A Large-Displacement 3-DOF Flexure Parallel

Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems October 9 - 15, 2006, Beijing, China

A Large-Displacement 3-DOF Flexure Parallel Mechanism with Decoupled Kinematics Structure Xueyan Tang

I-Ming Chen

School of Mechanical and Aerospace Engineering Nanyang Technological University Singapore 639798 Email: [email protected]

School of Mechanical and Aerospace Engineering Nanyang Technological University Singapore 639798 Email: [email protected]

Abstract— This paper proposes an XYZ -flexure parallel mechanism (FPM) with large displacement and decoupled kinematics structure. The large-displacement FPM has large motion range more than 1mm. Moreover, the decoupled XYZ-stage has small cross-axis error and small parasitic rotation. In this study, the typical prismatic joints are investigated, and a new largedisplacement prismatic joint using notch hinges is designed. The conceptual design of the FPM is proposed by assembling these modular prismatic joints, and then the optimal design of the FPM is conducted. The analytical models of linear stiffness and dynamics are derived using pseudo-rigid-body (PRB) method. Finally, the numerical simulation using ANSYS is conducted for modal analysis to verify the analytical dynamics equation. Experiments are conducted to verify the proposed design for linear stiffness, cross-axis error and parasitic rotation.

I. I NTRODUCTION In many applications such as chip assembly in semiconductor industry and cell manipulation in biotechnology, there is an urgent need for manipulators with high precision and rapid response. Conventional mechanisms with assembled joints and rigid links cannot meet this demand due to their coarse precision caused by friction, backlash, etc. Flexure mechanisms of monolithic and miniature piece can provide highly accurate micro-scale motion because they have less wear, backlash and no friction. Moreover, advanced electro-discharge machining (EDM) makes flexure mechanisms feasible. In most flexure mechanisms, two fundamental flexible components have been used extensively, leaf spring and notch hinge. However, both of them have limitations. Though leaf spring can reach large displacements, it is prone to buckling under compressive axial load and stiffening in the presence of tensile loads. Moreover, thickness and material selection are limited due to the possibility of manufacturing [1]. Notch hinges are also used extensively because of its easy manufacturablity and high off-axis stiffness. However, stress concentration near the thinnest portion of the notch hinge is its drawback, and it results in limitation on the motion range of the flexure mechanism using notch hinges. For example, Kim [2] worked out a single-DOF flexure mechanism with the motion range of 200µm. The XYθZ -stage developed by Ryu [3] [4] worked within an area of 40µm × 40µm. The workspace of the 6-DOF compliant manipulator by Chao [5] is 120µm × 130µm × 18mrad. The XYZ-flexure stage studied

1-4244-0259-X/06/$20.00 ©2006 IEEE

by Li [6] has the workspace of 140µm × 140µm × 140µm. In our project, the proposed XY Z-flexure stage using notch hinges has advantages of large displacement more than 1mm× 1mm × 1mm and easy manufacturablity. In recent research, 6-DOF, XYθZ and XYZ-flexure stages have been developed. Yamakawa [7], McInroy [8], Liu [9], etc. have done much work on 6-DOF stages. Lee [10], Chang [11], Ryu [3], Yi [12], etc. focused studies on XYθZ -stages. In some applications, such as optical alignment, XYZ-stage is required. Tsai [13] proposed a 3-limb parallel structure, but the motion range is small and the three translational motions are coupled. Arai [14] designed a decoupled serial stage, but its stiffness is not satisfied. Clavel [15] and Li [6] have developed two decoupled parallel XYZ-stages. In both flexure mechanisms, the cross-axis error and the parasitic rotation exist. Therefore, it is necessary to develop a parallel and decoupled XYZ- stage with small cross-axis error and small parasitic rotation. Hence, our proposed FPM has advantages of large displacement more than 1mm and decoupled kinematic structure with small cross-axis error and small parasitic rotation. It has potential applications such as optical alignment and assembly of micro parts. In this paper, the typical prismatic joints with motion range and cross-axis error is investigated, and a new large-displacement prismatic joint is designed in Section II. Using these modular prismatic joints, the conceptual design of the FPM is proposed in Section III, together with the analytical model and the design optimization. Then the numerical simulation for modal analysis are presented in Section IV. Finally, the experiment is conducted for verification in Section V. II. S TUDY ON P RISMATIC J OINT For the proposed FPM, 1-DOF prismatic joints are fundamental components. Motion range and decoupled characteristics of the FPM are determined by performance and arrangement of the prismatic joints. There are two typical prismatic joints shown in Fig. 1. The first type is the conventional parallelogram ,but it has the problem in the limited motion range or the cross-axis error. If the motion stage is constrained along the Y-axis, when a force along the Y-axis FX is applied, the axial tensile load along the notch hinge will be generated, and the stiffness will increase significantly. As a result, the motion range will be reduced. If

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Fig. 3. (a) Conventional parallelogram Fig. 1.

Notch hinge

(b) Double parallelogram

The nominal stress can be calculated using the equation Typical prismatic joints

6M , (2) t2 b where Kt is the concentration factor and is defined as 9 t −9 ) 20 = (1 + β) 20 . (3) Kt = (1 + 2R In our design, a specified angular displacement θ of the notch hinge can be calculated from the desired translational motion (as shown in Fig. 6), and is given as σ = Kt

Fig. 2.

Large-displacement prismatic joint

there is no constraint on the motion stage, the cross-axis error along the Y-axis will be created when FX is exerted. The second type, the double parallelogram, eliminates these two disadvantages. When FX is applied, the secondary stage is mobile and compensates for the Y-axis motion, and no tensile axial load occurs. Hence, the motion range of this joint is large. However, its drawback is the asymmetric structure which does not have uniform thermal expansion property. In order to eliminate asymmetry in the double parallelogram, a new large-displacement prismatic joint with symmetry is proposed (Fig. 2). This design is equivalent to the double parallelogram. When a force along the Y-axis FY is applied on the motion stage, the two secondary stages compensates for the X-axis motion, and there will be no axial stress. No axial tensile load at the notch hinge means that no stiffening occurs, and the motion range of the prismatic joint using notch hinges is large. No axial compressive load means that no buckling occurs. In this case, the notch hinge can be regarded as a 1DOF revolute joint, instead of a 3-DOF joint with one revolute and two translational motions. Considering compactness of the FPM, the double parallelogram and the new large-displacement prismatic joint are used in the conceptual design of the FPM. In our design, the dimensions of these two joints are the same. For both of the prismatic joints, the motion ranges are limited by the notch hinges. The thinnest portion of the notch hinge is prone to stress concentration and will generate plastic deflection, when a large load is exerted. As shown in Fig. 3, the approximate solution for the angular compliance was first presented by Paros and Weisbord [16], and is given as

u/2 . (4) L Thus, the nominal stress of the notch hinge is given as θ=

9

σ=

2Ebt 2 9πR

1 2

.

When the stress at the thinnest portion of the notch hinge reaches the yielding limit σb , plastic deflection occurs. Only elastic deflection of the notch hinge is permitted within the motion range of the prismatic joint. Hence, the motion range of the prismatic joint is the maximum displacement within elastic deflection 2Lσb β 2 f (β) (7) umax = 9 . E(1 + β) 20 III. D ESIGN AND M ODELING OF FPM A. Conceptual Design of FPM The double parallelogram and the proposed largedisplacement prismatic joint are used as the modules and are assemble to configure the XYZ-FPM. The XYZ-FPM is designed to be a parallel combination of the three limbs perpendicular each another shown in Fig. 4. Each limb has a P-P-P (prismatic-prismatic-prismatic) structure. It is a serial combination of Ui , Vi and Wi joints. The Ui joint and the Wi joint use the double parallelogram (Fig. 1(b)), and the Vi joint uses the large-displacement prismatic joint (Fig. 2). Based on the screw theory, the DOFs of the FPM is the intersection of those of the three limbs (U1 , V1 , W1 ) ∩ (U2 , V2 , W2 ) ∩ (U3 , V3 , W3 )

(1)

Under a pure bending moment, the maximum stress occurs at each outer surface of the thinnest part of the notch hinge.

(5)

s 3 + 4β + 2β 2 2+β 6(1 + β) −1 f (β) = + ( ). (6) 1 tan 2 1+β β (2β + β ) 2

5

Kr = KθZ −MZ ≈

E(1 + β) 20 θ, β 2 f (β)

= (Y, Z, X) ∩ (X, Z, Y ) ∩ (Y, X, Z) = (X, Y, Z) The working principle can be explained based on Fig. 4. When a force along the X-axis FX is directly applied on W1

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F

 

C

W L

 

t

A

Z

KWr/Ur

V

B

 

R

uW/U

U X

V

 U

W

Endeffector



W/U



U



MW/U

Y F

 V

KW/U

 W

Fig. 4.





F

(b) V joint

Fig. 6.

PRB models of prismatic joints

Schematic Diagram of FPM Z

W2 V2

PRB theory, the notch hinge can be simplified as a revolute joint when there is no axial compressive or tensile loads. In our design, the notch hinges do not support any axial load, and a revolute joint with constant stiffness is equivalent to the notch hinge when no stress concentration occurs. The links connecting two notch hinges are regarded as rigid, and the length is the distance between the centers of every two notch hinges. The equivalent rigid model can be further simplified as a second-order system with a mass and a linear spring. The simplification is shown in Fig. 6. The equivalent mass and the equivalent stiffness can be derived based on the theory of energy conversation.

X

U3 W3

U2

V3

(a) W/U joints

U1

Y

V1 W1

Fig. 5.

Prototype of FPM

joint, W1 joint deforms to generate the X-axis motion and transfers the force to U1 joint and V1 joint. Theoretically, U1 joint and V1 joint move along the X-axis without deformation, and at the same time U2 joint and V3 joint deflect to create the X-axis motion. Briefly, when FX is applied, W1 , U2 and V3 joints deflect to generated the X-axis motion, U1 and V1 joints move the same displacement as the end-effector without deflection, and other prismatic joints remain stationary. The motions along the Y- and Z-axes are identical. The prototype of the FPM has been manufactured shown in Fig. 5. Each prismatic joint is produced by wire cutting using a monolithic piece. The material Al7075 is selected because of good elasticity, low internal stress, and high strength. The actuation of the flexure mechanism is accomplished by three linear voice coil actuators connected directly with Wi joints. MGV 52-25 (miniature Guide Voice Coil Module) is used, together with MC4000 LITE control card.

MP/U =

(5MP/UA + MP/UB + 4MP/UC )L2P/U + 4JP/UA 4L2P/U KP/U

MV =

(8)

2KP/Ur = , L2P/U

(9)

(4MVA + 5MVB + 2MVC )L2V + 4JVA , 4L2V KV =

,

2KVr , L2V

(10) (11)

where KP/Ur = KVr = KθY −MY , and KθY −MY is the angular stiffness of the notch hinge. C. Stiffness Modeling As shown in Fig. 4, the designed FPM possesses a parallel structure composed of three limb, and each limb is serial combination of Ui , Vi and Pi joints. Therefore, the stiffness of the flexure mechanism can be given as

B. Pseudo Rigid Body Modeling Since the flexible components possesses infinite flexibility, it is difficult and complicated to model them precisely. The PRB method is a feasible solution for approximate modeling. In the

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0 B K=B 

KP1 +KV2 +KU3

0

0 0

KU1 +KP2 +KV3

0 0

0

KV1 +KU2 +KP3

1 CC. A

(12)

TABLE I R ESULT OF DIMENSION OPTIMIZATION

In the FPM, the flexure hinge (Fig. 3) is sensitive to the moment about the Z-axis, and the rotation about the Z-axis is the major deflection. The precise stiffness model is derive by Lobontiu [17] 1 KθZ −MZ

=

24R2 (6R2 + 4Rt + t2 ) + Ebt2 (2R + t)(4R + t)2 q 144R3 (2R + t) arctan 1 + + Ebt2 .5(4R + t)2 .5

4R t

.(13)

In the operation of the flexure mechanism, the flexure bears the moment about the Y-axis, and the rotation about the Y-axis is calculated as q q t t arctan 4(2R + t) 4R+t 4R+t − πt 1 = . (14) KθY −MY Eb3 t/6 In order to reduce the parasitic motions, the notch hinge should be sensitive to the moment about Z-axis and immune to the moment about Y-axis. The ratio λ of KθZ −MZ to KθY −MY should be small λ = KθZ −MZ /KθY −MY .

(15)

D. Dimension Optimization Flexure mechanisms depend on deflection of flexible components to achieve desired motion. The dimensions of flexible components are critical to static and dynamic performance of flexure mechanisms, such as stiffness, motion range, parasitic motion, natural frequency, etc. Ability of actuation, possibility of manufacturing and permissible volume usually limit the achievable dimensions. All of the factors should be considered in determination of the dimensions of the flexure mechanism. Therefore, optimal design is necessary. Based on the above PRB model, the optimal design can be formulated as follows • Objective: Minimize λ • Design variables: R, t, b, L • Constraints: 0.3mm ≤ t ≤ 1mm, 10mm ≤ b ≤ 20mm, 20mm ≤ L ≤ 30mm, 1mm ≤ umax , K ≤ 50N/mm. The objective of the optimal design is to develop an XYZFPM with minimum parasitic motion. The design variables are the dimensions of the notch hinges and the distance between every two notch hinges. The constraints consider the factors as follows. The flexure mechanism is cut by wire EDM. The thinnest portion of the notch hinge is not less than 0.3mm, because the tolerance of ±0.01mm cannot be ensured when the thickness is smaller than 0.3mm. For compactness, the size of the XYZ-FPM should not be large, thus the distance between every two notch hinges is limited. In order to achieve the desired motion range without plastic deformation, the maximum displacement should be larger than the required motion range of 1mm. With limitation of the output of the

R

t

b

L

3.5mm

0.4mm

10mm

25mm

voice coil actuators, the linear stiffness of the XYZ-FPM cannot exceed 50N/mm. The Gradient projection method (GPM) is used to search the optimal points in the workspace. The final values of the design variables are listed in Table I. The linear stiffness and the motion range of the XYZ-FPM based on PRB method can be calculated as K = 31.3N/mm, umax = 2.3mm. E. Dynamics Modeling Based on the simplified PRB model, the dynamics equation of the FPM is derived using Lagrangian method. The equation of motion can be obtained as       ¨ X X FX  FY  = M  Y¨  + K  Y  , (16) Z FZ Z¨

2 66 66 6 M =6 66 66 4

MW1 +MU2 +MV3

0

0

MW2 +MU2 +MU3

0

mU1 +mV1

0

mU1 +mV2

0

0

MW3 +MV1 +MV2

3 77 77 77, 77 77 5

mU3 +mV3

where M and K are the equivalent mass and the equivalent stiffness matrices of the XYZ-FPM, and mWi , mUi and mVi are the actual masses of Wi , Ui and Vi joints. Using the derived equation of motion, the natural frequency of the translational mode shape can be calculated as p K/M f= ≈ 78Hz 2π Note that the damping effect has not been considered in deriving (16). It is very difficult to estimate the damping coefficient analytically, which depends on the deformation of flexure hinges, the frequency of the external stimulation, and so on. Therefore, the experimental measurement is a feasible solution for obtaining this parameter. IV. FEM (F INITE E LEMENT M ETHOD ) S IMULATION Following design and modeling of the flexure mechanism, FEM analysis using ANSYS is conducted for verification of analytical dynamics modeling. The modal analysis is conducted for studying the activated mode shapes and the corresponding natural frequency. According to the selected material of Al7075, in ANSYS the material properties are defined with Young’s modulus E of 71GP a, Poisson’s ratio of 0.33 and density of 2700kg/m3 .

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TABLE II R ESULT OF MODAL ANALYSIS Mode shape

Natural frequency (Hz)

1

80

7

258

9

325

Then solid model is established according to the above design. The solid model is required to be meshed for further static analysis and modal analysis. The element for meshing is selected as 20-node SOLID95 for ensuring high accuracy of the model. The displacement constraints are applied according to the future experimental setup before performing analyses. Three voice coil actuators will mounted to three prismatic joints, and each of the actuators will activate one-axis translational motion and eliminate two other axis translational motions. Thus, each prismatic joint is allowed for its sensitive-axis motion and constrained for two other axes. W1 joint is allowed to move along the X-axis and fixed along the Y- and Z-axes. Similarly, W2 joint is allowed to move along the Y-axis and avoided along the X- and Z-axes. W P3 joint is permitted to move along the Z-axis and fixed along the X- and Y-axes. The modal analysis is then implemented for evaluating dynamics performance. Since flexure hinges have infinite flexibility, not only one-axis motion is activated. Usually the motion of the flexure mechanism is the combination of several mode shapes. The modal analysis just studies what kinds of mode shapes are activated and how fast they response. By checking the modal results, it is found that the following three mode shapes contribute to the operation of the flexure mechanism. The first mode shape describes the translational motion, which is desired, shown in Fig. 7(a). The seventh and ninth mode shapes (as shown in Fig. 7(b)) are facile to occur when the translational motion operates. These two order mode shapes partly explain the cross-axis errors and the parasitic rotations at the end-effector. In the above analytical modeling, the first mode shape is considered, but the seventh and ninth mode shapes are difficult to be modeled. Therefore, the seventh and ninth mode shapes are regarded as the unmodelled uncertainties. In the future control algorithm design, the unmodeled uncertainties and the external disturbances will be solved. The corresponding natural frequencies are listed in Table. II.

(a) First mode shape

(b) Seventh mode shape

(c) Ninth mode shape Fig. 7.

Mode shapes

Linear actuator Force sensor

Laser sensor

V. E XPERIMENT V ERIFICATION The experiments were conducted to verify the proposed FPM for stiffness, cross-axis error and parasitic rotation. Fig. 8 shows the experimental setup. The six-axis force sensor FT05270 (ATI) was mounted between the actuator and one input end of the FPM to measure the actuation force. The laser sensor LC2430 and LC2440 (KEYENCE) with the resolutions of 0.02µm and 0.2µm were used to measure the displacement of the end-effector located at the center of the FPM.

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Z End-effector

Fig. 8.

X

Y

Experimental setup

550

joints are assembled to the XYZ-stage using the parallel combination. PRB method is used to formulate the FPM, and the dimension optimization is implemented. The analytical models of the stiffness and the dynamics equation are derived. Finally, FEM simulation by ANSYS is conducted to evaluate the analytical dynamics equation, and the experiments are done to verify the analytical model of the stiffness and the decoupled characteristics. The FEM and experimental results show that the proposed FPM possesses large motion and decoupled kinematics structure.

500 450

Slope: 39.4 um/N Displacement (um)

400 350 300 250 200 150 100

Experimental data Fitting line

50 1

2

3

4

5

6

7

8

9

10

Force (N)

(a) Linear stiffness

R EFERENCES

100 Y/X-Axes Displacement %

1.9 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1

2

3

4

5

6

7

8

9

10

X-Axis Force (N)

(b) Cross-axis error Fig. 9.

Experimental results

The first experiment was done for measuring the linear stiffness of the FPM. The force along the X-axis was applied using the linear actuator, and the force sensor measured the Xaxis force. The laser sensor LC2430 sensed the displacement along the X-axis. Based on the input force and the output displacement, the linear stiffness along X-axis can be calculated. The experimental result is showed in Fig. 9(a). The slope of the fitting line is 39.4µm/N , and its reciprocal is the linear stiffness of 25.4N/mm. The linear stiffness along Y- and Zaxes is identical. The second experiment verified the decoupled characteristics. When the X-axis motion was activated, the displacement along the X- and Y-axes was simultaneously tested to study the cross-axis error. The experimental results showed that the cross-axis error is smaller than 1.9%, as shown in Fig. 9(b). At the same time, the parasitic rotation θZ was calculated based on the measured displacement along the Z-axis. It is proved that the parasitic rotation is smaller than 1.5mrad. Therefore, it can be verified that the FPM is indeed a decoupled kinematic structure. VI. C ONCLUSION This paper proposed an FPM with large displacement of 2.3mm and decoupled kinematics structure. The FPM can achieve decoupled X-, Y- and Z-axes translational motions with small cross-axis error less than 1.9% and small parasitic rotation less than 1.5mrad. To meet the requirements of large displacement and decoupled kinematics structure, the prismatic joints are investigated in this paper, and a new largedisplacement joint is designed. Then these modular prismatic

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