nano manipulation

6th IFAC Symposium on Mechatronic Systems The International Federation of Automatic Control April 10-12, 2013. Hangzhou, China

A novel flexure-based 3-DOF micro-parallel manipulator with a gripper for micro/nano manipulation Shunli Xiao ∗ Yangmin Li ∗ Qinmin Yang ∗∗ ∗

Department of Electromechanical Engineering, University of Macau, Taipa, Macao SAR, China. (Corresponding author Y. Li’s e-mail: [email protected]) ∗∗ Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China.

Abstract: This paper presents the design and analysis of a novel compliant flexure-based micro-parallel manipulator with gripper, which is with translational 3-DOF and driven by electromagnetic actuators. The stage is constructed with a symmetrical structure by employing three parallel PUU legs, a moving platform and a fixed platform. The mobility characteristics of the stage is analyzed and proved via FEA method. The kinematics and dynamic modeling of the mechanical system of the stage are conducted by resorting to compliance matrix method, and analytical models for electromagnetic forces are also established, both mechanical structure and electromagnetic model are validated by finite element analysis(FEA) performed with ANSYS. The mechanical structures is analyzed in a multi-physics environmental simulation and electromagnetic actuators are applied in ANSYS too. Both FEA and the analytical models well demonstrate that the movement of the stage is purely translational. Based on the designed parameters, this micro-parallel manipulator can have a large workspace, a very high resolution, and a heavy payload ability. 1. INTRODUCTION

the parallel manipulators can be designed and integrated with a 3-DOF micro-gripper.

The increasing demands of the modern industry require manipulating or assembling workpieces automatically, see Li [1997]. In micro/nano fields, micro parts can only be assembled by all kinds of micro/nano-position and gripping devices. It is essential to develop a new kind of assembling technology which can be used to handle very small parts. In view of the increasing functionalities of such systems, special micro-gripper is also a very active research field. In this research, a parallel manipulator is adopted to construct a 3-DOF micro gripper. The problems of handling very small micro-optical, micro-electronic and micro-mechanical elements in micro/nano-meter range can be solved by using micro/nano-position stages, see Tsai [1996]. It has been proved that a spatial closed loop parallel manipulator has its great potential and advantages over the traditional serial manipulator. The parallel manipulator is able to provide a very complicated motion for the end-effector in a flexible way with many degrees of freedom(DOF) by using a combination of prismatic, revolute, universal joints, etc. see Xiao [2013a] and Li [2012]. So,

There are many micro-manipulation applications which need a pure translational gripping motion. In addition, many researchers are studying parallel manipulators for these applications recently, due to its high stiffness and high load capacity. Tsai [1996] studied kinematics of a UPU(Universal-Prismatic-Universal) parallel manipulator. Gregorio [2004] studied carefully about the kinematic condition of pure wrist parallel robot. Yun [2012] presented a kind of dual parallel micromanipulator for positioning. Most of these manipulators are realized in macro size such as famous Delta 3-DOF translational motion robot.

⋆ This work was supported in part by National Natural Science Foundation of China (Grant No. 61128008), Macao Science and Technology Development Fund (Grant No. 016/2008/A1), Research Committee of University of Macau (Grant no. MYRG203(Y1-L4)FST11-LYM) and the State Key Laboratory of Industrial Control Technology Fund of Zhejiang University (Grant No. ICT1104).

978-3-902823-31-1/13/$20.00 © 2013 IFAC

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In this research, a 3-DOF translational micro/nano gripper with flexure hinges is constructed. A flexure mechanism with notched hinges has been widely used for microprecision machinery, like a micro-positioning stage, accelerometer, and micromanipulator. The flexure hinge is made of circular or rectangular notches, which is compliant in bending around its axis. So it works approximately as an angular revolute joint. The merits of the hinge are vacuum compatibility, no backlash property, no non-linear friction, simple structure and easy manufacture. Flexure parallel mechanism(FPM) possesses high stiffness and high natural frequency. There are no error accumulations and no need for lubrication. Due to the advantages of the FPM, it is widely employed as the ultra-precision manipulation 10.3182/20130410-3-CN-2034.00097

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

system in various fields such as fiber optic coupling, micromachining and semiconductor production. Many micro motion drivers such as piezoceramics (PZT) and shape memory alloy(SMA) can be selected as the actuators. Although the PZT can provide a very high-precision driving output. As for the SMA, it can have a large moving range but have long time response and heating/cooling problems. For both large moving range, high-resolution and large output force, electromagnetic actuators are selected as the drivers for 3-DOF translational flexure-based micro parallel manipulators. Compared with the PZT actuated micromanipulator, it behave a larger workspace. Compared with the maglev micro manipulator, it can own a heavier payload capability. Besides, the proposed electromagnetic driving 3-DOF translational micro gripper can be fabricated easily at very low-cost. 2. ARCHITECTURE DESCRIPTION OF THE MANIPULATOR

Fig. 1. A 3-PUU parallel mechanism system.

The schematic diagram of a 3-PUU translational micro parallel manipulator is shown in Fig.1. It consists of a mobile platform with gripper, a fixed base, and three limbs with identical kinematic structure. Each limb connects the fixed base to the mobile platform by a P¯ (prismatic) joint and two U (universal) joints in sequence, where P¯ joint is the active joint driven by a linear actuator assembled on the fixed base. Thus, the mobile platform is attached to the base by three identical P¯ U U linkages. To facilitate the analysis, as shown in Fig.1 and Fig.2, we assign a fixed Cartesian frame O{x, y, z} at the centered point O of the fixed base, and a moving Cartesian frame P {u, v, w} on the mobile platform at the centered point P , along with the z- and w-axes perpendicular to the platform, and the x- and y- axes parallel to the u- and v-axes, respectively. Fig.2 also shows the i-th leg,i = 1, 2, 3, of the 3-PUU mechanism. In the figure, sij , (i = 1, 2, 3, j = 1, 2, 3, 4) is the unit vector of the axis of the j-th revolute pair of −−→ the i-th leg and si0 is the unit vector of OAi . Point Ai is the intersection of the two revolute pairs of the U joint that connect the i-th leg to the P joint before fixing to the base. Point Bi is the center point of the axes of the two revolute pairs of the universal joint that connects the i-th leg to the mobile platform. Finally the length di is the distance of Ai Bi , here d1 = d2 = d3 = d, since the three legs are identical. In order to obtain pure translational motion characteristics of the platform with respect to the base, with these notations, the platform and the base have to comply with the following geometric conditions, see Xiao [2012],Kim [2003]. s11 · s21 = s13 · s23 s11 · s31 = s13 · s33 s21 · s31 = s23 · s33 and si2 = si4 ,si1 = si3 ,(i = 1, 2, 3) for simplicity, we assemble out parallel mechanism as si1 ⊥si0 , si1 ⊥si2 , si1 = si3 (si1 //si3 ), si2 = si4 (si2 //si4 ) and △B1 B2 B3 is equilateral triangle and △A1 A2 A3 is equilateral triangle at the initial time, that will be fulfilled the requirement mentioned above.

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Fig. 2. The i-th leg of a 3-PUU parallel mechanism. 2.1 Compliant Modeling of Flexure Mechanisms Based on Matrix Method Various methods can be adopted in modeling the compliant mechanisms, see Howell [2001]. One of those typical methods is pseudo-rigid body modeling, which is based upon simplification analysis that only deformation around one axis is considered while other members of the mechanism except hinges are assumed as rigid body. Stiffness or compliance matrix method, which has been well developed in architecture to analyze a rigid structure like bridges, is widely used into analyzing compliant mechanisms. This method has advantages since it can deal with full set of deformations of all mechanical members. Although finite element method is also a kind of matrix method, the stiffness or compliance matrix method can be understood easily and less computational power is needed. The linear relationship between force and rotational displacement can be established by hooke’s law and then the amplifier ratio can also be conducted by this method. As mentioned above, stiffness or compliance matrix has already been well developed, so we will adopt this method to analyze our micromanipulator and validate it in the following section by FEA software.

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Fig. 3. A flexure with coordinates and parameters. Here, it is assumed that the linear relation is established between force and deformation. When forces and moments in/around certain axes are exerted on certain points, the infinitesimal translational and rotational displacements of that point are formulated as Eq. (1) and Eq. (2). Note that the set of forces and moments as well as the set of translational and rotational displacements are expressed as F and X, respectively. Therefore, the force and moment, translational and rotational displacement will not be distinguished in expression. X = CF

(1)

F = KX

(2)

Where the matrix C is the compliance matrix, the matrix K is the stiffness matrix. C and K are inverse matrices of each other, and both are regular and symmetric. Eq.(3) shows the displacement of the tip of a general beam when external forces are exerted on it. fn and δn are the force and translational displacement in n-axis, respectively and Mn and θn are the moment and rotational displacement around n-axis, respectively. [δx , δy , δz , θx , θy , θz ]T = Ch [fx , fy , fz , Mx , My , Mz ]T (3) And the Ch is:  c1 0 0  0 c2 0   0 0 c5 Ch =  0 0 0  0 0 c9 0 c10 0

0 0 0 c6 0 0

0 0 c4 0 c7 0

 0 c3   0 0  0 c8

The compliance factors ci , (i = 1 : 10) in the matrix, which are frequently used in flexural mechanisms, have been derived in Howell [2001], and the equation with best accuracy that we adopted in our calculation can be found in Xiao [2011]. Additionally, as shown in Fig.3, the local compliance of the hinge is defined as Ci0 = Ch , where the upper-right superscript describes the coordinate with respect to which the compliance is described throughout this paper, and ”0” indicates the ground that will be omitted for the clarity of representation. The compliance Ci can be transferred onto another frame Oj by Cij = Tij Tx (γ)Ty (β)Tz (α)Ci [Tz (α)]T [Ty (β)]T [Tx (γ)]T (Tij )T (4) where the translational transformation matrix takes on the following form:   j Tij = I S(ri ) (5) 0 I 0 = 03×3 , I = I3×3 , rij is the position vector of point Oi expressed in reference frame Oj , and S(r) represents the skew-symmetric operator for a vector r = [rx ry rz ]T with the notation:

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Fig. 4. The structure of a flexure P joint with parameters. " S(r) =

0 −rz ry rz 0 −rx −ry rx 0

# (6)

where Tz (α), Ty (β) and Tx (γ) are the rotational transformation matrices, and   Rz (α) 0 Tz (α) = (7) 0 Rz (α)   Ry (β) 0 Ty (α) = (8) 0 Ry (β)   Rx (γ) 0 Tx (γ) = (9) 0 Rx (γ) Where Rz (α), Ry (β), Rx (γ) is each axis rotational matrix of coordinate Oi with respect to Oj . Generally, a flexural mechanism consists of n individual flexure elements connected in serial or parallel manners. In order to obtain the compliance model of the entire mechanism, the local compliances need to be transformed to a common(global) frame chosen to describe the mechanism. Then, compliances connected in serial and stiffness connected in parallel can be added together to generate the entire one. 2.2 Compliance Modeling of the P-joint The compliance C1D is defined as the compliance of point O1 with respect to the ground point D. All the limbs and hinges are connected in serial or parallel. For the purpose of calculating the compliance of the stage, compliance of every flexural hinge with respect to the ground CiD can be derived by CiD = TiD Ci (TiD )T (10) Considering the symmetrical structure of the P-joint, each limb LO1 O3 , LO2 O4 , LO5 O7 , and LO6 O8 is a serial chain with two flexural hinges. The local compliance matrices of each flexural hinge are C, for limb O1 O3 , we have rAO1 = [0, 0, −l2/2] and rAO3 = [0, l, −l2/2], the transformation  matrix is  1 0 0 0 l2 /2 0 0 1 0 −l2 /2 0 0    0 0 1 0 0 0   TOA1 =  0 0 1 0 0  0  0 0 0 0 1 0  0 0 0 0 0 1 and

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Table 1. Parameters of the mechanism Architectural parameters (mm) t w d |OAi | |OBi | l1 l1 0.5 14 135 30 101 35 40 Material parameters E σy ν ρ 71.7GPa 503MPa 0.33 2.81 × 103 Kg/m3 r 2

Table 2. Kinetostatics performance of a gripper Fig. 5. A flexure-based U joint structure.

Performance Matrix model FEA Deviation(%)

Output stiffness(N/µm) 0.22 0.19 15.8

Input stiffness(N/µm) 6.95 5.92 17.4

reference point can be described as that the leg rotates around y-axis first, then rotates around z-axis and moves translational along y-axis to connect with the P-joint. π π CLOi = TAO Ty (βi )Tx (γi )Tz ( )CL [Tz ( )]T [Tx (γi )]T [Ty (βi )]T 2 2 (16) For each PUU leg, we can have the whole compliance matrix as follows: CLimb−i = CLOi + CPOi (17) Fig. 6. A U-U leg  1 0 0 1  0 0 A TO3 =  0 0 0 0 0 0

structure with coordinate. 0 0 1 0 0 0

0 l2 /2 −l1 1 0 0

−l2 /2 0 0 0 1 0

l1 0 0 0 0 1



2.4 Compliance modeling of the whole manipulator

     

The whole manipulator can be treated as three symmetric legs connected in parallel and the compliance matrix can be derived by KPO = [CLimb−1 ]−1 + [Tz (2π/3)CLimb−2 [Tz (2π/3)]T ]−1

Then, the compliance matrix with respect to the point A can be derived by CLAO1 O3 = TOA1 C ′ [TOA1 ]T + TOA3 C ′ [TOA3 ]T (11) where C ′ = Tz (π/2)C[Tz (π/2)]T . Just the same as obtaining of CLAO O , we can get the 1 3 compliance matrix of CLAO O , CLBO O , CLBO O . Then the 2 4 5 7 6 8 compliance of P with respect to the ground point O described as CPO can be derived by KPO and

= {TAO [(CLAO1 O3 )−1 +{TBO [(CLBO5 O7 )−1

+ +

(CLAO2 O4 )−1 ]−1 [TAO ]T }−1 (CLBO6 O8 )−1 ]−1 [TBO ]T }−1

CPO = [KPO ]−1

(12) (13)

2.3 Compliance Modeling of the U-joint and the Leg As shown in Fig.5, an U joint can be treated as two parallel R joints, and the two R joints just rotate π/2 with respect to x-axis, the local compliance matrix of an U joint can be derived by (14) CU = C + Tx (π/2)C[Tx (π/2)]T The local compliance of the leg can be derived by ′ ′ (15) CL = CU + TAB′ CU [TAB′ ]T As shown in Fig.6, rA′ B ′ = [d, 0, 0] for TAB′ . Then the compliance matrix of the leg with respect to the ground

+[Tz (4π/3)CLimb−3 [Tz (4π/3)]T ]−1 (18) and

CPO = [KPO ]−1 (19) Then, the input stiffness can be calculated. The output stiffness can be defined as the stiffness from point Ai to O, O it is noticed that CA can be obtained as the same process i as the calculation of the input stiffness. 3. INVERSE KINEMATIC ANALYSIS OF THE 3-DOF MICRO GRIPPER As a typically micro parallel positioning stage based 3DOF micro gripper with three pure translational motions, the inverse kinematic model can be derived as follows, see Merlet [2006]: → − − − → → Bi = R b i + P (20) → − −−→ − → −−→ − → −−→ where: R = I3×3 , b i = P Bi , P = OP , Bi = OBi , → − −−→ A i = OAi As the proposed 3-PUU micromanipulator is with pure translational movement, 2 2 d2 = rA + rB − 2rAi rBi cos(θi ) (21) i i 6 where: θi = Ai OBi , rAi = kAi k, rBi = kBi k, cos(θi ) =

− → Bi rBi

·

− → Ai rAi

, then

′

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rAi =

2rBi ±

q 2 − 4r2 cos(θ ) 4rB i Bi i 2

(22)

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Fig. 7. The FEA model of a P joint. According to the configuration of the structure, we can obtain q 2 − 4r2 cos(θ ) 2rBi + 4rB i Bi i rAi = (23) 2

Fig. 8. FEA model and analysis of the 3-PUU micro parallel flexure-based 3-DOF micro gripper.

4. COMPLIANCE MODEL VALIDATION VIA FEA The established models for the assessment of input and output compliance of the micro-parallel 3-DOF micro gripper are validated by the FEA through ANSYS software. The architecture parameters of the stage are described in Table 1 where all the hinges are designed as identical dimensions, and the mesh model is created with the element SOLID185. The FEA model of P joint is built and simulated in the ANSYS software. When a force is applied at the input surface of the P joint, the corresponding output motion of the P joint can be seen in Fig.7. The FEA result of the stage can be seen in Fig.8, the relationship between the input force and the output motion displacement can be obtained to determine the input and output compliance. The stage performances evaluated by the derived models and FEA are elaborated in Table 2. Taking FEA results as the benchmark, we can observe that the maximum deviation of the derived model from the FEA results is around 20 percent. The offset mainly comes from the accuracy of the adopted equations for the compliance factors and the neglect of compliances of the links between flexure hinges since these links are assumed to be rigid in the matrix model. When the FEA model in ANSYS is applied by the inverse model conducted in the last section, it moves totally in translational along x-,y- zdirection. Given arbitrary input, the responses are shown in Fig.8 and Fig.9. It shows clearly that the mobile platform is mainly translational motion with respect to its initial position. Furthermore, the modal analysis is conducted in FEA model via ANSYS. The results of natural frequencies are listed in Table 3. It can be seen that the first mode is mainly up and down translational movement, the second mode mainly is horizontal movement, the third mode is mainly rotational movement and the fourth mode is wrap of the end-plate movement. The mode frequencies of the first two modes are 69.5Hz and 139Hz respectively, which are far lower than the rotational and wrapped ones. That is to say, when the stage is driven, the motion is mainly translational one, even if a very high dynamic driven signal or disturbances can lead to some rotations and wrapped movements.

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Fig. 9. Translational motion characteristic analysis through FEA model.

(a)

(b)

(c)

(d)

Fig. 10. The first 4 mode shapes of a 3-DOF micro gripper. 5. MODELING AND ANALYSIS OF THE ELECTROMAGNETIC NON-CONTACT LINEAR ACTUATOR To predict the static characteristics of a single electromagnetic actuator, the relationship between the magnetic field, the mechanical movement and the electrical supply must be quantified. The equation for the static analysis of the stage is F = kx

(24)

IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China

Table 3. Natural frequencies of the 3-DOF micro gripper Modal sequences FEA (Hz)

1 69.5

2 139

3 319.5

4 326.5

5 588.5

6 996.5

where k is the stiffness of the stage, and F is the driving force by the actuator, x is the moving distance of the stage. For each single actuator, the approximate calculation method for the electromagnetic force, see Xiao [2013b]: Φ (25) S0 By Ampere’s circuital law and Kirchhoff’s law, we have NI 2δ la lb Φ= /( + + ) (26) µ0 µ0 S 0 µ0 µr1 S1 µ0 µr2 S2 where N is the turns of the coil, I is the exciting current, δ is the length of the air gap, la is the length of the armature, lb is the length of the back-iron, µ0 is the vacuum permeability, µr1 µr2 is the relative permeability of the armature and back-iron respectively, S0 is the area of the air gap and S1 or S2 is the cross-sectional area of the armature and back-iron, respectively. Because our air gap is small enough, we can assume that the magnetic area for the gap is equal to the back-iron. The basic formulation for the electromagnetic force can be established by virtual work principles as follows: B 2 S0 F = (27) 2µ0 The formulations mentioned above are for an ideal simple electromagnet, which are quite different from the practical working situation. According to the practical experiences, there are always several factors for the formulations above. A more accurate solution may be obtained by assuming several values of B by interpolation and taking the nonlinearly relationship between the magnet flux density B and the magnet field intensity H into consideration. B=

Since both F and Φ are functions of the current i and the air gap δ, and the actuator is partially saturated, a finiteelement analysis is carried out and the axial symmetry in the current situation needs only two-dimensional computations. The amount of both flux linkage and the force by using Maxwell’s stress tensor at discrete positions of the armature will be obtained. For a single actuator, when the air gap is 2mm, the length of the whole flux path is about 180mm, the section area of the back-iron is 100mm2 , the exciting current is 0.25A and the turns of the coil are 1200, the driving force can reach 266N by virtual work principle and 248N by Maxwell stress tensor principle respectively from FEM analysis via ANSYS. According to the input and output compliance from the Table 2, the driving force is enough to obtain a large working volume. 6. CONCLUSION A conceptual design of a novel 3-DOF flexure-based parallel manipulator with micro gripper driven by electromagnetic actuators is presented. The stage possesses a mainly translational mobility characteristics, simple symmetrical structure and large movement range. The compliance

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matrix-based method is applied in the statics modeling of the mechanical structure. All statics and models of the mechanical system and the electromagnetic models are validated by FEA method via ANSYS. As stated in section II, the input compliance of the mechanical system is 0.19N/µm. Aiming for generating a movement range of ±500µ of the mobile finger, at least 95N driving force is needed. The force that the electromagnet designed in section V can generate 268N, which is enough for driving the stage. In the near future, three laser displacement sensors(Microtrak II head model: LTC-025-02 from MTI instrument,Inc.) with a resolution of 0.1µ and linearity better than 0.05% over a measurement range of 2mm associated with a high-accuracy multifunction data acquisition and control board (model: NI PCI-6289 from National Instruments, Inc., which is combined with 16 channels 18-bit analog input and 4 channel 16-bit analog output) will be used to construct the experimental setup. Forward and inverse kinematics equations will be constructed and robust control strategy will be adopted to obtain a high accurate micro-parallel 3-DOF gripper. REFERENCES Y. Li (1997). Hybrid control approach to peg-in-hole problem. IEEE Robot. Auto. Mag., 4(2): 52-60. S. Xiao and Y. Li (2013a). Optimal design, fabrication and control of an XY micro-positioning stage driven by electromagnetic actuators. IEEE Trans. Ind. Electr., DOI: 10.1109/TIE.2012.2209613. Y. Li and S. Staicu (2012). Inverse dynamics of a 3-PRC parallel kinematic machine. Nonlinear Dynamics, 67(2): 1031–1041. L. W. Tsai (1996). Kinematics of a three-DOF platform with three extensible limbs. in: J. Lenarcic, V.ParentiCastelli(Eds.), Recent Advances in Robot Kinematics, 401–410. S. Xiao and Y. Li (2012). Mobility and kinematic analysis of a novel dexterous micro gripper. IEEE Int. Conf. on Rob. and Auto. (ICRA), May 14-18, St. Paul, Minnesota, USA, pp.2523-2528. R. Di Gregorio (2004). Statics and singularity loci of the 3-UPU wrist. IEEE Trans. Robot., 20(4): 630–635. Y. Yun and Y. Li (2012). Modeling and control analysis of a 3-PUPU dual compliant parallel manipulator for micro positioning and active vibration isolation. Journal of Dynamic System, Measurement, and Control, Transactions of ASME, 134(2): 021001-1-9. D. Kim, W. K. Chung (2003). Kinematic condition analysis of three-DOF pure translational parallel manipulators. ASME J. of Mech. Des., 125(2): 323–331. L. L. Howell (2001). Compliant Mechanisms. John Wiley & Sons, Inc. S. Xiao, Y. Li and X. Zhao (2011). Optimal design of a novel micro-gripper with completely parallel movement of gripping arms. Pro. of the 5th IEEE Int. Conf. on Rob., Auto. and Mech. (CIS-RAM 11), Qingdao, China, Sept. 17-19, PP. 2127-2132. J.P. Merlet (2006). Parallel Robots. Springer. S. Xiao and Y. Li (2013b). Modeling and high dynamic compensating the rate-dependent hysteresis of piezoelectric actuators via a novel modified inverse Preisach model. IEEE Trans. Contr. Syst. Tech., DOI: 10.1109/TCST.2012.2206029.