim University College, Anyang, Korea. Since 2011, he has been a Professor in the Department of Mechanical Design, Yuhan University, Bucheon, Korea. His research interests include the analysis and control of precision systems using air pressure, piezoelectric materials, and mechanical linkages.