DESIGN OF LARGE-RANGE XY COMPLIANT PARALLEL ...

Proceedings of the ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2013 August 4-7, 2013, Portland, Oregon, USA

DETC2013-12206 DESIGN OF LARGE-RANGE XY COMPLIANT PARALLEL MANIPULATORS BASED ON PARASITIC MOTION COMPENSATION Guangbo Hao*

Qiaoling Meng

Yangmin Li

Department of Electrical and Electronic Engineering, School of Engineering, University College Cork (UCC), Cork, Ireland Email: [email protected] *Corresponding author

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira, Taipa, Macao SAR, China. Email: [email protected]

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira, Taipa, Macao SAR, China. Email: [email protected]

ABSTRACT This paper presents a large-range decoupled XY compliant parallel manipulator (CPM) with good dynamics (no under-constrained/non-controllable mass). The present XY CPM is composed of novel parallelogram flexure modules (NPFMs) that are parallel four-bar mechanisms as prismatic (P) joints with four identical monolithic cross-spring flexural pivots, flexure revolute (R) joints. The parasitic translation of the NPFM is compensated via the rotational centre shift of the flexure R joint thereof based on the prior art. The optimization function and optimised geometrical parameters are investigated for the NPFM at first to achieve the largest translation. The design of a large-range XY CPM is then implemented according to the fully symmetrical 4-PP parallel kinematic mechanism (PKM) and through using the optimised NPFMs. Finally, the simplified analytical stiffness modelling and finite element analysis (FEA) are undertaken for the static and/or dynamic characteristics analysis of the 4-PP XY CPM. It is shown from FEA in the example case that the present 4-PP XY CPM has good performance characteristics such as large-range motion space (10 mm × 10 mm with the total dimension of 465 mm× 465 mm), no non-controllable mass, monolithic configuration, maximal kinematostatic decoupling (cross-axis coupling effect less than 1.2%), maximal actuator isolation (input coupling effect less than 0.13%) and well-constrained parasitic rotation (less than 0.4 urad). In addition, the stiffness-enhanced NPFM using over-constraint is proposed to produce a first/second modal frequency of about 100 Hz for the resulting XY CPM.

translates in the XY plane by the deformation of the compliant members. Compared with its rigid-body counterparts, the XY CPM has plenty of merits such as eliminated backlash and friction, no need for lubrication, reduced wear and noise, and reduced part number up to monolithic configuration [3]. While typical dynamic ranges of 105 are easily achievable in flexure/compliant stages, large specific range is still the most challenging issue in high-precision (such as nanopositioning) compliant mechanisms [1] with its typical value of 10-3 for most designs. Large range of motion is generally affected by the following factors: a) system size (beam length), b) beam thickness, c) material selection (high yield strength/Young’s Modulus ratio), d) linear actuator, and e) conceptual-level design. The last factor is the most effective and best way to raise the motion range by using the distributed-compliance and/or novel configuration design (such as serial multi-level modules) for given material and actuators. This is because enlarging the length of beams can make the configuration bulky and reducing the thickness of beams may result in the decrease of stiffness significantly and other issues such as manufacturability. In addition, the number of non-controllable motion mass should be reduced from the good dynamics point of view, which is another limitation for large range of motion. Most existing XY CPMs [4-6] are obtained from the kinematically decoupled 4-PP (P: prismatic) parallel kinematic mechanisms (PKMs) and 2-PP PKMs as shown in Fig. 1 by replacing each rigid P joint with an appropriate flexural counterpart. In addition to the system construction, the flexural P joints play vital roles on the performance of the resulting XY CPMs such as motion range, cross-axis coupling (for output motion stage unless indicated otherwise), and actuator isolation. The prior art shows that flexural P joints mainly stem from the parallel four-bar mechanisms, i.e., parallelogram flexure modules (PFMs). Three typical examples are shown in Fig. 2. The basic PFM is an in-parallel two-plate spring (Fig. 2(a)). This module configuration is simple enough and can achieve large primary motion owing to using distributed-compliance but suffers from the inevitable undesirable parabolic parasitic translation and small parasitic rotation which contribute to the cross-axis coupling and/or actuator non-isolation effect in the resulting XY CPM. In order to eliminate parasitic motion effect, a parallel double PFM (Fig. 2(b)), composed of two basic PFMs in mirror symmetry, is frequently revisited. It, however, as the actuated P joint, sacrifices the large motion

1. INTRODUCTION There is an increasing demand for designing large dynamic range and large specific range 1 XY compliant parallel manipulators (CPMs) for precision engineering applications such as atomic force microscope scanning tables [1, 2]. The XY CPM, a flexure stage, is typically composed of a fixed base and a motion stage connected by compliant members actuated by actuators indirectly. Its motion stage only 1

‘Dynamic range’ is the ratio of the motion range to the minimum motion resolution. ‘Specific range’ is the ratio of the total motion range to the mechanism size. Under a given minimum motion resolution, the large specific range and large dynamic range can refer to ‘small size and large-range motion’. 1

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range of each basic PFM due to the significant load-stiffening effect2, resulting from the purely elastic effect compensating the parasitic translation in the symmetric configuration. This load-stiffening effect along X-/Y-direction may introduce a cross-axis motion of Y-/X-input stage in the resulting XY CPM. The serial double PFM (Fig. 2(c)) [7] is also frequently adopted to raise the motion range and eliminate the parasitic translation because of the presence of motion compensation from the secondary stage. However, the secondary stage involved in the serial double PFM is under-constrained or non-controllable and can undergo free vibration along the unconstrained directions [8, 9]. Therefore, the resulting XY CPMs can only behave well under quasi-static/low speed motion mode, in which the secondary stages do not vibrate uncontrollably. As a result of these shortcomings, the above three types PFMs are not to be employed in the work of this paper towards a well-behaved large-range XY CPM for an appreciable speed motion mode without under-constrained mass.

modelling and finite element analysis (FEA) are undertaken for the static and dynamic characteristics analysis. Section 5 further investigates an efficient strategy for achieving larger natural frequencies. Finally, conclusions are drawn. Motion stage

Secondary stage

Motion stage

Motion stage

b) Parallel double PFM in symmetry

a) Basic PFM

c) Serial double PFM

Figure 2. THREE TYPES OF PFMS AS FLEXURE P JOINTS IN DISTRIBUTED-COMPLIANCE

λLP (λ>1)

uL W

Actuated P joint

Actuated P joint

Y

Passive P joint

LP

Motion stage

Y X

X

2α

Motion stage

L

L: valid length W: valid width

aL

Motion stage Actuated P joint

L1

a) NPFM configuration with geometry Passive P joints

(a) 4-PP PKM

Y

(b) 2-PP PKM

Figure 1. KINEMATICALLY DECOUPLED XY PKMS

X

In addition, a family of novel parallelogram flexure modules (NPFMs) acting as large-range flexural P joints were recently proposed and thoroughly analyzed by Zhao et al [10, 11], each of which is composed of four identical generalized cross-spring flexural pivots [12], flexure revolute (R) joints. The rotational centre shift of the flexure R joint is utilized to compensate the parasitic translation of the resulting NPFM in the parallelogram without uncontrollable mass. Therefore, this family of NPFMs are good candidates as the large-range flexural P joints with eliminated parasitic translation in principle. One out of these synthesized configurations with geometry indication is shown in Fig. 3, which will be detailed in the next section. This illustrated configuration is adopted as a flexural P joint of the large-range XY CPM to be presented in this paper since it has the best comprehensive characteristics such as large mechanism motion range and monolithic manufacturing [11]. It is therefore an objective of this paper to propose a new XY CPM that is able to produce large range of motion without under-constrained mass (i.e. good dynamics). The remainder of this paper is organised as follows. Section 2 derives the optimization function and obtains the optimised geometrical parameters to achieve the largest translation. The large-range XY CPM is then designed according to a 4-PP PKM and through using the optimised NPFMs proposed in Section 3. In Section 4, the simplified analytical stiffness

Δcm θ

Motion stage

: R joint : P joint

ΔCM: rotational

L1 Δcm

centre shift in the parasitic motion direction

b) Parasitic motion compensation schematic diagram via demonstrating single chain PRBM

Figure 3. A SELECTED NPFM AS A FLEXURE P JOINT (TOP VIEW) 2. OPTIMISATION OF A NPFM 2.1. Parasitic Translation Compensation Condition The NPFM (flexural P joint) is studied in this section in details (Fig. 3), whose height parameter L is specified as the characteristic length. It is composed of four flexure R joints with the rotational radius length not less than the flexure plate’s length of LP. The angle between two symmetrical flexure plates is 2α, the distance between two rigid stages of the flexure R joint is uL, and the distance from the rotational center to the nearer rigid stage is aL. In order to compensate the parasitic translational motion from the parallelogram configuration, the rotational centre shift along the parasitic translational direction should be equal to half of the parasitic translation. The parasitic translational motion under a small angle assumption from the parallelogram configuration can be first

2

The load-stiffening effect 1) significantly increases the primary motion stiffness resulting in the use of only small motion linear actuators such as PZT actuators, and 2) significantly increases the tensile stress causing the ease of yield under large range of motion. 2

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LP σ s 1 ) (6) 3(1 + a / u ) − 1 T E ' Equation (6) can be re-written as follows: u / cos α L σ s ] (7) (1 + 2a ) L × sin[ 3(1 + a / u ) − 1 T E ' If T, L, σs and E' are all designated as constant, the maximal motion range index of the NPFM is determined by three scalar parameters as (1 + 2a )u 2 (8) max[ ] (2u + 3a) cos α Since cosα is a monotonic function over (0, 900), it is easy to observe from Eq. (8) that the larger the value of α, the larger the motion range. Combining with the suggestion of Fig. 4, we may propose to use the small value of a to obtain large motion range of the NPFM. A more detailed quantitative analysis is demonstrated in Fig. 5, which shows that the largest u (up to 0.5) and the smallest a (up to 0) can produce the largest motion range. However, in the above situation, the large angle of α (up to 700) will be caused (Fig. 4), which may bring over a very large size W (Fig. 3) of the NPFM in the valid width direction. Therefore, the specific range concept can be used here, which refers to the ratio of the motion range index to the normalized width (W/L) as follows: (1 + 2a)u 2 (9) /[ 4(a + u ) tan α ] (2u + 3a ) cos α The specific range index (Eq. (9)) is shown in Fig. 6. The largest u and the smallest a (including the corresponding α subsequently obtained from Eq. (3)) are proved to be the optimised values to produce both the largest motion range and the large specific range.

derived as

L1 sin θ max = L1 sin(

2

ΔPT = L1θ / 2 (1) where L1=(1+2a)L, which is the link length of the parallelogram configuration. The dominant rotational centre shift along the parasitic translational direction can be obtained based on [10] as follows: LP (9λ2 − 9λ + 1)θ 2 ΔCM = (2) 15 cos α where LP=uL/cosα, and λ=1+a/u. Let ΔCM = ΔPT / 2 , and LP and L1 be denoted by characteristic length L and the three scalar parameters, a, u and α (angle in radian), we have the following three-variable compensation equation:

cos α =

4u 2 + 36au + 36a 2 15(1 + 2a )u

(3)

where three variables are subject to ⎧0 < α < 900 ; ⎪ (4) ⎨0 ≤ a; ⎪0 < u < 0.5 ⎩ Equation (3) can be illustrated in Fig. 4, which shows that the variable, a, has a significant effect for the change of the angle, α, of the flexure R joint compared with the contribution of the variable, u, and its increase causes the decrease of the angle. Please note that the change range of the variable, a, depends on the value of the variable, u. For example, when u=0.25, a≤0.29. The smaller the variable u, the smaller the change range of the variable a. 80 u=0.45 u=0.40 u=0.35 u=0.30 u=0.25

70 60 u decrease

0.7

50

0.5

40

motion range index

alpha (deg)

u=0.45 u=0.40 u=0.35 u=0.30 u=0.25

0.6

u decrease

30 20

u decrease 0.4

0.3

0.2 10 0

0.1 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

a

0

Figure 4. COMPENSATION EQUATION VARIABLE RELATIONSHIP

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

a

Figure 5. MOTION RANGE INDEX OF THE NPFM

2.2. Maximal Motion Range Optimisation

0.7

u=0.45 u=0.40 u=0.35 u=0.30 u=0.25

0.65

Our objective in this section is to find one or more combination(s) of the three variables controlled by Eq. (3) to achieve the maximal translational motion range of the NPFM. Using the result obtained in Appendix A, the rotational motion range of the flexure R joint can be determined by LP σ s 1 Lp σ s 1 = θ max = (5) 3λ − 1 T E ' 3(1 + a / u ) − 1 T E ' where T is the in-plane thickness of the flexure plate. σs is yield strength of the material, and E' is the Plate Modulus (E'=E/(1-v2) with E of Young’s Modulus and v of Poisson’s ratio). Using Eq. (5), we have the translational motion range of the NPFM:

Specific range index

0.6 0.55

u decrease

0.5 0.45 0.4 0.35 0.3 0.25

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

a

Figure 6. SPECIFIC RANGE INDEX OF THE NPFM 3

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When the values for T and LP are given to be 0.4 mm and 25 mm for a common slender flexure plate, considering the actual monolithic manufacturing and the rigidity need of non-flexural parts, the following a set of values are chosen as the optimised geometrical parameters (Table 1). The resulting optimised NPFM as a flexure P joint is shown in Fig. 7, and a corresponding complex NPFM (well-constrained parasitic rotation as actuated flexure P joint) composed of two optimised NPFMs in mirror symmetry is also shown in Fig. 8.

via rotational centre shift, which will be indirectly verified in the following sections. 3. DESIGN OF A LARGE-RANGE XY CPM

Both types of kinematically decoupled XY PKMs (Fig. 1), 2-PP PKM and 4-PP PKM, can be used to obtain the large-range XY CPM. However, only the 4-PP XY PKM configuration (fully symmetrical design) (Fig. 1(a)) is adopted for designing the XY CPM in this paper sacrificing the specific range, which attributes to the following two main reasons. 1) The well-constrained parasitic rotational yaw is the desired characteristic for the precision engineering applications. This good characteristic can be achieved easily by the fully symmetrical design. 2) Thermal stability is also a nice feature for precision flexure systems to be worthy addressing. The large range of motion requires a large-range linear actuator, which cannot be a PZT actuator. Although amplifiers as actuated compliant P joints can be combined with the PZT actuator to enlarge the motion range [9], adversely, they lead to relatively low off-axis stiffness and augment the minimum incremental motion of the actuators, i.e. poor resolution. Thus, one needs to use the linear Voice Coil (VC) actuator for millimeter-level actuation range. Because of the 4-PP configuration is fully symmetric design, the motion stage will not drift much due to the aluminium alloy’s thermal variations caused by VC actuators. A large-range 4-PP XY CPM (Fig. 9) can be obtained by replacing each passive P joint with an optimised NPFM (Fig. 7) and replacing each actuated P joint with an optimised complex NPFM (Fig. 8) in the 4-PP PKM using appropriate arrangements for large motion stage size and better actuation isolation characteristics. The complex NPFM used here as the actuated P joint is for the better actuator isolation. Two VC actuators are used to exert actuation forces on the X-input stage and the Y-input stage. The present XY CPM has an overall dimension of 465 mm × 465 mm with the motion range up to 10.54 mm in bi-direction of the X-/Y-axis. However, considering the safety factor in practice, the motion range is conservatively specified to 10 mm in the bi-direction (i.e. a specific range of 0.0215). Apart from the main merits of large-range motion and good dynamics performance (no non-controllable motion mass), the present XY CPM (Fig. 9) is also approximately kinematostatically decoupled, which means that one primary output translational displacement is negligibly affected by the actuation force along the cross-axis direction, which describes the relationship between the input force and output motion. This decoupling is also called the minimal cross-axis coupling (also maximal cross-axis decoupling) in CPMs, which is the sufficient condition of kinematic decoupling. The proposed XY CPM has reduced part number compared to the rigid counterpart, and can even be fabricated using planar precision machining technologies such as electrical discharge machining (EDM) and water jet from a piece of plate with out-of-plane thickness of 20 mm monolithically. Alternatively, each leg can be separately manufactured and then all parts are assembled together if the monolithic fabrication is difficult. The selected material AL7075 has a high σs/E ratio, low internal stresses, good strength and phase stability [14]. The adopted linear VC actuator used here has merits such as large-range nanopositioning (the large range of motion and high nanometric resolution), linear model, and force-control along with hysteresis-free, frictionless and

Table 1. OPTIMISED GEOMETRICAL PARAMETERS FOR T=0.4MM AND LP =25MM

u

a

0.40

0.05

α ( 0) 62.05

λ 1.125

L (mm) 29.30

L1 (mm) 32.22

W (mm) 107.00

Motion stage

a) Before deformation

b) In large deformation

Figure 7. OPTIMISED NPFM (TOP VIEW)

Motion stage

Figure 8. OPTIMISED COMPLEX NPFM (TOP VIEW)

Substituting the results in Table 1 into Eq. (6), the motion range of the optimised NPFM and complex NPFM (Figs. 7 and 8) can be calculated to be up to 5.27 mm in the uni-direction for selected material AL7075 with E=72 GPa, v=0.33 and σs=505 MPa, i.e. a total motion range of 10.54 mm in the bi-direction. Apart from the large motion range, the optimised NPFM and complex NPFM have better off-axis translational stiffness (i.e. no dramatic decrease) over the entire motion range compared to the flexure P joint composed of two serial double PFMs (Fig. 2(c)) [13]. It deserves to note that the optimised complex NPFM has a nearly constant primary translational stiffness that reflects the contribution of the above parasitic translation compensation 4

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cog-free motion. In addition, linear optical encoders can be chosen for input sensing and capacitive probes can be chosen

for output sensing by measuring the relative displacement [1].

Y stage2

Rendered Perspective View

Actuated flexure P joint

One leg

Y

X stage1: X-input stage

X stage2 Fx

Output motion stage

SC

X

AX

Rendered Top View

Passive flexure P joint

Y stage1: Y-input stage

AY

Fy

Figure 9. THE LARGE-RANGE 4-PP XY CPM WITHOUT UNDER-CONSTRAINED MASS 4. THEORETICAL CHARACTERISTICS ANALYSIS

In addition, dynamic characteristics such as natural frequencies and modal shapes will be investigated.

4.1. Characteristics Descriptions 4.2. Stiffness Modelling

Under the action of two input forces, Fx and Fy, being exerted at two input points, AX and AY, as indicated in Fig. 9, the following main static characteristics will be investigated. 1) Relationships between the input-force and output displacements for single loading and two-axis loading. Output displacement is specified at SC point on the motion range as shown in Fig. 9. These relationships can not only reflect the nominal primary motion stiffness (single loading), but also the cross-axis coupling effect denoted by (XSC-XSC|Fy=0)/XSC|Fy=0 or (YSC-YSC|Fx=0)/YSC|Fx=0. 2) Parasitic rotation for single-axis and two-axis loading. Parasitic rotation is specified for the rotation of the motion stage, which can be obtained by the displacement difference of two points on the motion stage. 3) Actuator non-isolation effect (i.e. input coupling effect) for single-axis and two-axis loading. Input-coupling is specified for the transverse motion at the two actuation points, AX and AY. Input-coupling effect can be denoted as YAX/YSC or XAY/XSC. 4) Lost motion effect for single-axis and two-axis loading. Lost motion is specified for primary motion difference between points AX and SC or between AY and SC. Lost motion effect can be expressed by (XAX-XSC)/XSC or (YAY-YSC)/YSC.

Neglecting the load-stiffening effect, the rotational stiffness for each flexure R joint in the NPFM is simply estimated as [15] 8(3λ 2 − 3λ + 1) E ' I 8(3λ2 − 3λ + 1) E ' (UT 3 / 12) KR = = Lp uL / cos α (10) 2 3 2(3λ − 3λ + 1) EUT cos α =

3uL(1 − v 2 )

where U is the out-of-plane thickness of 20mm of the flexure plate. E' is the plate modulus of AL7075-T6, which is equal to E/(1-v2), with a Poisson ratio v of 0.33. Using the result of Eq. (10), the translational stiffness for the NPFM can be obtained using the energy method. 4K 8(3λ2 − 3λ + 1) EUT 3 cos α K P = 2R = L1 3uLL12 (1 − v 2 ) (11) 8(3λ2 − 3λ + 1) EUT 3 cos α = 3u (1 + 2a) 2 L3 (1 − v 2 ) Moreover, the translational stiffness for the present 4-PP XY CPM (Fig. 9) along each primary direction without considering the cross-axis coupling and lost motion can be 5

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derived as 2

expected the first two modal shapes illustrate the free vibrations along the two primary motion axes, respectively, and the third modal shape shows the free rotation vibration about the central axis of the motion stage. All free vibration modes involve the motion of the input stages and/or the middle motion stage, and all motion mass are controllable. A different XY CPM design with non-controllable mass and its modal shapes are shown in Appendix B as a contrast. Figure 11 shows that the difference between the FEA nominal primary stiffness (Fy=0) and the analytical stiffness (Eq. (13)) (|analytical-FEA|/analytical) is about 1.04%, and the analytical values are a little bigger. In Fig. 12, it is shown that there is negligible cross-axis coupling effect ((YSC-YSC|Fx=0)/YSC|Fx=0) which is within 1.2%. The cross-axis coupling effect is dramatically influenced by Fx and slightly affected by Fy. The simulated curve is not symmetrical with regard to the vertical axis through Fx=0. The curve trends indicate that there is no cross-axis coupling effect, i.e. (YSC-YSC|Fx=0)/YSC|Fx=0=0, when Fx=0 although there is certain FEA result fluctuation at this point.

3

16(3λ − 3λ + 1) EUT cos α (12) u (1 + 2a ) 2 L3 (1 − v 2 ) Substituting all the known conditions into Eq. (12), we can obtain the primary motion stiffness value: K CPM = 90.58 N/mm (13) Equation (13) suggests that the input force over [-450N, 450N] is needed to produce a 10mm’s total motion range in the bi-direction of the X-/Y-axis. Therefore, the VC actuator LA30-48-000A (from BEI Kimco Magnetics) can be adopted for large-range requirement. The analytical equations of other characteristics such as cross-axis coupling and parasitic rotation are left for the future work, but to be demonstrated by the nonlinear FEA results (Comsol). K CPM = 6 K P =

4.3. FEA Simulations and Characteristics Analysis

FEA simulations (Figs. 10-15) are conducted in this section for determining the static characteristics and dynamics characteristics. Herein, refining meshing size and plane strain analysis are set up.

6 FEA Fy=0N FEA Fy=90N FEA Fy=180N FEA Fy=270N FEA Fy=360N FEA Fy=450N Analytical results

4

Xsc (mm)

2

0

-2

-4

a) Motion mode (top view)

b) Modal shape 1 (top view)

-6 -500

-400

-300

-200

-100

0 Fx (N)

100

200

300

400

500

400

500

Figure 11. PRIMARY MOTION -3

14

x 10

FEA Fy=90N FEA Fy=270N FEA Fy=450N

12

c) Modal shape 2 (top view)

(Ysc-Ysc|Fx=0)/Ysc|Fx=0

10

d) Modal shape 3 (top view)

8 6 4 2 0 -2 -500

-400

-300

-200

-100

0 Fx (N)

100

200

300

Figure 12. FEA RESULTS OF CROSS-AXIS COUPLING EFFECT e) Modal shape 4 (perspective view)

Lost motion effect ((XAX-XSC)/XSC) obtained from FEA is shown in Fig. 13 with the maximal value less than 0.8% occurring at the point very near to Fx=0. This relatively big lost motion effect may result from the inaccuracy of FEA in very small deformation. It is also shown that the lost motion effect dramatically reduces with the increase of Fx and increases with the increase of Fy if Fx>0. When Fx 1) LP

Y1 Mz Fy

X1 θ Δy

Figure A.1. FLEXURE R JOINT: MONOLITHIC CROSS-AXIS PIVOT

The rotational motion range for a flexure R joint: monolithic cross-axis flexure pivot is derived in this section. In the local coordinate system X1Y1Z1 for a single plate as shown in Fig. A.1, the transverse displacement has a relationship with the rotational angle: (A.1) Δy = −(λ − 1) LPθ Based on the beam constraint model derived by Awtar [14], we have ⎧⎪ Fy /( E ' I / L2P ) = −12(λ − 1) LPθ / LP − 6θ = (−12λ + 6)θ (A.2) ⎨ ⎪⎩M z /( E ' I / LP ) = 6(λ − 1) LPθ / LP + 4θ = (6λ − 2)θ Equation (A.2) can be simplified as −12λ + 6 Fy LP = Mz (A.3) 6λ − 2 Equation (A.3) suggests that the maximal moment occurs at the mobile end of the flexure plate with a value of (6λ − 2)θE ' I −12λ + 6 since is within (-2, -1.5] 6λ − 2 LP −12λ + 6 + 1 |< 1 ) as indicated in Fig. A.2. 6λ − 2 Therefore, the maximal normal stress from the bending (dominant effect) can be obtained as (6λ − 2)θE ' I / LP (3λ − 1)θE ' T σ max = = ≤ σ s (A.4) 2I / T LP

(|

From Eq. (4), we have the rotational motion range: 1 Lp σ s θ max = 3λ − 1 T E ' 9

(A.5)

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Appendix C: An XY CPM without Using Parasitic Translation Compensation

-1.45 -1.5 -1.55

In order to demonstrating the advantages of the parasitic translation compensation of the NPFM as a flexure P joint, a different XY CPM composed of flexure P joints with non-compensated parasitic translation can be obtained in Fig. C.1. The relationship between the input-force and the output-displacement obtained from the nonlinear FEA results for the XY CPM is shown in Fig. C.2. It implies an increased primary stiffness over motion range.

(Fy x Lp)/Mz

-1.6 -1.65 -1.7 -1.75 -1.8 -1.85 -1.9 -1.95

1

1.5

2

2.5

3 lamda (>1)

3.5

4

4.5

5

Figure A.2. RATIO OF THE MOMENT CONTRIBUTED BY TRANSVERSE FORCE AT THE FIXED END TO THE PURE MOMENT

Y stage2 Base

Base

Appendix B: A Large-range XY CPM with Non-controllable Mass

Y X stage1

A large-range XY CPM composed of serial double PFMs as flexure P joints is shown in Fig. B.1. The serial double PFM used herein (similar to Fig. 2(c)) includes a secondary motion stage that is under-constrained. Modal shape 4 shown in Fig. B.1 proves that the under-constrained mass can freely vibrate without causing the motion of any input stage and the middle motion stage.

Fx

X stage2 AX

X

SC XY Motion stage

Base

Base Y stage1 AY

Fy Figure C.1. THE XY CPM WITHOUT USING PARASITIC TRANSLATION COMPENSATION (TOP VIEW)

a) XY CPM design

c) Modal shape 3(23.08 Hz)

b) Modal shape 1 (23.07 Hz)

d) Modal shape 4(45.39 Hz)

Figure C.2. LOAD-DISPLACEMENT RELATIONSHIP

e) Modal shape 5 (60.24 Hz)

f) Modal shape 6(62.20 Hz)

Figure B.1. THE XY CPM AND ITS MODAL SHAPES (ALL IN TOP VIEW) 10

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