新型電磁致動與減震之三自由度撓摺結構精密定位平台

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國立臺灣大學電機資訊學院電機工程學研究所碩士論文

Graduate Institute of Electrical Engineering

College of Electrical Engineering and Computer Science

National Taiwan University Master Thesis

新型電磁致動與減震之三自由度撓摺結構精密定位平台 A Novel Electromagnetic Actuated and Damped 3-DOF Precision

Positioning Stage with Flexure Suspension

林志憲 Chih-Hsien Lin

指導教授：傅立成博士共同指導：陳美勇博士 Advisor: Li-Chen Fu, Ph.D.

Mei-Yung Chen, Ph.D.

中華民國九十七年六月

June, 2008

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致謝

寫了又刪，刪完再重寫，等待了很久的寫致謝的這天，卻又好像什麼也寫不出來。在台大的這兩年，跟各位師長的相處，和實驗室許多學長與同儕之間的回憶，又怎是這一頁就能道盡的。

感謝我的指導老師傅立成教授，在這兩年的研究生生涯，您對我的影響不可言喻，老師對理論的嚴謹、對研究的熱忱，我會永遠記在心中。感謝口試委員顏家鈺教授、胡竹生教授、

陳永耀教授以及劉昌煥教授，對我的論文提供寶貴的意見，使得內容更加完整。

感謝亦師亦友的美勇、勤鎰和紹剛學長，你們對我研究上的指點，是我得以完成這份論文的基石；感謝正民、明理、智富和聖化學長，你們不管上是在學業上、實驗室生活上都是我最好的學長，你們在實驗室的時候會讓我特別安心；感謝銘全和定國學長，在我有過迷惘的時候，是學長們適時地拉了我一把；感謝威文、政昌、國宏和俊緯學長，你們總是給我一些不同的想法來面對事情。

感謝怡孜跟琬琳，不僅是在研究跟課業上幫我許多，也在我人生低潮的時候適時鼓勵我；

感謝顯真，你將自行車帶入我的生命，讓我的生活得以豐富，難以用言語表達我的感謝；感謝貫豪跟晏榆，這兩年和你們相處很有趣；感謝實驗室的學弟妹們，繕琮、世勳、羿如、柏男、柏徐、志鵬和孟勳，有你們的實驗室總是充滿著歡樂；感謝助理郁璇、懿萱、小寧跟立婷，妳們在行政上面真的幫我太多忙了。

感謝我的家人，特別是我的母親，您在最後這段時間忍受我的任性，包容我的不成熟。

我永遠都以能生為妳的兒子而感到幸運。

志憲於電二 236 實驗室

2008.07.25

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摘要

本論文提出一種以電磁力產生驅動與減低結構振動的平面式定位平台，可用於精密定位相關應用。平台本體採用一體成型之平行撓性結構做為導引機構，其運動來自結構本身的彈性變型。藉著一個近似均勻強度的磁場與四組線圈，本論文提出一種線性電磁制動器的設計，做為提供驅動平台移動與轉動的力量來源。

為了減輕撓性結構天生的共振問題，亦提出了以渦電流效應提高系統阻尼的電磁式阻尼器，其非接觸式的特性較一般接觸式的阻尼器更適合用於本論文所訴求之精密運動控制系統。本新型精密定位平臺達成三大目標：第一，擁有大移動行程的能力（此指公釐的範疇）; 第二，精密定位的能力; 第三，採用簡潔的機構設計。

為了要在系統參數變化與外在擾動干擾下達到系統穩定與強健性的目標，本論文提出一個強健適應性滑動模式控制器，提升系統在定位與循跡需求上的效能。

控制器包含具有強健性的滑動模式控制器，與線上估測系統參數同時調變控制器

的適應律。本論文設計的平台最大行程可達到 3x3mm²，定位解析度為 10μm。透

過在時域與頻域上的分析，撓性結構的共振現象可以有效地被渦電流阻尼器減輕。

實驗結果亦證實本論文所設計之平面精密定位平台具有所訴求之精密定位與循跡之能力。

關鍵字：精密運動控制, 結構減震, 平行式撓性機構, 電磁制動器, 渦電流阻尼器, 適應性滑動模式控制

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Abstract

This thesis proposes a novel planar electromagnetic actuated and damped positioning stage for precision positioning applications. The moving stage is suspended by the monolithic parallel flexure mechanism, whose motion comes from the elastic deformation of the flexure. A linear electromagnetic actuator which consists of a near-uniform magnetic field and four coils is designed and implemented to provide the propelling force and torque for 3-DOF motions. In order to suppress the vibration of the flexure suspension mechanism, an eddy current damper is designed and integrated with the electromagnetic actuator. Since the electromagnetic damper experiences no contact, it is obviously more adequate than other kinds of contact damper to be incorporated into precision motion control. The three salient features of the novel system design in the research include: (1) to have large moving range (in mm level), (2) to achieve precision positioning, and (3) to design a compact mechanism.

For the purpose of gaining system robustness and stability, a robust adaptive sliding-mode controller is proposed to enhance the system performance for both regulation and tracking tasks. The developed robust adaptive control architecture consists of two components: 1) sliding mode controller, and 2) robust adaptive law.

With the designed controller, the stage can achieve high positioning resolution, where

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the tracking error in each axis is kept within 10μm. Experiment results show the vibration of the flexure mechanism can be suppressed by the eddy current damper successfully in a series of time-domain and frequency domain tests. Besides, the designed traveling range of the positioning stage is 3mm x 3mm in planar motion, and tracking and contouring performance are also examined to assure the appealing dynamic property of the stage.

Keywords: Precision motion control, Vibration suppression, Parallel flexure mechanism, Electromagnetic actuation, Eddy current damper, Adaptive sliding mode control

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List of Figures

Fig. 1-1 The exploded view of the flexure and evaluation stages [11] ... 4

Fig. 1-2 Translation stage proposed by Chang [12] ... 5

Fig. 1-3 Schematic of the 3-DOF microparallel mechanism [13] ... 5

Fig. 1-4 The complex flexure mechanism with high stiffness [14] ... 5

Fig. 1-5 The 2-DOF compliant parallel manipulator [15] ... 6

Fig. 1-6 The sponge-like damper used to suppress vibration ... 7

Fig. 2-1 Lorentz force principle ... 13

Fig. 2-2 The eddy current phenomenon ... 14

Fig. 2-3 Eddy current damper to suppress vibration [19] ... 15

Fig. 2-4 B-H curve of a typical ferromagnetic material ... 16

Fig. 2-5 Axis-symmetric beam ... 23

Fig. 2-6 Common flexure types: (a) simple cantilever, (b) clamped-clamped, (c) crab-leg, (d) folded-flexure, (e) serpentine ... 26

Fig. 2-7 Cantilevered beam and its free body diagram ... 27

Fig. 2-8 Quad-symmetric clamped-clamped flexure and its free body diagram ... 28

Fig. 2-9 Quad-symmetric crab-leg flexure and its free body diagram ... 31

Fig. 2-10 Abbe error ... 34

Fig. 2-11 Cosine error ... 35

Fig. 3-1 The VCM actuator of our previous research [8] ... 41

Fig. 3-2 The steel structure and permanent magnets ... 42

Fig. 3-3 The near-uniform magnetic field ... 42

Fig. 3-4 The numerical simulation of magnetic field ... 43

Fig. 3-5 The assembly of the VCM actuator ... 44

Fig. 3-6 The forces that are generated by Lorentz force principle ... 44

Fig. 3-7 The eddy current inside the copper mounting. ... 45

Fig. 3-8 Parallel kinematical XY flexure mechanism ... 47

Fig. 3-9 The detailed view of the thin flexure ... 47

Fig. 3-10 The stiction on outer frame ... 49

Fig. 3-11 The triangulation measurement method ... 50

Fig. 3-12 Perspective view of the measuring system ... 50

Fig. 3-13 The exploded view of the positioning stage ... 51

Fig. 3-14 The arrangement of four propelling forces ... 52

Fig. 4-1 The allocation of the forces ... 53

Fig. 4-2 The arrangement of sensors ... 56

Fig. 4-3 Abbe error in the measurement system ... 57

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Fig. 4-4 The free body diagram of the system ... 59

Fig. 4-5 The frequency response of the standard 2^nd order system ... 62

Fig. 4-6 Frequency response in the x-axis ... 64

Fig. 4-7 Frequency response in the y-axis ... 64

Fig. 5-1 Direct adaptive scheme ... 66

Fig. 5-2 Simulation of adaptive sliding mode controller’s regulation. ... 78

Fig. 5-3 The estimation of (a) kx, (b) bx, (c) m, (d) wc ... 79

Fig. 6-1 The flexure mechanism and inlaid structure ... 81

Fig. 6-2 The noise of the displacement sensor ... 82

Fig. 6-3 The positioning stage and sensors ... 83

Fig. 6-4 The vibration suppression in impulse response ... 84

Fig. 6-5 The vibration suppression in step response ... 85

Fig. 6-6 The vibration suppression in frequency response ... 85

Fig. 6-7 The measurement equipment of the force variation ... 86

Fig. 6-8 The variance of the force in the traveling range ... 87

Fig. 6-9 1mm step response on X-axis ... 88

Fig. 6-10 The transient of the step response ... 89

Fig. 6-11 The steady state of step response ... 89

Fig. 6-12 The remainder states of step response, (a) displacement in y-axis, and (b) rotation in θ-axis ... 90

Fig. 6-13 0.5Hz sinusoidal tracking in x-axis ... 91

Fig. 6-14 Sinusoidal tracking error ... 91

Fig. 6-15 The remainder states of sinusoidal tracking, (a) displacement in y-axis, and (b) rotation in θ-axis ... 92

Fig. 6-16 Trajectory of circular contouring ... 93

Fig. 6-17 Contouring error of circular motion ... 93

Fig. 6-18 Trajectory of spiral contouring ... 94

Fig. 6-19 Contouring error of spiral motion ... 95

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List of Tables

Table 2-1 Characteristics of Magnet ... 19 Table 2-2 Specifications of NdFeB ... 20 Table 6-1 Specifications of the PC-based controller. ... 83

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Chapter 1 Introduction

1.1 Motivation and goal

Therole of the multi-DOF ( Degree-of-freedom ) and high precision positioning stages becomes more and more important during the development of the micro- and nano-scale technology [1]. Positioning stages with high resolution, good repeatability and load capacity are essential to the applications of the most advanced researches. For the researches of Scanning Probe Microscope ( SPM ) or Atomic Force Microscope ( AFM ) [2][3], in order to create the three-dimensional surface topographic images of the specimens with resolutions of all dimensions down to the nanometer or Angstrom scales, the multi-DOF positioning stages providing capabilities of ultra-high resolution motions are necessary. For the modern fabrication, inspection and package processes, in which precision positioning and manipulation of the very small objects are purposed [4][5], and hence the multi-DOF precision positioning stages are also imperative.

In precision positioning, friction is the main obstacle, i.e., it is one of the most important factors limiting the performance in the precise positioning application.

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Ball-screws drivers are popularly used in traditional positioning stages. Unfortunately, it will cause undesirable disturbances and backlash due to contact of bearing element.

These unfavorable factors will be likely to affect our efforts to achieve high precision motion. Considerable effects to model and compensate the effect of friction have been made for years [6][7]. But there is no way more direct and effective than the avoidance of contact. Two modern non-contact approaches are usually utilized to avoid the effect of friction: the magnetic levitation and flexure hinge suspension.

In magnetic levitation (Maglev), the stage is levitated by the electromagnetic force.

The Maglev technology has been widely used on transportation system, and it is also useful in precision positioning. Another modern approach uses the flexure mechanism for suspension and guidance of the positioning stage, and the principle is to fully utilize the elastic deformation of the very thin cantilever beams. Without contact, friction, stiction, and backlash are removed, and hence the flexure mechanism facilitates smooth and high resolution motion of the positioning stage. It is worthwhile to note that the actuation adapted in most researches involving positioning stage with flexure mechanism utilizes piezoelectric actuators. However, typical piezoelectric actuators can merely travel in a linear range of micrometers due to the hystersis property, which severely limits the traveling range of the positioning stage. Most piezoelectric actuators

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can only handle small range motion and may not be suitable to achieve precision motion with large moving range.

In our previous researches [8][9], electromagnetic actuations are used. There are many advantages associated with electromagnetic actuating technology, such as no contamination, no friction, fast response, high acceleration, large travel range, low noise, and low cost. The flexure mechanism positioning stage with electromagnetic actuators thus has the potential to achieve high precision control and to achieve a longer traveling range typically ten or fifteen times of that from the same stage but using piezoelectric actuator.

The aiming target of this thesis is to develop a low-cost and compact positioning stage with large traveling range, which is suspended by flexure mechanism and driven by the electromagnetic actuators.

In the following section, the literature on the related work and the results of our previous researches will be introduced.

1.2 Literature Survey

1.2.1 Precision positioning utilizing flexure mechanism

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The positioning stage proposed in this thesis work utilizes flexure mechanism based on the property of elastic deformation of the flexure material itself. A general discussion on designing flexure hinge mechanisms was given in detail by Smith [10].

The precision devices use flexure mechanism as replacement of conventional hinges, which in turn prevents the problems of friction and backlash. One degree-of-freedom (DOF) nanopositioning system (developed in Asylum Research, Santa Barbara, Univ. of California) has been proposed [11], as shown in Fig. 1-1. The work equipped with the symmetrical geometry to achieve nanometer resolution with x-axis or y-axis translation or even z-axis rotation has been reported [12], as shown in Fig. 1-2. Similarly, a planar three DOF parallel-type structure has also been designed in [13], as shown in Fig. 1-3.

Besides the mechanisms mentioned above, there are still considerable efforts devoted to the mechanism design to enhance the stiffness of the precision device, which together lead to more complicated [14], as shown in Fig. 1-4.

Fig. 1-1 The exploded view of the flexure and evaluation stages [11]

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Fig. 1-2 Translation stage proposed by Chang [12]

Fig. 1-3 Schematic of the 3-DOF microparallel mechanism [13]

Fig. 1-4 The complex flexure mechanism with high stiffness [14]

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Moreover, Li and Xu [15] proposed a 2-DOF compliant parallel manipulator as shown in Fig. 1-5. The workspace of the stage turns out to be rougnly 180 x180 μm.

They optimized the kinematic design to enlarge the workspace by finite element analysis.

Fig. 1-5 The 2-DOF compliant parallel manipulator [15]

1.2.2 Our previous research

In the previous researches of our lab, precision positioning stages using flexure mechanism have been developed. Huang [8] first proposed a 3-DOF flexure mechanism and Wu [9] next proposed a 6-DOF flexure mechanism. The electromagnetic actuators were instead of the common piezoelectric ones used in these researches in order to extend the traveling range of the positioning stage. However, vibration of flexure mechanism becomes significantly severe under the circumstances without support of the rigid piezoelectric actuators. It is noteworthy that such vibration is a serious issue to

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control the positioning stage.

Fig. 1-6 The sponge-like damper used to suppress vibration

Therefore, sponge-like material was used to act as the mechanical damper to suppress the vibration, which is as shown in Fig. 1-6. But such damper is a contact device, and its use contradicts to the underlying designing principle of a noncontact and frictionless precision positioning stage. An important objective of this thesis is to seek an adequate contactless damper nether than a contact one to serve as vibration remover.

1.3 Contribution

In this thesis, a novel planar positioning stage including mechanism, control, and analysis are successfully presented. In particular, there are several main goals that have been achieved here: (1) to design a linear electromagnetic actuator and a built-in contactless electromagnetic damper to suppress the vibration of flexure mechanism, (2)

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to integrate the electromagnetic devices and the parallel flexure mechanism for planar positioning system, (3) to derive a precise measurement methodology for the reliable and accurate measurements of the position and posture of positioning stage, (4) to develop an advanced adaptive sliding-mode controller, (5) to perform numerical simulation to validate the satisfactory performance, and (6) to perform extensive experiments to validate the excellent performance.

1.4 Thesis Organization

There are totally seven chapters in this thesis. It starts with an introductory chapter which motivates this research and introduces the state-of-the-art research results in precision positioning devices. In Chapter 2 , we review some basic theories of electromagnetism, properties of the permanent magnet, the analysis of the energy method, and the introduction on flexure mechanism. The following chapter, Chapter 3 , describes the design concept for fulfilling the desired motion behavior through description of the detailed specifications of various components in the novel 3-DOF positioning stage. Next, the force allocation, the sensing methodology, the mathematic model, the dynamic behavior and the system identification of the 3-DOF positioning stage will be analyzed and derived in Chapter 4 . Then, proper controller design, which based on adaptive sliding-mode control technique, and the numerical simulations are

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conducted in 0. Subsequently in order to validate the effectiveness and appealing performance of the design, extensive experimental results are provided in Chapter 6 . Finally, we make some conclusions to sum up the results in this thesis in Chapter 7 .

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Chapter 2 Preliminary

Equation Chapter 2 Section 1

2.1 Basic Theories of Electromagnetic

In this section, we will briefly review the basic theories of electromagnetic that will be used in our system design and analysis, specifically concerning Lorentz force principle and eddy current phenomenon.

2.1.1 Lorentz force principle

The Lorentz force equation is the basis for governing all magnetic forces. Magnetic fields are a description of the relativistic effects that occur among moving charges, which are a direct result of the Lorentz transformation of the Coulomb force.

The force on a current element immersed in a magnetic field B is given as:

dF Idl

= ×

dF I Bdl (2.1)

Note that cannot exist by itself as it must be part of a complete loop or circuit. On such a loop, the total summed force is

Idl

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B

=

∫v

×

F Idl (2.2)

As shown in Fig. 2-1, in order to simplify (2.2), a segment with length L of a long straight wire is assumed to be exposed to a uniform magnetic field B that is perpendicular to the wire, and the return path of the wire is to be outside the field. Then, this integral can be expressed as a scalar solution

F ILB (2.3)

where I is the current carried on the conduction wires, L is the length of the conduction wires through the magnetic field, and B is the external magnetic flux density. If there are N-turn wires through the magnetic field, then

=

F NILB (2.4)

It is important to note that the force on the conductor is given only by (2.4) if the field due to the current I can be neglected.

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Fig. 2-1 Lorentz force principle

The electromagnetic actuator designed based on Lorentz force principle are often applied when high bandwidth dynamic are to be achieved. Examples are voice coil actuators, loudspeakers, synchronous brushless DC motors, and so on.

2.1.2 Eddy current phenomenon

When a non-ferromagnetic conductor moves in a magnetic field, or a moving and varying magnetic field intersects a non-ferromagnetic conductor, the relative motion causes a circulating current within the conductor. Figure 2-2 shows that when a conductor is moving in the magnetic field B with the velocity v, the eddy current i is generated within the conductor. The interaction between the current and the magnetic field will generate a force to resist the relative motion between the magnetic field and the conductor. Because of the resistance of the conductor, the eddy current will dissipate and energy of the system will be transformed into heat.

N B S

L

F

wire

I

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Fig. 2-2 The eddy current phenomenon

For the researches on, say, electric motors, the eddy current phenomenon will reduce the efficiency and should be avoided. However, it can be used to remove the excessive energy from the system to physical contact. There are lots of practical applications of eddy current devices [18], including electromagnetic braking, magnetic damping, and passive vibration control.

Bae et al. [19] proposed the design and modeling of a passive eddy current damper shown in Fig. 2-3. The eddy current damper (ECD) consists of a copper plate and a pair of magnets. The experimental results show that the vibration of the cantilever beam has successfully been suppressed by the additional ECD. The magnitude of the damping force is also described in [19] as:

2

Fd = −σδvB Sα (2.5)

where σ is the thickness of the conductor in the magnetic field, δ is the conductivity, v

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is the velocity, B is the magnetic flux density, S is the effective area, and α is the dimension parameter.

Fig. 2-3 Eddy current damper to suppress vibration [19]

2.2 Properties of Permanent Magnet

Since we use the electromagnetic actuators, the magnet characteristics cannot be neglected. In this section, we will review briefly some basic properties of permanent magnet (PM), and then present the detailed data of the used magnets, such as maximum energy product, coercive force, and temperature coefficient, etc.

According to Gauss Law, the magnetic flux continuity law can be described as follows:

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0

0H M

μ

∇ ⋅ G = −∇ ⋅ G

μ (2.6)

When a magnetic field HG

is applied to a ferromagnetic substance, the material will be magnetized with the internal flux density BG

given by

MG =NmG

BG =μ0 HG

(2.7) (2.8)

(

⁺^M^G

)

where MG

is the induced magnetization density, defined as the magnetic dipole moment per unit volume, and N is the number of dipoles per unit volume. By (2.7) and (2.8), we can obtain the B-H curve by varying the field HG

and measuring the flux densityBG

. Figure 2-4 is a typical B-H curve of a ferromagnetic material.

P

O Hc

B

H B_r

Fig. 2-4 B-H curve of a typical ferromagnetic material

From Fig. 2-4, the curve OP is the initial magnetizing curve, and point P is the saturation point that means the material reaches its maximum magnetization. Once we vary HG

from positive value to negative value and then back to positive value again,

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the B-H curve forms a loop called hysteresis. The intersection of the loop and the BG axis is known as the remanence, residual magnetization, or residual flux density, denoted as BG^r

, which is the magnetic flux density inside the magnet when the external field HG

is reduced to zero. Moreover, HGc

is known as the coercivity or coercive force which is the external field needed to completely demagnetize the substance.

By (2.6), when external HG

is removed, the residual flux density inside the ferromagnetic material is

BGr MG

(2.9)

=

MG which indicates that the material has become a PM with residual flux density . Then, the magnetization of this PM can be expressed as:

MG BGr

(2.10)

=

G =

Therefore, the dipole moment resulting from the definition of magnetization is then given as:

rV m BG

(2.11)

where V is the volume of this PM.

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r

In the following, comparisons among different magnets with their specific properties shown in Table 2-1 will be given. Among the listed magnet materials, Ferrite, also known as ceramic magnet, provides the lowest maximum energy product

and the lowest residual induction

BHmax

BG

. Ceramic materials are hard and brittle and are extensively used in consumer products, and on the back of popular refrigerator magnets.

Rare-earth elements are the most popular materials used to produce the strong magnets. One of the strong magnets is Samarium Cobalt, which has highBG^r

, highHGc

, relatively high maximum energy product ( ), and also higher cost than NdFeB.

Commonly, its energy product ranges from 18 BHmax

MGOe (Mega Gauss Oersteds) to about 32 MGOe . The most familiar one of the strong magnets is NdFeB or, for more accurately, sintered NdFeB magnet, whose property is similar to that of SmCo but which belongs to the most powerful class and is commercially available today. Its energy product ranges from 2.8 MGOe to about 48 MGOe . Therefore, NdFeB magnet is the most reliable choice to provide high magnetic force in our system.

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Table 2-1 Characteristics of Magnet

Property Unit

AlNiCo Ferrite Rare Earth

AlNiCo Magnet

Sintered Ferrite Magnet

Bonded Ferrite Magnet

Sintered SmCo Magnet

Bonded SmCo Magnet

Sintered NdFeB Magnet

Bonded NdFeB Magnet Residual

Induction (Br) kG 11.5 4.4 3.1 11.6 8.5 14.2 7.3

Coercive

Force (bHc) kOe 1.6 2.8 2.4 10.1 7.6 11.7 5.7

Intrinsic Coercivity (iHc)

kOe

Maximum Energy Product (BH)max

MGOe 11 4.6 2.2 32 17 48 11

Temperature Coefficient α(Br)

%/K -0.02 -0.18 -0.0.3 -0.03 -0.11 -0.10

Temperature Coefficient β (iHc)

%/K ~0 +0.4 -0.2 -0.2 -0.6 -0.4

Curie Temperature Tc

℃ 845 460 795 795 335 335

Flexure

Strength kgf/mm² 28 13 12 25

Density ρ g/cm³ 7.3 5.0 8.4 7.0 7.5 6.0

Hardness Hv 650 530 550 80-120 600 80-120

Electrical Resistivity

μΩ‧

cm 60 >1010 80 44000 150 26000

(Data from Spin Technology Corp. in Taiwan.)

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We can conclude that NdFeB material is the best choice at present, since AlNiCo has low coercivity, Ferrite has low remanence, and Samarium Cobalt magnets are still expensive. Table 2-2 indicates several characteristics of the NdFeB magnets.

Table 2-2 Specifications of NdFeB

Specifications NdFeB

Remanence (T) 1.29

Coercivity (kA/m) 990

Maximum energy product (kJ/m³) ³²⁰

Density (g/cm³) 7.49

Curie temperature (⁰C) ³¹⁰

Resistivity (μΩm) 6

2.3 Basic Theories of Energy Methods

In this section, we will introduce how to apply energy method to solve problems involving deflection. Then, Castigliano’s theorem is an important theory to be referred to, which is used here to determine the stiffness of the flexure mechanism. For more details, readers are suggested to refer to the work [17].

2.3.1 External work and strain energy

First, we will define the work caused by an external force and a couple moment.

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Work of a force

In mechanics, a force does work when it undergoes a displacement dx that is in the direction the same as that of the force. The work done is a scalar, defined as . If the total displacement is x, the work becomes

dUe =Fdx

x

0 x

Ue =

∫

Fd ^(2.12)

Work of a couple moment

A couple moment M does work when it undergoes a rotational displacement dθ along its direction of action. The work done is defined as dU_e=Mdθ . If the total angle of rotational displacement is θ rad., the work becomes

0

Ue M d

θ θ

=

∫

^(2.13)

When loads are applied to a body, they will deform the material. Provided no energy is lost in the form of heat, the external work done by the loads will be converted into internal work called strain energy. This energy, which is always positive, is stored in the body and is caused by the action of either normal or shear stress.

Normal Stress

In general, if the body is subjected only to a uni-axial normal stress σ, acting in a

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specified direction, the strain energy in the body is

i 2

V

U ₌ σε dV

∫

^(2.14)

Also, if the material behaves in a linear-elastic manner, Hooke’s law suggests σ =Eε , whereby we can express the strain energy in terms of the normal stress as

2 i 2

V

U E

=

∫

σ ^dV ^(2.15)

where E is the Young’s module.

2.3.2 Strain energy for bending moment

Since a bending moment applied to a straight prismatic member develops normal stress in the member, we can use (2.15) to determine the strain energy stored in the member due to bending. Considering a bending applied to the axis-symmetric beam as shown in Fig. 2-5, the internal moment here is M, and hence the normal stress acting on the arbitrary element at a distance y from the neutral axis is σ =^{My I}^/ . If the volume of the element is , where dA is the area of its exposed face and dx is its length, the elastic strain energy in the beam is

dV =dAdx

2

1 2

( )

2 2

i V V

U dV My d

E E I

=

∫

σ =

∫

Adx (2.16)

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The integral over the volume can be expressed as the product of an integral over the beam’s cross-sectional area A and an integral over its length L. Thus,

2

2 0 2 2

L

i A

U M dx y

EI

⎡ ⎤ ⎡

= ⎢⎣

∫

⎥ ⎣⎦

∫

dA⎤⎦ (2.17)

Realizing that the area integral represents the moment of inertia I of the beam about the neutral axis, the final result can be re-expressed as:

2

0 2

L i

U M

=

∫

EI dx (2.18)

Fig. 2-5 Axis-symmetric beam

2.3.3 Castigiano’s theorem

The internal strain energy for a beam is caused by both bending and shear.

However, if the beam is long and slender, the strain energy due to shear can be neglected compared with that of bending. Assuming this to be the case, the internal strain energy for a beam is given by (2.18). Substituting it into δ = ∂U/∂ , we can get: P

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2

0 0 ( )

2

L M L M dx

dx M

P EI P EI

δ = ^∂ =

∂

∫ ∫

^∂∂ ^(2.19)

Where the variables used are defined below:

δ = displacement of the point caused by the real loads acting on the beam,

P = external force of variable magnitude applied to the beam in the direction of δ, M = internal moment in the beam, expressed as a function of x and is caused by both the force P and the loads on the beam,

E = modulus of elasticity of the material,

I = moment of inertia of cross-sectional area computed about the neutral axis.

If the slope at a point on the elastic curve is to be determined the partial derivative of the internal moment M with respect to the external couple moment M’ acting at the point must be found. For this case,

0^L ( M dx)

M M EI

θ= ^∂

∂ ′

∫

^(2.20)

2.4 Flexure Mechanism

The most important advantage of flexure mechanism is frictionless and stictionless which relies on the elastic deformation of material. Sliding and rolling effects are completely eliminated in the devices using flexure mechanism. Flexures have been used

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(1) as bearings to provide smooth and guided motion, when in precision motion stages;

(2) as springs to provide preload, when in the brushes of a DC motor or a camera lens cap; (3) to avoid over-constraint, as in the case of bellows or helical coupling; (4) as clamping devices, for example, the collets of a lathe; (5) for elastic averaging as in a windshield wiper; and (6) for energy storage such as, in a bow or a catapult. Above all, it encompasses applications with regard to the transmission of force, displacement as well as energy.

In our work, flexure mechanism is used as the suspension of precision positioning stages. The motion is generated due to molecular level deformation, which results in two primary characteristics of flexures – smooth motion and small range of motion, and the phenomena of friction, stiction and backlash are completely eliminated. On the other hand, flexure mechanisms allow for very clean and precise motion.

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FGx

FGy

FGx

Fig. 2-6 Common flexure types: (a) simple cantilever, (b) clamped-clamped, (c) crab-leg, (d) folded-flexure, (e) serpentine

There are several kinds of flexure type such as simple cantilever beam, clamped-clamped flexure, crab-leg flexure, folded flexure, and serpentine flexure as shown in Fig. 2-6. These kinds of flexures generate one DOF motion along the direction of force. In the following, we will give some analyses of characteristics of these flexure mechanisms.

Consider the simple cantilever beam in Fig. 2-7 with a rectangular cross section, and let it be subjected to a load P at its end. Now, we want to determine the displacement of the load.

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Fig. 2-7 Cantilevered beam and its free body diagram

It is assumed that we have known the material characteristics, and EI is constant.

Here, E means the Young’s modulus, and I means the inertia mass. Then, clearly, a small deflection on the elastic beam will result in

2 2

( ) d y M x

dx = EI (2.21)

According to free body diagram, we can get

0 Force= ⇒ V = −P

∑

0

(2.22) Moment= ⇒ M =

∑

⁻^Px^(2.23)

Using (2.21) ~ (2.23), and applying the boundary conditions, we can obtain the deflection curve as

3 2

( 3 2

6

y P x L x

= EI − + − L (2.24) ³)

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so that the tip deflection can be easily derived as ₀ ³

x 3

y P L

EI k

=

− F

= − . As a result, the

spring constant k can be obtained as

3

k 3EI

= L (2.25)

After discussing the simplest cantilevered beam, now we will discuss the quad-symmetric clamped-clamped flexure mechanism. Due to its symmetric structure, this mechanism only generates one DOF motion. As shown in Fig. 2-8, we can model the structure as four guided-end beams.

Fig. 2-8 Quad-symmetric clamped-clamped flexure and its free body diagram

According to the free body diagram, an external force Fx and a bending moment M0 can be applied to a body to find the displacement δx and the corresponding spring constant kx. Therefore, the bending moment of the beam is found to be

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0 x

M =M −Fξ (2.26)

By (2.18), the associated strain energy of the beam is

2

0 2

L M

U =

∫

EI dξ^(2.27)

Now, rewrite the Castigliano’s Theorem, mentioned in (2.19) and (2.20), as follows:

2 0 0

0

( ) , , 2

LM x U

U dx

EI δ ^∂F_δ θ M

= =

∂ ∂

∫

⁼ ^∂^U

0

(2.28)

which together with the constraint θ = and (2.26), will lead to the following 0 relation:

0 0

1 ( ) 0

L

x

U M M

M EI M d M F d EI

θ ξ

ξ ξ

∂ ∂

= =

∂ ∂

= −

∫

⁼

(2.29)

Now, substituting ₀ 2 F Lx

M = into the (2.26), we can then obtain

(2

x

M =F L− (2.30) ξ)

Again, through use of Castigliano’s Theorem, (2.28), and its substitution into (2.30), we finally derive the displacement as:

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0

3 2

0

( )

2 ( )

2

( )

2 1

x x L

x

L x

x L x

U F

M M d EI F

F L L

EI d

F L F

EI d EI

δ

ξ ξ

2 L ξ ξ

ξ ξ

= ∂

∂

= ∂

∂

= − −

= − =

∫

(2.31)

As a consequence, the spring constant of the beam and of the quad-symmetric clamped-clamped flexure can readily be obtained as:

, 3

x 12

x beam x

F EI

k ⁼δ ⁼ L

, 3

4 48

x x beam

k k EI

= ⋅ = L (2.32)

Besides the above two kinds, we will analyze another type of flexure, called quad-symmetric crab-leg flexure mechanism, as shown in Fig. 2-9. The same as the clamped-clamped flexure, this mechanism also only generates one DOF motion. It can be modeled as four crab-leg flexure, and we divide every crab-leg into two parts, thigh (beam a) and shin (beam b).

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Fig. 2-9 Quad-symmetric crab-leg flexure and its free body diagram

For the purpose of analysis, likewise we apply Fx , Fy , and M0 at the end of the thigh. According to the free body diagram, the bending moments of the thigh (Ma) and shin (Mb) can be respectively expressed as:

0

a y

M =M −F

b b

x (2.33)

0 _x ( 0 _{y a}) _x

M =M −Fξ = M −F L −Fξ (2.34)

We now use Castigliano’s Theorem, (2.28), and apply the boundary conditions (θ₀ = ,0

y 0

δ = ) to find M0 and Fy as:

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0

0 0

2 2

0

1 ( ) ( )

1 ( ) 0

2 2

a b

L L

a a b b

L L

y y a

y a x b

a b y a b

U M

M M M M

dx d

EI M EI M

M F x dx M F L Fx d EI

F L F L

M L L F L L

EI θ

ξ

ξ ξ

= ∂

∂

∂ ∂

= +

∂ ∂

⎡ ⎤

= ⎢⎣ − + − − ⎥⎦

⎡ ⎤

= ⎢ + − − − ⎥

⎢ ⎥

⎣ ⎦

∫ ∫

=

(2.35)

so that

2 2

0

2

2( )

b x a b y a y

a b

L F L L F L F

M L L

+ +

= + (2.36)

and

0 0

2 3 2

0 2

0

1 ( )( ) ( )( )

1 0

2 3 2

a b

y y

L L

a a b b

y y

L L

y y a x

a y a x

a b y a b

U F

M M M M

dx d

EI F EI F

a

a b

M F x x dx M F L F L d

EI

M L F L F L L

M L L F L L EI

δ

ξ

ξ ξ

= ∂

∂

∂ ∂

= +

∂ ∂

⎡ ⎤

= ⎢⎣ − − + − − − ⎥⎦

⎡ ⎤

= ⎢− + − + − ⎥=

⎢ ⎥

⎣ ⎦

∫ ∫

(2.37)

so that

3 2

( 4

b x

y

a a b

F L F

L L L

= + ) (2.38)

Note that M0 and Fy are both functions of Fx .

Now, we use Castigliano’s Theorem again to derive δ as:

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0 0

1 ( )( )

a b

b

x x

L a a L b b

x x

L

y a x

U F

M M M M

dx d

EI F EI F

M F L F d

EI δ

ξ ξ ξ ξ

= ∂

∂

∂ ∂

= +

∂ ∂

⎡ ⎤

= ⎢⎣ − − − ⎥⎦

∫ ∫

∫

2 2

1 0

2 2

b

y a b x b

M L F

F L L EI

⎡ ⎤

= ⎢− − +

⎣ ⎦

L ⎥ (2.39)

After substituting (2.36) and (2.38) into (2.39), we readily have

2 2

0

2 2 2 2 2

3

1

2 2

2 3

1

2( ) ( 4 ) 2

( )

,

3 ( 4 )

b x b

x y a b

b x a b y a y a b x x b

a b

a b a a b

b a b x

a b

M L F L

F L L EI

L F L L F L F L L F F L

EI L L L L L L L

L L L F EI L L

δ = ^⎡⎢− − + ^⎤⎥

⎣ ⎦

⎡ + + ⎤

= ⎢− − + ⎥

+ +

⎢ ⎥

⎣ ⎦

= +

+

(2.40)

from which the spring constant of the quad-symmetric crab-leg flexure can be readily derived as:

3

3 ( 4 )

( )

x a b

x

x b a b

F EI L L

k δ L L L

= = +

+ (2.41)

2.5 Measurement Error

There is no perfect measurement system in the real world. The error comes from imperfect sensor assembly and alignment, measurement methodology, and signal noise.

This section would discuss two common errors, namely, Abbe error and cosine error.

The compensation of the two errors will be described in the following section so that an

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accurate and reliable measurement methodology can be properly derived.

2.5.1 Abbe principle and Abbe error

When the axis to be measured and the axis of measurement are not coaxial, a measurement error will occur due to the offset between two axes. In late 1800s, Dr.

Ernest Abbe investigated the issue and proposed this principle: the measuring system should be placed coaxially with the axis which is to be measured. Figure 2-10 shows the Abbe error, which can be estimated as:

Abbe tan

e =D θ (2.42)

where D is the offset between the two axes.

Fig. 2-10 Abbe error

2.5.2 Cosine error

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Cosine error occurs when the axis of measurement and axis which is to measured are not completely parallel. In Fig. 2-11, the included angel θ causes that the measured displacement x’ is different from the real displacement x. The cosine error hence results from inadequate alignment between the motion stage and the sensor.

Fig. 2-11 Cosine error

From the above figure, the relationship between the measured displacement x’ and the real displacement x can be expressed as:

' 1

x sin x

= θ (2.43)

Then, the cosine error can be estimated as:

cos

(

' 1

e x x sin x x x θ

= − = θ − = ^sec −^{1 (2.44)}

)

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Chapter 3 Mechatronic Design

Equation Chapter (Next) Section 1

The aiming target of our research is a positioning stage with high positioning accuracy, large moving range with multiple DOFs. To realize these properties, we adopt the planar parallel flexure mechanism as the suspension of the moving stage due to its frictionless effect, electromagnetic actuator for its low cost, and appropriate arrangement of measurement system in order to precisely measure the 3 DOF displacements of the three degrees of freedom of the designed system.

The related researches and needed background knowledge have been reviewed and introduced in the previous chapter. In this chapter, the design concept of the proposed positioning stage will be introduced, including the flexure suspension mechanism, electromagnetic actuator and damper, measuring system, and the integration of all the components.

3.1 Design Strategies

Now, we list all the design objectives that we want to accomplish as follows:

1. high positioning accuracy,

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2. long planar stroke, 3. fast positioning, 4. compact system.

The following subsections will translate these goals into actuator level requirements. Since most objectives are strongly coupled, we are not able to consider respective design separately.

3.1.1 High positioning accuracy

To attain high positioning accuracy, either the system needs a high disturbance rejection, or the external noise sources need to be shielded off. Moreover, the bits-resolution of AD/DA cards and the resolution of sensor are also some major factors to be concerned. Therefore, instead of investing on installation of expensive equipments, we set our design goal on how to utilize commonly available sensors and AD/DA cards on how to optimize the integrated performance up to respective performance limits of individual components.

To reject the large disturbance and obtain a high bandwidth, performance of the actively controlled system is usually limited by the controllability, linearity, and response time of the actuator and controller.

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3.1.2 Long planar stroke

A larger positioning range within the same outer dimensions is advantageous.

Normally, to reduce costs of production, parallel or batch processing is often applied to increase throughput of samples or products. Consequently, those larger specimens require larger strokes without the decreasing in speed and acceleration.

In the target application, the traveling range in using piezoelectric actuators is just 100 μm [13] due to the constraint of piezoelectric material’s characteristics. In our system, because of fewer physical constraints, we choose electromagnetic actuator to enlarge the traveling range up to several mm.

3.1.3 Fast positioning

For the same reason mentioned in previous section, to increase the throughput of production, motion of any positioning system should be fast as possible so that it can become commercially attractive. For the transient response specification, the rise time and the settling time must be short. To fulfill such objective, we must have both a strong and high speed actuator and a well-designed mechanism which have a higher bandwidth.

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3.1.4 Compact system

The positioning system is usually a part of a larger system, which leads to restrictions on the space available for the motor and bearings. These restrictions are even more severe under the circumstance of an expensive ultra clean or vacuum environment. Generally, the outer diameter of a positioning system will be less when there are more moving parts. The implication of the above is that the total positioning should be as compact as possible.

3.2 Electromagnetic Actuation and Damper

This section discusses the design of electromagnetic actuator and damper which will be utilized in this proposed positioning system. Generally, using a rotary motor to generate the linear reciprocating motion must incorporate the use of the transmission mechanism, such as ball-screws, gears, racks, etc., to change the rotating motion into the linear motion. However, the transmission mechanisms usually have problems of backlash and friction which severely influence the precision of positioning. Therefore, it is better to use electromagnetic actuator, so-called voice coil motor (VCM), due to its frictionless contact and non-backlash nature.

VCM is one kind of linear direct-current motor and is composed of permanent

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magnets (PMs) and coils which are assembled appropriately. It can produce the linear thrust proportional to the current which flows through the magnetic field.

Fig. 3-1 The VCM actuator of our previous research [8]

Figure 3-1 shows the VCM actuator which is used in our previous research [8].

Two magnets are placed side by side with opposite polarity. The current-carrying coil would generate the force as shown in the above figure. But the assumption on the magnetic field produced by the magnets is not realistic. The real distribution of the magnetic field is highly non-uniform, so that the actuator suffers from the nonlinearity and the current/force relationship of the actuator will vary with the relative position between the magnets and the coils. This nonlinearity also limits the stroke of the actuator. In this section, we will discuss the design of a stronger linear electromagnetic actuator and the built-in electromagnetic damper.

Coils Magnets

S

magnet

BK

magnet

BK

F

N S

N

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3.2.1 Near-Uniform Magnetic Field

Since the nonlinearity of the electromagnetic actuator comes from the non-uniform magnetic field, intuitively a construction of uniform magnetic field will solve the issue.

Figures 3-2 and 3-3 show the concept of constructing a near-uniform field. Two plate permanent magnets are installed inside a steel structure. A near-uniform magnetic field will be generated between the two plate permanent magnets and the outer magnetic flux will be confined within the ferromagnetic steel structure to form a closed loop flux path, as shown in Fig. 3-3. The dimension of the plate magnet is 60mm x 60mm x 2mm and its material is neodymium-iron-boron (NdFeB), the powerful rare earth permanent magnet material.

Fig. 3-2 The steel structure and permanent magnets

Fig. 3-3 The near-uniform magnetic field

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To verify the near-uniform property of the constructed magnetic field, a finite element analysis is conducted as shown in Fig. 3-4. The outcome of such analysis does validate our viewpoint and the design.

Fig. 3-4 The numerical simulation of magnetic field

3.2.2 The proposed electromagnetic actuator

Figures 3-5 and 3-6 show the configuration and assembly of the proposed electromagnetic actuators. A copper mounting is manufactured with the shape shown in Fig. 3-5 and is placed in the middle between the upper part and the lower part of the steel structure. Note that the two parts of the steel structure are attracted to each other by magnetic force and are connected through four rectangular holes opened on the copper mounting. The clearance between the steel structure and copper mounting is designed for the motion of the moving stage and is sufficient for the needed traveling range. Four square coils are placed on the copper mounting. When current is fed into the coils, the

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interaction between the current and the intersected near-uniform magnetic field would generate the electromagnetic force according to Lorentz force principle.

Fig. 3-5 The assembly of the VCM actuator

Fig. 3-6 The forces that are generated by Lorentz force principle

It is noteworthy that only half of the coil intersects the magnetic field, as shown in Fig. 3-6. The part of the coil intersecting the magnetic field can actually be divided into three sub-parts, which result in three corresponding electromagnetic forces, F1, F2

and F3. Due to symmetry, the magnitudes of F2 and F3 are equal but the directions are

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opposite. Since the two forces cancel each other, the effective force generated by the coil is only F1. It is important to see that the magnitude of the effective force of a coil is invariant to the translation of the steel structure because the intersection part of the coil corresponding to F1 is always constant and the varying forces F2 and F3 are always equal but opposite. When the stage rotates, the magnitudes of F2 and F3 are different.

The purpose of the positioning stage is the precision motion in x- and y-axis, and the rotation angle of the stage is always regulated to zero. The difference of the magnitudes of F2 and F3 can be neglected when the rotation angle is small.

3.2.3 Eddy current damper

One should also beware the magnetic field intersects the copper mounting as well.

When the steel structure and the magnetic field move relative to the copper mounting, the eddy current will be generated inside the copper structure. Hence, the interaction results in a damping force which resists the mentioned relative motion.

Fig. 3-7 The eddy current inside the copper mounting.

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3.3 3-DOF Flexure Mechanism

Numerous multi-DOF flexure mechanisms have been presented in the literature.

There are two well-known configurations adopted in the design of multi- DOF flexure stages – serial kinematics [20]-[25] and parallel kinematics [26]-[28]. Each configuration has its advantages and disadvantages.

In serial design, multiple DOFs are achieved by stacking multiple single DOF systems, one on another. The technical literature has presented several such designs.

Serial kinematical mechanisms are relatively simple to design and have significantly higher inertia, but their weak points are the resulting center of gravity is relatively higher, and the off-axis errors are harder to be corrected. Besides those, the cables which are connected to every stage are sources of disturbance, which is detrimental for nanoscale positioning. Moreover, the actuators, especially when large range of motion is desired, are bulky and may reduce the motion bandwidth of the axes of DOF.

However, parallel kinematical designs are free of these problems due to ground mounted actuators, and are also usually more compact. On the other hand, they provide smaller ranges of motion and exhibit significant cross-axis coupling. Furthermore, the stiffness of one axis varies with motion or force along the other axis. This affects the

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static as well as dynamic performance of the mechanism.

In our work, we use the parallel kinematical XY flexure mechanisms as shown in Fig. 3-8 and Fig. 3-9. The thin flexure mechanism is fabricated by electrical discharging wire cutting (EDWC), where the resulting height is 5mm and the width is only 0.3mm.

Fig. 3-8 Parallel kinematical XY flexure mechanism

Fig. 3-9 The detailed view of the thin flexure

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Note that the stiffness of the structure should be proportional to the area moment of inertia I. Recalling the cantilever as shown in Fig. 2-7, we can calculate the area moment of inertia I as:

3

12

I =bh (3.1)

Since the height is greater than the width, the stiffness of our designed structure in vertical direction is much greater than that in the lateral direction. Therefore, we assume that the structure is always rigid in the vertical direction even when it has deformations in the lateral direction.

Figure 3-8 shows that the flexure mechanism is made of two kinds of material: the brown outer frame means the frame is made of copper and the gray blue inner flexure means the material is steel, which is used to construct the near-uniform magnetic field.

The reason why the outer frame cannot be made of the same material as the inner flexure is that if the material of outer frame is also ferromagnetic, the moving stage will be attracted and stuck on the outer frame when it moves too close to the outer frame, as shown in Fig. 3-10. Thus, the outer frame should use non-ferromagnetic material. In our design, we choose copper as the material of the outer frame, and then the inner flexure is inlaid into the outer frame.

新型電磁致動與減震之三自由度撓摺結構精密定位平台

國立臺灣大學電機資訊學院電機工程學研究所 碩士論文

National Taiwan University Master Thesis

新型電磁致動與減震之三自由度撓摺結構精密定位平台 A Novel Electromagnetic Actuated and Damped 3-DOF Precision

Positioning Stage with Flexure Suspension

林志憲 Chih-Hsien Lin

指導教授：傅立成 博士 共同指導：陳美勇 博士 Advisor: Li-Chen Fu, Ph.D.

Mei-Yung Chen, Ph.D.

中華民國九十七年六月

June, 2008

致 謝

摘要

Abstract

Table of Contents

List of Figures

List of Tables

Chapter 1 Introduction

1.1 Motivation and goal

1.2 Literature Survey

1.3 Contribution

1.4 Thesis Organization

Chapter 2 Preliminary

2.1 Basic Theories of Electromagnetic

∫v

2.2 Properties of Permanent Magnet

(

)

2.3 Basic Theories of Energy Methods

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫ ∫

∫

2.4 Flexure Mechanism

∑

∑

∫

∫

∫

∫

∫

∫

∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫

2.5 Measurement Error

(

)

Chapter 3

Mechatronic Design

3.1 Design Strategies

3.2 Electromagnetic Actuation and Damper

3.3 3-DOF Flexure Mechanism

國立臺灣大學電機資訊學院電機工程學研究所碩士論文

指導教授：傅立成博士共同指導：陳美勇博士 Advisor: Li-Chen Fu, Ph.D.

致謝