compliant manipulators - nanyang technological university€¦ · actuators and sensors which the...

Compliant Manipulators

Tat Joo Teoa*, Guilin Yangb and I-Ming ChencaMechatronics Group, Singapore Institute of Manufacturing Technology, Singapore, SingaporebInstitute of Advanced Manufacturing, Ningbo Institute of Materials Technology and Engineering of the ChineseAcademy of Sciences, Zhenhai District, Ningbo, Zhejiang, People’s Republic of ChinacSchool of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, Singapore

Abstract

Compliant manipulators are advanced robotic systems articulated by the flexure joints to deliverhighly repeatable motion. Using the advantage of elastic deflection, these flexure joints overcomethe limitations of conventional bearing-based joints such as dry friction, backlash, and wear and tear.Together with high-resolution positioning actuators and encoders, the compliant manipulators aresuitable ideal candidates for micro-/nanoscale positioning tasks. This chapter presents the relevantknowledge of several fundamental topics associated with this advanced technology. After reviewingits evolution and applications, the principal of mechanics is used to explain the limitations of thesemanipulators. Subsequent topic covers various theoretical modeling approaches that are generallyused to predict the deflection stiffness of flexure joints and stiffness characteristics of compliantmanipulators. Next, various fundamental design concepts for synthesizing the compliant mecha-nism will be introduced and several examples are used to demonstrate the effectiveness of theseconcepts. The topic on actuation, sensing, and control summarizes the types of high-resolutionactuators and sensors which the compliant manipulators use to achieve high-precision positioningperformance. Performance trade-offs between various actuators and among different sensors arediscussed in detail. With this relevant knowledge, this chapter serves as a guide and reference fordesigning, analyzing, and developing a compliant manipulator.

Introduction

A compliant manipulator is a motion system that consists of a compliant mechanism driven by high-resolution positioning actuators. Unlike traditional rigid-link mechanisms, a compliant mechanismgains its mobility from the deflection of flexible members (Howell 2001), which is termed as theflexure joints. Using the advantages of elastic deflection, a flexure joint overcomes the limitations ofa conventional bearing-based joint such as dry friction, backlash, and wear and tear (Smith 2000).Consequently, a compliant mechanism offers high repeatable motion due to the frictionless charac-teristics of these flexure joints. Driven by high-resolution positioning actuators such as the piezo-electric (PZT) solid-state actuators and the electromagnetic (EM) actuators, the compliantmechanisms become high-precision manipulators that are ideal candidates for micro-/nanoscalepositioning tasks (Teo et al. 2010a; Yang et al. 2008; Ryu et al. 1997; Lee and Kim 1997; Jyweet al. 2008). Figure 1a shows an example of a compliant manipulator developed by the Singapore

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Handbook of Manufacturing Engineering and TechnologyDOI 10.1007/978-1-4471-4976-7_102-1# Springer-Verlag London 2014

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Institute of Manufacturing Technology (SIMTech) to automate the out-of-plane alignment andimprinting tasks for a Nanoimprint Lithography (NIL) process. This three degree-of-freedom(DOF) compliant manipulator is driven by Lorentz-force EM actuators and has achieveda positioning and angular resolution of 10 nm and 0.05 arcsec (0.242 106 rad), respec-tively, throughout a workspace of 5 5 5 mm.

Compliant manipulator can be classified as a partially or fully compliant manipulator. A partiallycompliant manipulator has limbs that consist of both rigid bodies and flexure joints, while a fullycompliant manipulator has continuous flexible limbs. For example, the manipulator shown in Fig. 1ais considered as a partially compliant manipulator since each limb is articulated by a combination ofrigid bodies and flexure joints. On the other hand, the mechanism shown in Fig. 1b is considered asa fully compliant manipulator because each limb is formed by a continuous flexure joint without theneed of rigid bodies. Other than gaining performance, these compliant manipulators also benefitfrom the simple constructions of the flexure joints that lead to reduction of parts and assembly time.Taking the spatial joint compliant module shown in Fig. 2a as an example, it is a single monolithic-cut joint that has significantly less parts as compared to a bearing-based spherical joint as shown inFig. 2b. Having less parts may shorten the manufacturing and assembly time and eventually reduce

Fig. 1 (a) A 3-DOF out-of-plane motion compliant manipulator and (b) a 3-DOF in-plane motion compliant mecha-nism developed by SIMTech

Fig. 2 (a) A spatial joint compliant module developed by SIMTech and (b) a bearing-based spherical joint fromHEPHAIST SEIKO (S. HEPHAIST 2014)


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cost. In addition, parts reduction may also reduce assembly errors, which often affect the accuracyand motion repeatability of a manipulator.

Among various motion systems that utilize noncontact bearings such as the compliant, air,magnetic, fluid, and ultrasonic bearings, the compliant manipulators are cost-effective andmaintenance-free because the flexure joints do not require any electrical/fluid/air source, actuation,sensor, and complex control system to function as a noncontact bearing. Being maintenance-free isa significant merit because the compliant manipulators could operate in harsh environments that maydamage or degrade the joints. For example, flexure joints made of Teflon could be used in chemicalsolutions and even space exploration systems since no lubrication is required. The frictionlesscharacteristics of a compliant manipulator also suit the clean vacuum environment since no particleswill be generated through friction. Most importantly, reduction in part counts coupled with thesimple constructions of flexure joints becomes an attractive solution for developing microscalecompliant manipulators such as Micro-Electro-Mechanical Systems (MEMS) and macro-/microscale nanopositioner (Chen and Culpepper 2006). Figure 1b shows a 3-DOF macroscalefully compliant mechanism that benefitted from the simple construction of the flexure joints,which allow the mechanism to be fabricated using polymer material for harsh environment usage.

Although a compliant manipulator has an abundance of benefits, the usage of elastic deflectionfrom the flexure joints is also accompanied by limitations. As all flexure joints are required to deflector bend within the elastic region of the materials, the deflection of these flexible members limits themotion of the compliant manipulator. For example, a flexure joint cannot produce the continuousrotation motion of a ball-bearing rotary joint. In general, the workspace of a typical compliantmanipulator is limited to a few millimeters and degrees as the stress concentration of each flexurejoint must not exceed the yield strength of the material. Other than stress–strain characteristics of thematerials, the force-displacement (or stiffness) characteristics of a flexure joint are also crucial to thedevelopment of a compliant manipulator. Accurate prediction of the stiffness characteristics requiresin-depth knowledge of the principal of mechanics, deflection theory, mechanism synthesis, andsynthesis methods. Yet, from the recent advancements in compliant manipulator, flexure joints aretasked to produce larger deflections, which exhibit nonlinear behavior. Hence, classical beamequations derived from the small deflection theory are no longer valid. Even with sufficientknowledge, manufacturing tolerances can easily result in uncertainties to the actual performanceof the compliant manipulator.

Dealing with flexure-based (or compliant) bearing involves the transfer or transformation ofstored energy from input to output. To overcome the stored energy, the energy used to createa displacement tends to be higher as compared to other noncontact bearings. Under significantstress, prolonged stress, or constant exposure to high temperature, the stored energy will also causea certain degree of hysteresis in the stress–strain characteristics resulting in creep effects. Althoughstored energy can be reduced by lowering the displacement stiffness, this approach further reducesthe off-axis stiffness of the compliant manipulator, which is relatively lower as compared to motionsystems using other noncontact bearings. Thus, the compliant manipulators are not suitable forcarrying high payloads and any accidental overload may lead to instabilities, e.g., buckling. Fatigueis another crucial factor that determines the reliability of a compliant manipulator since the flexurejoints are deflecting cyclically with constant load. Unfortunately, existing theoretical model, whichpredicts the fatigue life, is only applicable to standard specimen shapes, e.g., rectangular andcircular. In addition, the material properties, geometries/dimensions of the flexure joints, andtypes/amount of loadings have significant effects on the fatigue life. As a result, tedious andmeticulous evaluations are required to determine the life span of any compliant bearing inperforming its prescribed tasks.


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This chapter introduces the essence of compliant manipulators by presenting several fundamentaltopics that are associated with this advanced technology. After the brief history on its evolution andapplications, the first topic covers the principal of mechanics, which is used to explain the limitationsof these manipulators. The next topic introduces the fundamental design concepts that are typicallyused to synthesize the compliant mechanism. Subsequent topic covers various theoretical modelingapproaches that are generally used to predict the deflection stiffness of flexure joints and the stiffnesscharacteristics of the compliant manipulators. Following these fundamental topics, the materialproperties, the types of fabrications, and the manufacturing limitations are discussed. The topic onactuation, sensing, and control summarizes the types of high-resolution actuators and sensors whichthe compliant manipulators use to achieve high-precision positioning performance. Lastly, the futureadvancement of the compliant manipulator technology is presented.

Brief History

Implementation of flexure joints into precision machines can be dated as early as 1826, when metalstrips were first used to replace torsional members of a classical torsion balance to increase itsprecision of measuring fine torque when subjected to mechanical or electrical loads (Jones 1962).Absence of the “sticking” effect made it possible to register very small changes in torque withmeaningful observations of 109 rad change in orientations. A cross-strip hinge, which was laterintroduced to increase the stiffness of non-actuating directions, was well adopted by subsequenttorsion balances. In 1902, such slender strips were used to support the ruling engine for gratingdiffraction (Fig. 3) so as to avoid the effects of friction (Jones 1988). This slender strip, which istermed as a leaf spring, is considered as the earliest form of a flexure joint. By the dawn ofWorldWarII, these shock-proof torsional leaf springs have been increasingly used in electrical instruments toreplace jeweled pivots (Jones 1962). During the war, the flexure joints became widely used indeveloping highly sensitive measurement instruments such as highly accurate load cells for forcemeasurement and the pendulum pivots of miniaturized force-balance accelerometers (Motsinger1964; Tuttle 1967).

After the war, the compliant mechanisms were widely used to develop metrology systems whenprecise positioning of optical lens or mirrors was needed (Jones 1951, 1952, 1955, 1956). Forexample, the optical slit mechanism of an infrared spectrometer shown in Fig. 4a is a compliantmechanism. Using leaf springs to support the slit jaws, the resolution of the micrometer is directlytranslated onto the optical slit mechanism since there is no backlash between the micrometer and the

Fig. 3 H. A. Roland with his ruling engine for diffraction gratings (Jones 1988)


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jaws. Due to their ability to provide direct transmission between the input and the output, leaf springswere gaining popularity in the development of highly sensitive metrology devices. For example,Fig. 4b illustrates a Michelson interferometer mirror positioner that is supported by a pair of leafsprings and Fig. 4c shows a seismograph that used these compliant bearings to achieve directmeasurement of the amplitude of earthquakes without losses through friction. Using leaf springs asflexure joints requires additional assemblies that generally affect the precision of a mechanism. Inaddition, the leaf springs have poor stiffness in other non-actuation directions, which will furtherdeteriorate precision when subjected to off-axis external loading.

To address these issues, the notch hinge was introduced as an alternative solution for creating thecompliant mechanism. A monolithic-cut approach was used to produce such notch hinges and theentire mechanism could be fabricated from a single piece of workpiece without involving anyassembly. Figure 5 illustrates two compound linear spring compliant mechanisms constructed by theleaf springs and notch hinges, respectively. The notch hinges were localized flexure joints thatformed only a small portion of the entire mechanism. As a result, the off-axis (or non-drivingdirections) stiffness of the monolithic compound linear spring compliant mechanism is more

Fig. 4 (a) An optical slit mechanism, (b) a parallel spring mechanism for positioning the Michelson interferometermirror, and (c) a gravimeter-vertical seismograph (Jones 1952)

Fig. 5 A leaf-spring compound linear spring mechanism versus a monolithic compound linear spring mechanism(Jones 1988)


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superior than its leaf spring counterpart. In addition, the notch hinges possess limited deflections,which exhibit linear characteristics. Hence, well-established mechanics theory was used to predictthe deflection stiffness of these notch hinges. Monolithic-cut fabrication also reduced the accumu-lative assembly errors that could potentially cause the motion of the compliant mechanism to beindeterministic. Thus, compliant mechanisms that used notch hinges were more predictable thanthose with leaf springs.

Between the late 1960s and the early 1990s, compliant mechanisms were mainly developed viathe notch hinges and many were employed as subnanometer positioning stages for laboratory usage(Paros and Weisbord 1965; Deslattes 1969; Haberland 1978; Becker et al. 1987; Smith et al. 1987;Nishimura 1991). The architectures of all these compliant stages mainly evolved from a parallellinear spring concept that resembles the classical four-bar linkage mechanism. To double thedisplacement range, a compound linear spring concept, which is formed by connecting a pair oflinear spring in series, was introduced. Examples of the compound linear spring compliant mech-anisms were illustrated in Fig. 5. To further enhance the robustness of these stages towards externaldisturbance, a symmetrical double compound linear spring concept was introduced. Figure 6a showsan example of a single DOF compliant stage that was developed via a double compound linearspring concept and notch hinges. Driven by a PZT actuator, it has a positioning resolution of 50 nmover a traveling range of 100 mm and was used for wafer-bump inspections (Ho et al. 2004).

To obtain a 2-DOF motion, the simplest approach was to stack a 1-DOF compliant stage on top ofanother in series. Another approach was to nest a 1-DOF compliant stage within another stage. Thisapproach was used to develop several multi-DOF nanopositioning stages (Her and Chang 1994; Gaoet al. 1999). Figure 6b shows an example of an X-Y translational optical lens steering stagedeveloped by the US National Institute of Standard and Technology (NIST) for space communica-tion purposes (Boone et al. 2002). This 2-DOF PZT-driven compliant stage was developed based ona nested approach where the y-axis stage is embedded inside the x-axis stage. Precisions of thesestacked and nested compliant stages are usually affected by the accumulative positioning errors. Theresponses of these stages are also slow because the lower stage needs to carry the inertial mass of theupper stage via the stacked approach and the outer stage carries the inertial mass of embedded stagefor the nested approach. Inertial masses of the upper and embedded stages from both approaches alsocaused these 2-DOF compliant stages to have nonsymmetric natural frequencies.

Towards the late 1990s, the parallel-kinematic architectures were widely adopted by moderncompliant manipulators to achieve a higher precision and better performance in multiple DOFmanipulations. This architecture plays an important role in the success of these manipulators due toits advantages of a lower inertia, programmable centers of rotations, superior dynamic behavior, and

Fig. 6 (a) A 1-DOF compliant stage developed by SIMTech (Ho et al. 2004) and (b) a 2-DOF nested compliant stagedeveloped by NIST (Boone et al. 2002)


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insensitivity to external disturbances, e.g., thermal expansion. Most importantly, the limited deflec-tions of flexure joints also suit the limited workspace characteristics of a parallel-kinematic archi-tecture. Figure 7a shows an example of a planar motion compliant manipulator developed based ona parallel-kinematic architecture. Being used as an X-Y-yz wafer positioning stage, this compliantmanipulator delivers a positioning resolution of 8 nm along the x- and y-axes and a rotationalresolution of 0.057 arcsec about the z-axis throughout a workspace of 41.5 mm 47.8 mm 322.8 arcsec (Ryu et al. 1997). Another form of planar motion compliant manipulator was developedbased on a three-legged revolute-revolute-revolute (3RRR) parallel-kinematic architecture(Yi et al. 2003). This compliant manipulator was used for positioning the wafer and delivers anX-Y translational workspace of 100 mm2with a rotational range of 0.1. A similar concept was foundin another X-Y-yz compliant manipulator as shown in Fig. 7b, which was used as a precision stage forpositioning the samples within a Scanning-Electron-Microscope (SEM) machine (Yong and Lu2009).

With the advancement of electrical discharged machine (EDM) in the 1990s, the dimensions offlexure joints can be fabricated more precisely. Hence, the modern flexure joints played an importantrole in realizing various types of spatial motion compliant manipulators (Han et al. 1991; Hudgensand Tesar 1988; Seugling et al. 2002; Portman et al. 2000; Canfield et al. 2002; Wang et al. 2003a;Mclnroy et al. 1999; Mclnroy and Hamann 2000). One example of these manipulators is the Delta3

developed by École Polytechnique Fédérale de Lausanne (EPFL) as shown in Fig. 8a. This X-Y-Zspatial motion compliant manipulator was constructed based on the kinematics of a Delta robotwhere each limb is formed by three rigid links coupled together via the universal flexure joints(Henein 2000). Driven by Lorentz-force EM actuators, this 3-DOF compliant manipulator delivers apositioning repeatability of 100 nm within a workspace of 1 cm3. Figure 8b shows anotherinteresting 3-axes translational motion compliant manipulator that was developed by NanyangTechnological University (NTU). Other than using a three-limbed parallel configuration toconstruct this compliant manipulator, each macroscale compliant limb is articulated by a 3RRRparallel-kinematic architecture. As a result, all three translational axes are kinematically decoupledand hence each PZT actuator will only generate a single-axis motion.

Using six sets of spherical-prismatic-spherical (SPS) serially connected compliant limbs,a compliant Steward platform was developed by Shizuoka University (SU) as shown in Fig. 9a.This 6-DOF compliant manipulator obtained a translational accuracy of 160 nm and rotational

Fig. 7 Planar motion compliant manipulators that were used as (a) a wafer positioning stage (Ryu et al. 1997) and (b)a positioner of SEM (Yong and Lu 2009)


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accuracy of 2 m rad (Oiwa and Hirano 1999). Due to the limited displacement stroke of PZTactuators, it could only achieve 10 mm in all translational directions, 200 mrad about the x- andy-axes, and 100 mrad about the z-axis. Figure 9b shows another example of a 6-DOF compliantmanipulator developed by Massachusetts Institute of Technology (MIT). It was constructed basedon a hybrid concept by stacking an X-Y-yz compliant mechanism on top of a Z-yx-yy compliantmechanism (Zuo et al. 2003). As a result, the planar motion is decoupled with the out-of-planemotion. Driven by three PZT actuators, the planar motion workspace is 140 mm 140 mm 7.6.The out-of-plane workspace was generated by three hybrid PZT-EM stepper actuators and achieveda workspace of 5 mm 2.4 2.4.

Today, the compliant manipulators can be fabricated in the micron level through the semicon-ductor photolithography process. Driven by electrostatic actuation, thermal actuation, and evenmicroscale gears, these compliant manipulators become MEMS devices such as microsensors andmicro-actuators. Figure 10a shows a MEMS-based micro-actuator developed by MEMS andNanotechnology Exchange. The compliant mechanism adopts a double compound linear springconcept with slender hinges with the translator driven by a pair of electrostatic actuators. The latest

Fig. 8 (a) Delta3 compliant manipulator developed by EPFL (Henein 2000) and (b) a 3-axes translational motioncompliant manipulator developed by NTU (Pham and Chen 2005)

Fig. 9 (a) A compliant Steward platform developed by SU (Oiwa and Hirano 1999) and (b) a hybrid 6-DOF compliantmanipulator developed by MIT (Zuo et al. 2003)


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and most noticeable development of microscale compliant manipulator comes from MIT, PrecisionSystems Laboratory. The research group presented a 6-axes microscale nanopositioner, termedmHexFlex, based on a fully compliant design concept as shown in Fig. 10b. The mHexFlex consistsof a central stage that is connected to the base via three parallel flexure joints where each joint isdriven by two microscale thermal mechanical actuators. As each thermal actuator has 2-DOFactuation, i.e., in-plane and out-of-plane deflections, a combination of six actuators delivers a6-DOF motion to the system. mHexFlex registered a positioning error of 10 nm over a workspacevolume of 8.4 12.8 8.8 mm3 and 19.2 17.5 33.2 mrad for the x-y-z axes and the yx-yy-yz,respectively.

Principles of Solid Mechanics

A compliant manipulator is articulated by the flexure joints that are considered as “springs” withhigh stiffness ratios. The basic working principles of these flexure joints are elastic bending andtorsion. The advantages of elastic bending or torsional motion include frictionless, contactless, andnon-hysteresis characteristics. The disadvantages include limited deflection, limited load capacity,and fatigue. To give a better understanding of the limitations of the flexure joints, this sectionreviews the characteristics of the elastic bending and torsional motion based on the principal of solidmechanics.

Strength and StiffnessThe strength and stiffness offer different insights to the deflection of the flexure joint. Strengthdetermines the stress a deflected flexure joint can withstand before failure and is associated with theproperty of the material. On the other hand, stiffness determines how much a flexure joint deflectsdue to a load. Based on the Bernoulli-Euler law, the bending moment is proportional to the beamcurvature

M ¼ EIdyds

(1)

whereM represents the bending moment, dy/ds is the rate of change in the deflection angle along thecurvature of the beam, and EI represents the bending rigidity with E representing the Young’smodulus (modulus of elasticity) of the material and I is the cross-sectional moment of inertia (thesecond moment of area).

Fig. 10 (a) A MEMS-based micro-actuator from MEMS and Nanotechnology Exchange and (b) a 6-axes microscalenanopositioner developed by MIT (Zuo et al. 2003)


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The bending rigidity is a function of its material properties and the geometries due to the presenceof the Young’s modulus and the cross-sectional moment of inertia. Figure 11 illustrates a cantileverthat is made of an isotropic material with equal Young’s modulus and strength in all directions.When subjected to a load along the x-axis,Fx, the bending rigidity of the cantilever beam is governedby EIyy. When subjected to a load along the y-axis, Fy, the bending rigidity will be governed by EIxx.As a result, the bending rigidity along the x-axis is much stiffer as compared to the bending rigidityalong the y-axis. This example explains that even with similar Young’s modulus and strength in bothdirections, the stiffness between both directions may not necessarily be the same. It also highlightsthat geometry has crucial influence on the stiffness characteristics of a flexure joint. In addition,different materials will affect its flexibility due to the variations in the Young’s modulus. Forexample, an aluminum (E ¼ 71 GPa) beam will be approximately 3 times more flexible thana steel (E ¼ 210 GPa) beam, while the flexibility of a Teflon (E ¼ 0.5 GPa) beam will be 142 timeshigher than the aluminum beam. These comparisons also highlight the importance of materialselection in the flexure joint design.

Stress FailureIn general, the deflection of a flexure joint is limited by the bending or torsional stress. As these jointsonly operate within the elastic region, the yield strength of the material becomes the maximumallowable stress and the maximum stress generated via the deflection must be kept below it. Intheory, the maximum stress, smax, due to bending is given as

smax ¼ Mc

l(2)

where c represents the location of the neutral axis from the loading point. By substituting Eq. 2 intoEq. 1 and let c ¼ h/2, the maximum stress due to bending moment about the x-axis is expressed as

smax ¼ EIxxh

2l

dyxds

(3)

By referring to Eqs. 2 and 3, the maximum bending stress is proportional to the maximum bendingmoment and geometry also plays an important role in reducing the stress. In addition, selectingmaterials with low Young’s modulus could also reduce the stress. Yet, materials with low Young’smodulus would have lower yield strength. Based on a stress–strain curve shown in Fig. 12, theYoung’s modulus of a material is calculated based on the linear stress–strain relationship, while theyield strength is beyond the proportional limit. Hence, choosing a material with lower Young’smodulus will effectively lower the maximum allowable stress. Yet in some cases, higher Young’smodulus is desirable for achieving higher stiffness characteristics. Consequently, the major

Fig. 11 A cantilever beam subjected to two independent loads along the x- and y-axes


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challenge in designing a flexure joint is to obtain the desired deflection that could fulfill the functionof the compliant mechanism while maintaining the stresses well below the yield strength ofthe material. Table 1 summarizes this section by listing some materials that are commonly used todevelop the compliant mechanism.

Fatigue FailureAny flexure joint that flexes to deliver motion is subjected to fatigue failure. The fatigue life of anymaterial is usually presented in an S-N diagram (Woehler strength-life diagram) as shown in Fig. 13.From this S-N diagram, the number of cycle can be classified into three regions, i.e., low cycle, highcycle, and infinite life (Howell 2001). For the low cycle category, the fatigue failure usually occursbetween 1 and 1,000 cycles. As for the high cycle category, the fatigue failure typically occursbeyond 1,000 cycles. The infinite life region is for the flexure joints that required to flex constantlyand only applies to some materials that do not fail regardless of the number of cycle.

The ultimate strength of the material is represented by Sut and Sf represents the fatigue strength ofthe material. The first limit that bounds the low cycle region is represented by SL, while the secondlimit that bounds the finite life is known as the endurance limit, Se. This limit is common in manylow-strength carbon and alloy steels, some stainless steels, irons, molybdenum alloys and titaniumalloys, and certain polymers (Dowling 1993). If the stress is kept below the endurance limit,

Fig. 12 Stress–strain curve of a ductile material

Fig. 13 S-N curve of steel material


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continuous cycle without fatigue failure is possible and the flexure joint has infinite life. From pastliterature (Juvinall 1967), SL can be approximated as

SL ¼ cf Sut (4)

where

cf ¼ 0:9 bending0:75 axial loading

(5)

For low cycle fatigue estimation, the maximum stress must not exceed SL. For high cycle fatigueestimation, Sf can be approximated as

Sf ¼ af Nbf (6)

where

af ¼cf Sut 2

Sebf ¼ 1

3log

cf SutSe

(7)

Based on Eq. 6, the number of cycle can be estimated by assuming that smax ¼ Sf. For materialswithout endurance limit, af and bf are expressed as

logaf ¼logN 2ð Þ logcf Sut

3logSf 2logN 2 3

bf ¼ 1

3 logN 2log

cf SutSf 2

(8)

where Sf1 ¼ cfSut, N1 ¼ 1 103, Sf2 ¼ Se, and N2 ¼ 1 106. For materials with endurance limit, Seobtained through these specimen tests are termed as uncorrected endurance limit, Se

0. For materials

without endurance limit, the Sf obtained via the specimen tests are known as uncorrected fatiguestrength, Sf

0. As information is often unavailable, Table 1 listed some initial approximations of Se

0and

Sf0for some materials that are useful for estimating the fatigue life (Norton 2000; Shigley and

Mitchell 1983; Forrest 1962).The initial approximation values can be used to predict Se and Sf (Shigley and Mischke 2001;

Marin 1962) using

Table 1 Initial approximation values of Se0and Sf

0for some materials

Material Classification Values Conditions

Steel Endurance limit Se0 0.5Sut Sut < 1,400 MPa

Se0 700 MPa Sut 1,400 MPa

Iron Endurance limit Se0 0.4Sut Sut < 400 MPa

Se0 160 MPa Sut 400 MPa

Aluminum No endurance limit, fatigue estimation for 5 108 cycles Sf0 0.4Sut Sut < 330 MPa

Sf0 130 MPa Sut 330 MPa


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Se ¼ csur f csizecloadcreliabS 0e (9)

Sf ¼ csur f csizecloadcreliabS0f (10)

where csur f represents the Marin correction factor for surface finishing, csize for size, cload forloading, and creliab for reliability. csur f can be approximated as

csur f ¼ aSbut if aSbut < 11 if aSbut 1

(11)

where the values of a and b are listed in Table 2. csize and creliab are listed in Tables 3 and 4,respectively. From past literature (Norton 2000), cload¼ 1 for bending load, cload¼ 0.7 for axial load,and cload ¼ 0.577 for torsion and shear load.

Theoretical Modeling Approaches

Many theoretical models and modeling methods were introduced over the past 50 years due to thecontinuous evolution of the flexure joints and the applications of the compliant manipulators. Fromthe very beginning, modeling of the flexure joints from classical bending-moment equation wassufficient when the flexure joints were expected to deliver small deflection motions. As the desire forlarge deflection increased, analytical models focusing on nonlinear deflection behavior of the flexurejoints were introduced by the late 1960s. When high-precision compliant manipulators were neededin the semiconductor industry in the 1990s, the evolution of notch-hinge flexure joints spurred theefforts in finding more accurate modeling approaches. These efforts continue till today due to newbeam-based flexure joints that were introduced for delivering large deflection motions. Afterunderstanding the limitations of the flexure joints and the design constraints, this section presentsa comprehensive library of theoretical models andmodeling methods that can be useful for modelingdifferent types of flexure joints.

Table 2 Curve-fitted a and b parameters for surface finishing

Surface finish a (MPa) b

Ground 1.58 0.086

Machined, cold rolled 4.45 0.265

Hot rolled 58.1 0.719

As forged 271 0.995

Table 3 Marin correction factors for size

CsizeaConditions dependent on the diameter of the sample, d

1 For d < 2.79 mmd0:3

0:1133 If d is in inches and 0.11 d 2 in.

d7:62

0:1133 If d is in millimeters and 2.79 d 51 mm

0.6 For d > 51 mmaFor rectangular shape subjected to zero-torsional bending, d ¼ 0:808

ffiffiffiffiffibh

p


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Small Deflection TheoremsLeaf spring is considered as the earliest form of a flexure joint. It can be modeled as a cantileverbeam based on the Bernoulli-Euler law. From Eq. 1, the beam curvature due to a bending momentcan be represented in rectangle coordinates

M ¼ EId2y=dx2

f x; yð Þ (12)

where f(x, y)¼ [1 + (dy/dx)2]3/2. Based on small deflection assumption, the square of slope, (dy/dx)2,is approximated as zero. This assumption allows f(x, y)¼ 1 and leads to the classical beam-moment-curvature equation given as

M ¼ EId2y

dx2(13)

For a cantilever beam subjected to an end load shown in Fig. 14a, the summation of moment givesM¼ P(l x). By solving Eq. 13 withM¼ P(l x), the maximum deflection along the y-axis occursat x ¼ l and is expressed as:

dmax ¼ Pl3

3EI(14)

By substituting Eq. 2 into Eq. 14 with M ¼ Pl and c ¼ h/2, the maximum stress is given as

smax ¼ 3Edmaxh

2l2(15)

For pure translationmotion as shown inFig. 14b, the summation ofmoment givesM¼P(l s x).By solving Eq. 13 withM ¼ P(l s x), the maximum translation motion along the y-axis, whichoccurs at x ¼ l and s ¼ l/2, is expressed as

dmax ¼ Pl3

12EI(16)

Using Eq. 2 with M ¼ Pl and c ¼ h/2, the maximum stress is given as

smax ¼ 3Edmaxh

l2(17)

The pure translation motion equation expressed in Eq. 16 is useful for finding the translationstiffness of a parallel linear spring mechanism shown in Fig. 15a. To avoid parasitic torsionalmotion, this compliant mechanism employs two parallel leaf springs to achieve a pure translation

Table 4 Marin correction factors for reliability for steel material

Reliability (%) 50 90 99 99.9 99.99 99.9999999

creliaba 1.000 0.897 0.814 0.753 0.702 0.520

aAssuming a standard deviation of 8 %


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(or prismatic) motion along the y-axis. Based on the same notations used to define the geometries ofa cantilever beam shown in Fig. 11, the translation stiffness along the y-axis,KdyPy, which is twice ofEq. 16, is expressed as

KdyPy ¼Py

dy¼ 24EI

l3(18)

To achieve maximum translation motion from a given load, Py, the loading point must be locatedat l/2 away from the base as illustrated in Fig. 15b. Although the stiffness is doubled due to theparallel configuration, the amount of deflection remains unchanged and the maximum stress issimilar to a spring leaf-spring configuration, which is given in Eq. 17. Based on Eq. 18 with Iyy, thetranslation along the x-axis, KdxPx , is expressed as

KdxPx ¼ 2Ehb

l

3

(19)

The torsional stiffness about the x- and z-axes is expressed as

KyxMx ¼Ebhe2

2l(20)

KyzMz ¼Ehe2

2

b

l

3

(21)

Any translation motion provided by the linear spring mechanism will be accompanied by

Fig. 15 (a) Parallel linear spring mechanism and its (b) front view and (c) side view

Fig. 14 (a) A beam that is subjected to an end load and (b) delivers a pure translation motion


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a parasitic height variation, l. Figure 16a illustrates a compound linear spring mechanism that caneliminate or minimize this error. Using two linear spring mechanisms connected in series, theparasitic height variation of the moving platform can be canceled out by the parasitic height variationof the intermediate platform. Due to the series connection, the translation stiffness is half of the linearspring mechanism. Figure 16b shows a double compound linear spring mechanism, which isa symmetrical concept of a compound linear spring mechanism. It was introduced to obtain superiorrectilinear motion and reduce sensitivity to external disturbance via the symmetrical concept. Hence,the translation stiffness is twice of the single compound linear spring mechanism. Table 5 lists thetranslation stiffness along the y-axis and maximum stress for both types of compound linear springmechanisms.

The discovery of notch hinge allows the compliant mechanisms to be fabricated in the monolithic(single piece) forms where no assembly is required. Hence, assembly errors can be minimized oravoided to make the compliant manipulators more deterministic. The simplest form of a notch hingeshown Fig. 17a has a circular shape profile, which incorporates a circular cutout on both side ofa blank to form a necked-down section. This necked-down section, which serves as a fixed center ofrotation, exhibits a pure rotational motion within a small dedicated range. In 1965, Paras andWeisbord (1965) presented a complete analysis of such notch hinges.

Assuming that the ratio h/(2R + t) is near to unity, which makes the notches nearly semicircular,the angular stiffness of the notch hinge is expressed as

KyzMz ¼Mz

yz¼ 2Ebt5=2

9pR1=2(22)

For circular shapes defined by t < R < 5 t, the angular stiffness is expressed as

KyzMz ¼Ebt3

24kR(23)

where the correction factor, k, is given as

k ¼ 0:565t

Rþ 0:166 (24)

The maximum stress is determine by

Fig. 16 (a) Compound linear spring mechanism and (b) double compound linear spring mechanism


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smax ¼ ktEt

4kRymax (25)

where the stress correction factor, kt, is given as

kt ¼ 2:7t þ 5:4R

8Rþ tþ 0:325 (26)

The monolithic translation motion compliant mechanisms shown in Fig. 18 are constructed viathe circular notch-hinge flexure joints. Unlike the leaf-spring versions, each limb is formed by a pairof notch-hinge joints connected in series. Table 6 lists the translation stiffness and the maximumstress of various forms of notch-hinge type of linear mechanisms.

The circular-shaped notch hinges usually lead to high stress concentrations during operations.Subsequently, various kinds of shapes were explored to avoid such high bending stresses (Xu andKing 1996; Tseytlin 2002; Lobontiu et al. 2002; Lobontiu et al. 2004; Yong et al. 2008). Examples ofthese flexure joints include the elliptical shape shown in Fig. 17b and the corner-filleted shape hinge

Fig. 17 Three types of notch flexure joints: (a) a circular shape, (b) an elliptical shape, and (c) a corner-filleted shape

Fig. 18 Notch-hinge types: (a) linear spring mechanism and (b) single and (c) double compound linear springmechanisms

Table 5 Translation stiffness along the y-axis and maximum stress for different types of compound linear springmechanisms

Type Translation stiffness, KdyPy Maximum stress, smax

Single 12EIl3

3Edmaxh2l2

Double 24EIl3

3Edmaxh2l2


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shown in Fig. 17c. Based on past literature (Lobontiu 1962), the angular compliance of the ellipticalnotch hinge can be expressed as

CyzMz ¼yzMz

¼ 12l

Ebt2 2cþ tð Þ 8c2 þ t2ð Þ 6c2 þ 4ct þ t2;þ 6c 2cþ tð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit 4cþ tð Þp arctan

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4c

t

r" #(27)

For the corner-filleted notch hinge, the angular compliance is given as

CyzMz ¼12

Ebt31 2r þ 2r

2r þ tð Þ 4r þ tð Þ3( )

t 4r þ tð Þ 6r2 þ 4rt þ t2 þ 6r 2r þ tð Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit 4r þ tð Þ

p; arctan

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4r

t

r" # (28)

Many studies were conducted to find the optimal elliptical and corner-filleted shapes. Figure 19plots the results obtained from one investigation (Henein 2006) that presented one graph that plotsthe stresses obtained from the elliptical shape hinge and another from the corner-filleted shape hinge,respectively. To avoid stress concentration, the results from the investigation suggest that the ellipseration, ry/rx, of the elliptical shape hinge must not exceed 0.025. Investigation results also suggestthat the fillet radius, r/tmin, must maintain below two. By comparing the normalized stress level,s/Ea, between both graphs, the stress level of the optimized corner-filleted shape hinge is 10% lowerthan the elliptical shape hinge of R ¼ 2tmin and 5 times lower than a circular shape hinge. Yet, it isstill 13 % higher than the ideal prismatic beam

Nonlinear Large Deflection TheoremsAs compliant manipulators progressed to larger displacement, the ideal beam shape becamea promising solution due to its low stress but large deflection characteristics. However, deliveringlarge deflection means that these flexure joints will experience parasitic shift in the pivot point andexhibit nonlinear characteristics. Figure 20a shows that with a fixed pivot point, the ideal deflectionpath is a concentric arc about the pivot point. Once there is a shift in the pivot point, the deflectionpath is altered causing a variation between the actual and targeted deflected position. This shift iscommonly known as the parasitic shift.

The effects of parasitic shift are accounted by the Bernoulli-Euler law expressed in Eq. 12. Insmall deflection theory, the slope dy/dx is assumed to be zero resulting in the derivation of theclassical bending-moment equation. However, this assumption is invalid for large deflectionanalysis. Considering the f(x, y) within Eq. 12, Table 7 lists the values from f(x, y) due to thechanges of slope.When the deflection angle is small, the effects of f(x, y) are negligible. At 26.6, the

Table 6 Translation stiffness and maximum stress for different notch-hinge type mechanisms

Type Translation stiffness, KdyPy Maximum stress, smax

Linear spring mechanism Ebt3

6kℒ2RktEtdmax4kRℒ

Single compound linear spring mechanism Ebt3


Double compound linear spring mechanism Ebt3



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value of f(x, y) increases to 40 %. By 45, the value of f(x, y) increases up to 2.8 times higher than theinitial value. Consequently, this investigation shows the importance of f(x, y) as it will account forthe nonlinearity behavior of the large deflection.

For large deflection analysis, the angular compliance of a cantilever beam subjected to a momentloading at free end (Fig. 21c) can be derived directly from Eq. 1 and written as

CyM ¼ l

EI(29)

By applying the cross-product rule on Eq. 1, the deflection along the x- and y-axes can beexpressed as

dx ¼ l l sin yy

(30)

0.05

σ/(E

α)

σ/(E

α)

Analytical (Gross stress)

AnalyticalFEM

FEM (Gross stress)FEM (Local stress)

0.025

0.03

0.04

0.05

0.06

0.12

0.1

0.08

0.06

0.04

0.02

0.035

0.045

0.055

0 0.1

Stress from elliptical shape

Corner-filleted

Ideal beam

Circular

R

l

Elliptical

Stress from corner-filleted shape

ry /rx

tmin

ry

rx

r

r / tmin

0.15 0.2 6 8 10420

Fig. 19 Stress concentration studies conducted on various forms of notch hinges (Henein 2006)

Fig. 20 (a) Concept of parasitic shift of center of rotation and (b) its effect on a cantilever beam


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dy ¼ l 1 cos yð Þy

(31)

Equation 29 can be used to find the angular compliance of a 1-DOF cross-spring pivot shown inFig. 21a by considering it as a pair of cantilever beams subjected to an external moment loading atfree end.

For analyzing a cantilever beam subjected to a perpendicular point loading at free end (Fig. 22a),the bending moment is expressed as

M ¼ EIdyds

¼ P l x dxð Þ (32)

By integrating Eq. 32 by s yields

dyds

¼ 2P

EIsin y0 sin yð Þ12 (33)

Solving Eq. 33 with a2 ¼ Pl2

EI and an assumption thatÐ0yds ¼ l yields

a ¼ 1ffiffiffi2

pðy00

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin y0 sin y

p (34)

Elliptic integrals or numerical integration can be used to solve Eq. 34. As there are many examplesand sources (Howell 2001; Frisch 1962; Byrd and Fredman 1954), this section will not go further toobtain the closed-form solution.

Table 7 Effects of slope on f(x, y)

dy/dx y (deg) f(x, y)

0.01 0.6 1.0001

0.10 5.7 1.0150

0.50 26.6 1.3976

1.00 45 2.8281

Fig. 21 (a) A 1-DOF cross-spring pivot that delivers (b) angular rotation via (c) large deflection of cantilever beams


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Pseudo-Rigid-Body ModelFor large deflection analysis, the elliptic integrals or numerical integration can offer closed-formaccurate solutions. However, it is observed that these methods are cumbersome during the designstage of compliant mechanism. To make analysis of large nonlinear deflection easy to predict yetaccurate, Howell (2001) introduced an approximation method known as the Pseudo-Rigid-Body(PRB) modeling.

Analysis of any flexure joint is based on a perception that deflection is generated with respect toa pivot point. In PRB modeling, this pivot point is predefined based on the types of loading and thenature of the flexure joint. Another uniqueness of this method is a torsional spring, K, which isattached to each pivot. This spring governs the torsional stiffness of the flexure joint. Revisit theproblem of a cantilever beam subjected to a perpendicular point loading at free end. Figure 22bshows the equivalent PRB model representation with a characteristic pivot and a characteristicradius that defines the deflection path. Based on PRB modeling (Howell 2001), the deflection alongthe x- and y-axes can be approximated by

c ¼ l 1 g 1 cosYð Þ½ (35)

dy ¼ gl sinY (36)

where g ¼ 0.85 for this specific case. For accurate prediction using these equations, the deflectionangle must maintain below 64.3. With the torsional spring governing the angular stiffness, the forceand angle are related by an applied torque about the characteristic pivot. Hence, the relationshipbetween the load and angle is given as

P ¼ KYgl sin fYð Þ (37)

where K is the spring constant andf represents the angle of the load (e.g., vertical load givesf ¼ p2).

The spring constant is expressed as

K ¼ bEI

l(38)

where b ¼ 2.25 for this specific case. For other cases of configurations or loading conditions, thePRB method offers specific representation, modeling approach, and parameters for each case. Some

Fig. 22 (a) A cantilever beam subjected to a perpendicular point loading at free end and (b) its correspondingrepresentation based on PRB modeling


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examples are shown in Fig. 23 where each individual configuration or loading condition hasa specific PRB modeling representation.

In the case of a small length flexure pivot configuration shown in Fig. 23a, the flexure joint iscoupled with a rigid link to amplify the deflection. Based on the PRB modeling (Howell 2001), thedeflection along the x- and y-axis can be approximated by

c ¼ l

2þ Lþ l

2

cosY (39)

dy ¼ Lþ l

2

sinY (40)

where l represents the flexure joint length and L represents the rigid-link length. The relationshipbetween the load and angle is given as

P ¼ KYLþ l

2

sin fYð Þ (41)

whereK is expressed in Eq. 38 with b¼ 1. For accurate analysis, L lmust be satisfied, e.g., Lmustbe at least 10 times greater l.

For a cantilever beam with pure translation motion, this case can be modeled as a fixed-guidedcantilever beam shown in Fig. 23b. Based on the PRBmodeling (Howell 2001), the deflection alongthe x- and y-axis can be approximated by

c ¼ l 1 g 1 cosYð Þ½ (42)

dy ¼ gl sinY (43)

where g¼ 0.8517 for this specific case with a constant vertical load and reaction moment. The springconstant for each torsional springs is given as

Fig. 23 PRB modeling representations for (a) a small length flexure pivot, (b) a fixed-guided cantilever beam, and (c)a cantilever lever beam subjected to a moment loading at free end


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K ¼ 2gKYEI

l(44)

where the characteristic stiffness, KY, is 2.67617 for constant vertical loading.In the case of a cantilever lever beam subjected to a moment loading at free end shown in Fig. 23c,

the deflection along the x- and y-axis can be approximated using Eqs. 35 and 36, respectively, withg¼ 0.7346. In this case, the spring constant is expressed in Eq. 38 with b¼ 1.5164. There are othercases that have been presented in past literature (Howell 2001; Howell et al. 1996; Howell andMidha 1994). This section only provides a few cases and the remaining cases can be found fromthese sources. One important note is that the PRB modeling method offers simplistic and accuratesolutions for analyzing various flexure configurations and loading conditions. Yet, the most signif-icant contribution is that the PRB modeling method has linked the classical linkage mechanism andthe compliant mechanism together. Using the unique concept of adding a torsional spring to eachpivot point to describe the angular stiffness, the knowledge of rigid-body mechanism can be used todesign a compliant mechanism.

Semi-Analytic ModelThe PRB modeling method has specific locations to place the pivot points, specific values for thetorsional spring constant, and specific representations for each flexure configuration and loadingconditions. As a result, any misjudgment and inappropriate selection of these PRBmodels often leadto inaccurate results, especially for large deflection analyses. Figure 24 shows different types offlexure joints found within a compliant manipulator. Analyzing each flexure joint with specificflexure configuration requires a person who is well verse in PRB modeling method. In addition,pairing a PRBmodel with a flexure configuration during the design stage is extremely restrictive andcould lead to inaccurate analysis. This is because designing a flexure joint often goes through aniterative process of changing the geometries of the rigid links or the flexure joints to achieve desiredstiffness and off-axis stiffness within a given size constrain. Considering the flexure joints withflexure configuration of L l shown in Fig. 24, the configuration could have change to L< l or L¼ lduring the design stage and different PRB models are required for different configurations. Hence,the main limitation of the PRB method is its inability to provide a simple and generic solution for allforms of flexure configurations.

A semi-analytic modeling method offers a generic, simple, and quick solution for analyzing thenonlinear characteristic of large deflection motion produce from any flexure configuration (Teoet al. 2010b). The term flexure configuration represents a flexure joint coupled with a rigid linkshown in Fig. 25a. The force, F, applied at the rigid-link end becomes amoment load at the end of theflexure joint shown in Fig. 25b. Based on the derivations presented in past literature (Teoet al. 2010b), the deflection of a generic flexure configuration along the driving direction can beapproximated as

Fig. 24 Different flexure configurations for different types of flexure joints found within one compliant manipulator


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D ¼ Lþ l

2ˆ

sin y (45)

where a represents the deflection angle and ˆ is a Sinc function, i.e., ˆ ¼ sin y/y. Referring toFig. 25b, this Sinc function accounts for the parasitic shifting of the pivot point, PP0. Subsequently,the in-plane parasitic deflection perpendicular to the driving direction can be approximated as

Dp ¼ Lþ l

2

Lþ l

2ˆ

cos y (46)

The angular stiffness is derived based on the hypothesis that the relationship between the appliedtorque and the deflection angle of the torsional spring is governed by a moment arm. This momentarm is formed by the rigid link and a portion of the flexure joint as shown in Fig. 26a. This portion ofthe flexure joint represents the distance from the center of the torsional spring to the coupling pointbetween the rigid link and the flexure joint. This portion of the flexure joint is termed as the changingarm, S, and can be expressed as

S ¼ rl

2ˆ(47)

where r is introduced as

r ¼ lffiffiffiffiffiffiffi1:8

p þ L

l þ L(48)

From Fig. 26b, the changing arm varies according to different flexure configurations even withsimilar flexure joint lengths. To address this issue, r is an empirical factor that is used to determine

Fig. 25 (a) A generic flexure configuration subjected to a loading at the rigid-link end and (b) a close-up on the largedeflected flexure joint with parasitic shift in the pivot point

Fig. 26 (a) Torsional spring with a moment arm that changes with respect to (b) different flexure configurations


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the changing arm with respect to any flexure configuration. Subsequently, the changing torque, Ty, isrecognized as a tangential force, Ft, applied to the moment arm and is expressed as

Ty ¼ FT Lþ rl2ˆ

(49)

The changing angular stiffness of a torsional spring is given as

Ky ¼ Tyy

(50)

By substituting Eqs. 49 and 29 into Eq. 50, the relationship between the force and angle isexpressed as

F ¼ EIy

l Lþ rl2ˆ

sin p

2 y (51)

In semi-analytical modeling, the changing angular stiffness of a torsional spring is expressed inEq. 51. WithFT ¼ F sin p

2 y

, the vertical force, F, applied on the moment arm can be determineddirectly based on a known deflection angle shown in Fig. 26a. Last but not least, the maximumbending stress, smax, is given as

smax ¼ F

I

l

2þ Lþ l

2ˆ

cos a

h

2

(52)

The uniqueness of the semi-analytic modeling is that it is generic for all flexure configurations. Incases where there is no rigid link, L will be zero. For any other case, the presence of r accounts forthe change in length of the moment arm. Together with Sinc function,ˆ, which governs the parasiticshift of pivot point, this method is a generic, simple, and quick tool for analyzing any flexureconfiguration.

Stiffness ModelingFlexure joints are considered as spring members within a compliant mechanism. Hence, the movingplatform of a compliant mechanism can be connected to a fixed base by a series of springs andparallel springs. When the springs are connected in series, the overall stiffness is expressed as

1

K total¼ 1

K1þ 1

K2þ þ 1

Kn(53)

or

Ctotal ¼Xni¼1

Ci (54)

where the compliance C ¼ K1. When the springs are connected in parallel, the overall stiffness isexpressed as


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K total ¼Xni¼1

Ki (55)

Example. Apply this concept to model the translation stiffness of a 5-DOF spatial compliant jointmodule shown in Fig. 27a. This module is formed by two identical segments where each segmentconsists of two parallel limbs as shown in Fig. 27b. As each limb can provide 3-DOF motions, i.e.,translation, bending, and torsion, combining two segments in series produces a 5-DOF spatialmotions, i.e., three rotational motions, Cx,y,z, and two translational motions, Dx,y.

For each segment to deliver a pure translation motion, each limb deflects in an “S”-shaped formand can be represented as two identical flexure joints with individual length being l/2 and thedeflection of each flexure joint is D/2 as shown in Fig. 27c. Hence, the translation stiffness ofa flexure joint, Kl, is expressed as

Kl ¼ 2F

D(56)

Based on semi-analytic modeling, the driving force, F, can be obtained from Eq. 51 with thedeflection angle, y, deriving from Eq. 45 based onD¼D/2, l! l/2, and L¼ 0.With each limb beingformed by two identical flexure joints connected in series, the translation compliance of each limb,Climb, is

Climb ¼ 1

K limb¼X2i¼1

1

Kl(57)

As each segment comprises of two parallel limbs, the linear translation stiffness of the spatial jointmodule is

KDFtotal ¼

X2i¼1

1

Climb(58)

Example. Considering the prismatic compliant joint module shown in Fig. 28a, it is formed bya compound linear spring module, which comprises of two parallel springs connected in series asshown in Fig. 28b. Each parallel spring is articulated by two parallel limbs and each limb comprisesof two flexure joints connected together via a rigid link. Hence, each limb can be represented by two

Fig. 27 (a) A 5-DOF spatial compliant joint module formed by (b) a pair of parallel limbs connected orthogonally inseries. (c) Representation of each flexure joint in a translational motion


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identical flexure configurations where each flexure joint is represented by a revolute joint witha torsional spring attached to it as shown in Fig. 28c.

The deflection of each flexure configuration is half of the desired deflection, i.e., Dz/2. Therefore,the deflection stiffness for one flexure configuration along the z-axis, Kfc

z , is expressed as

Kzfc ¼ 2

Fz

Dz(59)

Based on semi-analytic modeling, the driving force, Fz, can be obtained from Eq. 51 with thedeflection angle, y, deriving from Eq. 45 based on D¼ Dz/2 and L¼ L/2. With two identical flexureconfigurations connected in series to form each limb, the deflection compliance of each parallelspring along the z-axis of each limb, Cz

limb, is

Czlimb

¼X2i¼1

1

Kzfc

(60)

As each parallel spring is articulated by two parallel limbs, the linear translation stiffness along thez-axis, Kps

z , is

Kzps ¼

X2i¼1

1

Czlimb

(61)

With two parallel springs connected in series to form a compound linear spring, the translationstiffness of the compound linear spring along the z-axis, Ktotal

DzFz, is

1

KDzFztotal

¼X2i¼1

1

Kzps

(62)

This spring-based modeling concept can be extended to model the stiffness of the entire compliantmanipulator. Considering that a compliant manipulator comprises of a moving platform supportedby j number of parallel and symmetrical limbs, each limb can be formed by a group of compliantjoint modules connected in series by the links as shown in Fig. 29a. For a link in Cartesian space, themoment vector, m, is a cross-product of the link vector, r, and force vector, f, expressed as

Fig. 28 (a) A double compound linear spring module and (b) the schematic representation of this compliant jointmodule and (c) each parallel limb


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mx

my

mz

24

35 ¼

rxryrz

24

35

f xf yf z

24

35 ¼

0 rz ryrz 0 rxry rx 0

24

35: f x

f yf z

24

35 (63)

In addition, the linear displacement vector, dd, is a cross-product of the angular displacementvector, dy, and the link vector expressed as

ddxddyddz

24

35 ¼

dyxdyydyz

24

35

rxryrz

24

35 ¼

0 rz ryrz 0 rxry rx 0

24

35

dyxdyydyz

24

35 (64)

Within each limb, only the compliant joint module is assumed to be a spring member witha compliant matrix, Ci, established at the local coordinate frame attached to it. To establish thecompliant matrix of each limb, Climb, a Jacobian matrix, Ji, that maps the local coordinate frame ofeach compliant joint module to the local coordinate frame of the tip of each limb is required. Basedon Eq. 64, the displacement vector of each limb, X, can be expressed as

ddxddyddzdyxdyydyz

26666664

37777775¼

1 0 0 0 rzi ryi0 1 0 rzi 0 rxi0 0 1 ryi rxi 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

26666664

37777775:

xiyiziyxiyyiyzi

26666664

37777775¼ Jixi (65)

Based on Eq. 63, the force and moment applied to the end-effector, F, can be expressed as

Fig. 29 (a) A limb, which is constructed from a serially connected compliant joint module, is used to (b) form thesymmetrical limbs of a compliant manipulator


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Fx

Fy

Fz

Mx

My

Mz

26666664

37777775¼

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 rzj ryj 1 0 0rzj 0 rxj 0 1 0ryj rxj 0 0 0 1

26666664

37777775:

f xjf yjf zjmxj

myj

mzj

26666664

37777775¼ JT

j f j (66)

As each limb is formed by the compliant joint modules connected in series via the links shown inFig. 29a, the displacement of the limb is Xlimb ¼ J1x1 + J2x2 + J3x3 + . . . based on Eq. 65,reexpressed as ClimbF¼ J1f1 + J2f2 + J3f3 + . . . based on CF ¼ X. As Eq. 66 can be reexpressed asf ¼ JTF, the compliance matrix of each limb is given as

Climb ¼Xni¼1

JiCiJTi (67)

From Fig. 29b, the end effector is supported by a group of parallel limbs. Based on Eq. 66, thetotal force and moment applied to the end effector is F¼ J1

Tf1 + Jz Tf2 + J3

Tf3+...., reexpressed asKX ¼ J1

TK1x1 + J2 TK2x2 + J3

TK3x3 + . . . based on F ¼ KX. By reexpressing Eq. 65 intox ¼ J1X, the stiffness matrix of a compliant mechanism is given as

K ¼Xnj¼1

JTj Klimb, j J

1j (68)

Example. Consider the stiffness modeling of a 3-DOF compliant manipulator shown in Fig. 30a asan example. This compliant manipulator is formed by three symmetrical parallel limbs and eachlimb consists of spatial and prismatic compliant joint modules, as shown in Fig. 30b. All threeparallel limbs are placed 120 apart to support a moving platform with its end effector located at thecenter.

In this example, the compliance matrix of the prismatic compliant joint module, CA, and thespatial compliant joint module, CB, is given as

CA ¼

1:56e-61:41e-7 3:33e-7 SYM1:95e-8 0:23e-9 3:41e-40:15e-9 5:00e-9 5:58e-5 1:28e-30:60e-9 0:25e-9 1:70e-9 3:95e-8 4:61e-33:03e-5 3:03e-6 1:30e-9 0:01e-9 1:33e-8 6:65e-4

26666664

37777775

(69)

CB ¼

1:71e-61:03e-6 1:71e-6 SYM7:06e-8 7:06e-8 4:80e-93:19e-5 5:24e-5 2:17e-6 0:3385:24e-5 3:19e-5 2:17e-6 9:81e-4 0:3380 0 0 0 0 0:031

26666664

37777775

(70)

Next, individual local coordinate frame is attached to the center of the prismatic compliant module


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and the tip of the flexure joint within the spatial compliant joint module as shown in Fig. 30b. Theselocal coordinate frames have the same orientation but different offsets with respect to the localcoordinate frame attached to the tip of the limb. Hence, the Jacobian matrix for each module, Jm, isexpressed as

Jm ¼

1 0 0 0 hm 01 0 hm 0 0

1 0 0 01 0 0

zeros 1 01

26666664

37777775

(71)

wherem represents A or B for respective modules. In this example, hA¼ 87 mm and hB¼ 19.5 mm.The compliance matrices CA and CB are assembled to form

Ctotal ¼ diag CA;CBð Þ (72)

In addition, the Jacobian matrices JA and JB are assembled into

Fig. 30 (a) A compliant manipulator formed by three symmetrical parallel limbs where (b) each limb consists ofa spatial and prismatic compliant joint module to support (c) a moving platform at three symmetrical corners which are120 apart


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Jlimb ¼ JA JB½ (73)

From Eq. 67, the compliance matrix of each limb is derived as

Climb ¼ JlimbCtotalJTlimb

¼

1:65e-48:05e-7 1:43e-4 SYM8:94e-9 4:74e-6 3:41e-41:28e-5 6:79e-3 5:36e-5 0:3416:97e-3 5:10e-5 2:17e-6 9:81e-4 0:3443:03e-5 3:10e-6 1:30e-9 0:01e-9 1:33e-8 0:032

26666664

37777775

(74)

Prior to the derivation of stiffness matrix, the Jacobian matrix, Jj, that maps the local coordinateframe of each limb to this reference coordinate frame must first be established. Figure 30c illustratesthat the local coordinate frame attached to the tip of each limb has different orientations and a lineardisplacement w.r.t. the reference coordinate frame attached to the center of the moving platform.Here, the reference coordinate frame attached to the center of the moving platform falls on the sameplane as the local coordinate frame at the tip of each limb. Hence, there is no variation in the z-axisbetween both coordinate frames. The orientation of the local coordinate frame at the tip of each limbw.r.t. the reference coordinate frame in matrix form is expressed as

XY

¼ coscj cos bj

sincj sin bj

xjyj

(75)

where bj ¼ cj + p/2 and j represents either 1, 2, or 3. Based on Eqs. 75 and 65, the Jacobian matrixthat maps the local coordinate frame of each limb to the reference coordinate frame is

Jj ¼

coscj cos bj 0 0 0 r sin yjsincj sin bj 0 0 0 r cos yj0 0 1 r sin yj r cos yi 00 0 0 coscj cos bj 00 0 0 sincj sin bj 00 0 0 0 0 1

26666664

37777775

(76)

where c1 ¼ 3p/2, c2 ¼ p/6, c3 ¼ 5p/6, y1 ¼ p/2, y2 ¼ 7p/6, y3 ¼ 11p/6 and, in this example, r ¼64.83 mm. Based on Eqs. 68, 74, and 76, the stiffness matrix of the compliant manipulator is givenas


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K ¼X3j¼1

JTj C1

limb, jJ1j

¼

2:57e50 2:57e5 SYM4:3e-12 2:6e-12 8:83e31:63e2 5:14e3 5:7e-14 1:30e25:14e3 1:63e2 1:0e-13 5:9e-14 1:30e22:6e-11 1:6e-11 2:71e2 2:8e-13 5:4e-13 1:72e3

26666664

37777775

(77)

The obtained stiffness matrix was compared against the experimental values and results havedemonstrated that the variation between the theoretical and experimental values is less than 15 %.This shows that this stiffness modeling approach is accurate and able to provide reasonable analysison the stiffness characteristic of a compliant mechanism.

Fundamental Design Concepts

The most important design criteria of a compliant manipulator is to maximize the stiffness ratio, i.e.,between the off-axis stiffness and natural stiffness, of the compliant mechanism. Consider that thenatural stiffness quantifies the compliance in the desired driving directions while off-axis stiffnessquantifies how stiff the mechanism is in non-driving directions. Hence, a compliant mechanismmusthave high stiffness ratio since it directly affects the robustness of the compliant manipulator. Thissection highlights the several existing and new design concepts that can be used to synthesize thecompliant mechanisms.

Exact Constraint DesignExact constraint design approach excludes the principles of kinematics to synthesize a compliantmechanism. Its basic objective is to achieve desired degrees of motions or no motion throughapplying minimum number of constraints to a rigid body. Based on the principles of kinematics, anunconstraint rigid body has six degrees of motion, i.e., three translational and three rotationalmotions. Based on the principles of exact constraint design, a nonrigid body may have one ormore degrees of “flexibility”which serves as additional degrees of motions. Considering a box withlid as an example, it has six degrees of motions when the lid is on. Without the lid, the open box willhave seven or even eight degrees of motions due to the additional torsional motions. A total oftwelve fundamental ideas are associated with this design approach and can be found in the pastliteratures (Blanding 1992; Hale 1999; Slocum 1992). Most of these fundamental ideas are sum-marized in the matrix table shown in Fig. 31 illustrating how constrains can be added to achievedesired degrees of motions.

From Fig. 31, most of those fundamental ideas involve with orthogonal constraints. However,there are some that use non-orthogonal constraint. One of these fundamental ideas states that(Blanding 1992).

A constraint applied to a body removes that rotational degree of freedom about which it exerts a moment.

When a set of non-orthogonal constraints are used to constrain a translation motion as shown inFig. 32a, it creates an instantaneous center (I.C.) of rotation at the center of the circle. By addinganother constraint that reacts with a moment about the center, the rotation about the center of the


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circle is constrained. However, the circle may not be perfectly constrained because the length of theadditional constraint arm is not equal to the other non-orthogonal constraints. A perfect example thatreflects this fundamental idea is shown in Fig. 32b where all constraint arms are equal in length andsymmetrical. In some conditions, nonsymmetrical arrangement is acceptable, but all constraint armsmust remain equal as shown in Fig. 32c. This fundamental idea is useful for synthesizing compliantmechanism with constrained in-plane motion.

Another two fundamental ideas are related to flexible members (Blanding 1992).

An Ideal Sheet Flexure imposes absolutely rigid constraint in its own plane (dx, dz, and yy), but it allows threedegrees of freedom: Y, yx, and yz.

An Ideal Wire Flexure imposes absolutely rigid constraint along its axial (dx), but it allows five degrees offreedom: dy, dz, yx, yy, yz.

The fundamental idea of an Ideal Sheet Flexure that offers 3-DOF out-of-plane motions is shownin Fig. 33a. Using a leaf spring as an example, the in-plane motion is constrained and can berepresented by two vertical constraints and one diagonal constraint. When one set of constraints isused to support a rigid body, it permits 3-DOF motions, i.e., dy, yx, and yz, as shown in Fig. 33b.When another similar set of constraints are added at the opposite end of the rigid body, the only

Fig. 31 A matrix table of exact constraint design approach

Fig. 32 Examples of constraints been applied at the instantaneous center of a set of constraints to prevent rotationmotion


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degree of motion left is along the y-axis, dy, as shown in Fig. 33c. As each set of constraintsrepresents a leaf spring, having two in parallel forms a parallel linear spring mechanism.

This fundamental idea was also applied in synthesizing the spatial compliant joint module shownin Fig. 27a. The design concept of this spatial compliant joint module is also governed by thefollowing fundamental idea of exact constraint design (Blanding 1992).

When parts are connected in series (cascaded), add the degrees of freedom. When the connections occur inparallel, add constraints.

Referring to Fig. 27b, two parallel beam-based flexure joints, which forms one segment, increasethe deflection stiffness of the spatial compliant joint module. Yet, connecting two segments in seriesand orthogonal arrangement brings additional degrees of motions. Alternatively, the fundamentalidea of an Ideal Wire Flexure that offers 5-DOF motions is also useful for designing a spatialcompliant joint. In fact, classical spatial compliant joints were mainly realized through wire flexures.The last but most important fundamental idea of the exact constraint design states that (Blanding1992)

A constraint ℭ properly applied to a body (i.e., without overconstraint) has the effect of removing one of thebody’s rotational degrees of freedom (ℜ's). Theℜ removed is the one about which the constraint exerts a moment.A body constrained by n constraints will have 6 n rotational degrees of freedom, each positioned such that noconstraint exerts a moment about it. In other words, each ℜ will intersect all ℭ's.

This fundamental idea provides the definition of a mechanism being overconstrained or under-constrained. An extension of this idea was to generalize it to nonrigid bodies (Hale 1999). By addingnumber of DOF with due to the DOF of flexure joints, the mobility equation is rewritten as dof¼ 6 +f ℭ. Hence, ℭ must be sufficient to achieve the desired DOF. In addition, it is also important thatthere must be no redundantℭ. If removal of aℭ does not affect the DOF, the remainingℭ stays in themechanism. Lastly, the mechanism is exactly constrained if the removal of any singleℭ increases theDOF by one. With this final but most crucial fundamental idea, this section wraps up the review ofexact constraint design approach.

Parallel-Kinematic ArchitectureParallel-kinematic architecture plays an important role in the success of the compliant manipulatordue to its advantages of a lower inertia, programmable centers of rotations, superior dynamic

Fig. 33 (a) A leaf-spring flexure joint represented by two vertical constraints and one diagonal constraint. (b) This set ofconstraints is used to constraint a rigid body that eventually leads to (c) a parallel spring mechanism


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behavior, higher stiffness, and less sensitive to external disturbances as compared to its serialcounterpart. In addition, the limited deflection of the flexure joints suits the limited motion rangeof the parallel-kinematic architecture. Therefore, the parallel-kinematic architectures are commonlyused to synthesize the mechanisms within the compliant manipulators. In general, a parallelmechanism is made up of an end effector (located at the center of the moving platform) withn degrees of motions and of a fixed base that are both linked together by at least two independentparallel-kinematic chains (Merlet 2000). Each kinematic chain is articulated by a set of elementarykinematic joints that are connected in series. Such kinematic joints include the 1-DOF revolute(R) joint, the 1-DOF prismatic (P) joint, the 2-DOF universal (U) joint, and the 3-DOF spherical(S) joint. Parallel mechanism can be classified into two categories: planar and spatial motion.A planar motion parallel mechanism can provide up to 3-DOF of in-plane motions, while a planarmotion parallel mechanism can deliver out-of-plane motions.

Figure 34a illustrates a 3-DOF planar motion parallel mechanism that can only deliver twode-coupled X and Y translational motions and a coupled yz rotational motion. This parallel mech-anism consists of three kinematic chains. Starting from the fixed base to the moving platform, eachkinematic chain is formed by one P joint, another P joint orthogonal to the first, and one R joint. Thisis also termed as the 3-legged prismatic-prismatic-revolute (3PPR) parallel-kinematic architecture.To form a parallel manipulator, one of the P joints must be actively controlled, while the remainingP and R joints become the passive joints. To prevent high inertial and moving mass, it is always idealto fix the active P joints on the base. Figure 34b illustrates a spatial motion parallel mechanism thatproduces 3-DOF out-of-plane motions, i.e., Z, yx, yy. This parallel mechanism is formed by a 3RPSparallel-kinematic architecture and has an active P joint in each kinematic chain. A 6-DOF spatialparallel mechanism is illustrated in 34c. This parallel mechanism is formed by a 6UPS parallel-kinematic architecture with an active P joint in each kinematic chain. This parallel-kinematicarchitecture, which offers 6-DOF motions, is also known as the Steward platform. A compliantmechanism that uses this architecture was shown in Fig. 9a of section “Brief History”. In fact,section “Brief History” has introduced many compliant manipulators that are synthesized fromdifferent types of parallel-kinematic architectures.

With many variations of parallel-kinematic architectures, a systematic design methodology,which could identify the right architectures and conversion of kinematic chain to compliant limbs,is essential. This methodology is shown in Fig. 35 where one can easily follow to convert anylinkage-joint parallel mechanism into a parallel compliant mechanism. Termed as a task-orienteddesign approach, the first step (S1) is to identify the task or application, desired DOF, andworkspace. These data become the design criterions, which are used in the second step (S2), tosynthesize the right-type parallel-kinematic architecture, which is suitable to deliver the targeted taskspecifications. Upon selection of suitable architecture, the third step (S3) is to perform kinematic

Fig. 34 Different parallel-kinematic architectures


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analysis, i.e., forward and reverse kinematic analyses. The forward kinematic analysis works out theposition and orientation of the end effector (task space) based on given actuator displacements (jointspace). Hence, it is used to determine the required displacements from the actuators and thecompliant joint modules to achieve the targeted workspace. The inverse kinematic analysis providesthe actuator displacements based on given position and orientation of the end-effector. This analysisis a crucial analytical model when task-space control implementation is required.

The forth step (S4) is to convert all the rigid-body kinematic joints into the flexure joints or jointmodules. Here, a compliant joint module represents a specific conventional kinematic joint, e.g., therevolute joint, the prismatic joint, the spherical joint etc., which can be formed by a single flexurejoint, a series or a group of flexure joints. Next, the parametric analyses will be conducted based onthe forward or inverse kinematic solution to provide the estimated displacements required from eachcompliant joint module. Subsequently, the design of each module is conducted through analyticalmodeling of the required stiffness within those estimated displacements and material’s yieldstrength. The last step (S5) is to perform design optimization of the complete compliant mechanismthrough numerical simulation using finite-element modeling (FEM) platform, i.e., ANSYS, andanalytical stiffness modeling. S5 evaluates the achievable workspace, the stress concentration of theflexure joints, and the natural frequency of the FPM. An iterative process between S4 and S5 isnecessary should any of those parameters fall out of the desired specifications. Using this designapproach, a compliant mechanism is using the parallel-kinematic architecture systematically.

Example. Consider the design of a 3-DOF yx-yy-Z spatial motion compliant mechanism as anexample to apply the task-oriented design approach. This mechanism with active actuation andcontrol formed a compliant manipulator targeted to automate the imprinting and out-of-planealignment tasks within an Ultraviolet Nanoimprint Lithography (UV-NIL) process. The targetedworkspace was 5 5 5 mm, while the targeted imprinting force was ~200 N.

Type synthesis, which identifies an ideal parallel-kinematic architecture based on the taskrequirements, was conducted in S2. From past literatures (Merlet 2000; Tsai 1999), four possible

Fig. 35 Task-oriented design approach; a systematic methodology to convert any linkage-joint parallel mechanism intoa compliant-based parallel mechanism


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architectures were identified, i.e., 3RRS, 3PRS, 3RPS, and 3PPS. To design a high-precisionmechanism, P-joint, which provides stiffer and higher precision guide ways, is always preferredover the rotation counterpart. Considering the requirements of having high imprinting force and thecontact task, 3PPS parallel-kinematic architecture was selected as shown in Fig. 36a. To reduce themoving mass and inertia, the active P-joint was placed nearest to the base platform (note: theunderlined P represents active P-joint). To effectively translate the output force of the linearactuators into the desired imprinting force, the active P-joint in each leg was placed vertically,while the passive P-joint was placed horizontality with its axis of motion always pointing towardsthe center of the equilateral triangular base.

The forward and inverse kinematic models were derived in S3. Forward kinematic modeling isused to determine the moving platform pose based on the known active P-joint displacements andthe solutions are given as (Yang et al. 2011)

ex ¼ z2 z3a

(78)

ey ¼ ffiffiffi3

p2z1 z2 z3ð Þ

3a(79)

Fig. 36 (a) A 3PPS parallel-kinematic architecture synthesized from S2. (b) Projection of a rotation vector to representthe yx-yy-Z motion for kinematic and workspace analyses in S3


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Fig.3

7Initialconceptualdesign

ofa3P

PSparallelcompliant

mechanism


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zp ¼ z1 þ z2 þ z33

(80)

where a represents the edge length of the equilateral triangular moving platform while z1, z2, and z3represent the displacement of the active P-joints. Referring to Fig 36b, these forward kinematicsolutions were derived based on the notation that the 3-DOF motions can be represented bya rotation vector, o, with Z ¼ (0,0,1) and Z0 ¼ (ex, ey, ez) being the unit directional vectors of theoriginal and final Z axis of the moving platform frame, respectively. Hence, the projections of Z0

onto the x- and y-axes, i.e., ex and ey, are employed to uniquely define Z0 instead of using two rotationangles y and f. These kinematic solutions were used to determine the design parameters of themechanism based on the desired workspace. These parameters include the dimensions of the movingplatform, the desired translation, and rotation displacement from each joint. To obtain a verticaldisplacement of 5 mm, a long-stroke flexure-based electromagnetic linear actuator (FELA) wasemployed (Teo et al. 2008) as the active P-joints. Using the kinematic solutions and size constraint ofthe FELA, awas chosen to be 112.29 mm. Based on the selected design parameters, results obtainedfrom the workspace analysis suggested that the maximum orientation and translation displacementsof the moving platform were 5.1 and 5 mm, respectively (Fig. 36).

Subsequently, an initial conceptual design of a 3PPS parallel compliant mechanism was proposedin S4. It was articulated by three symmetrical compliant limbs where each limb consisted of a seriesof compliant joint modules that mirrored the PPS kinematic chain. The stiffness analysis of eachcompliant joint module was conducted via the semi-analytic modeling approach (see section “Semi-Analytic Model”) to determine the geometries of the flexure joints. This stage of design wasconducted using the analytical modeling approaches because it is a tedious and iterative process.Hence, using finite-element (FE) simulation is time consuming and computational intensive.

The FE simulation became useful in S5 as it was used to conduct final validation on the workspaceof the proposed parallel compliant mechanism and the maximum stresses within the flexure joints.Figure 38a shows that results from the FE analysis suggested that the passive P-joint compliantmodule and a segment of the S-joint compliant module were redundant because there was no stresswithin the flexure joints. Returning to S4, a simple 5-DOF spatial compliant joint module wasproposed to replace the passive S- and P-joint compliant modules. Reduction of the compliantmodules also simplified the entire mechanism design and increased the off-axis stiffness. Finally, theimproved version was evaluated through the FE simulation before the actual prototype wasdeveloped as shown in Fig. 38b.

Fig. 38 (a) FE analysis conducted on the initial proposed mechanism. (b) Final prototype with simplified compliantlimbs


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This example provides an overview on how the task-oriented design approach can be used todesign a parallel compliant mechanism. For S2 and S3, most of the parallel-kinematic architecturescan be found in many sources and past literatures (Merlet 2000; Tsai 1999). Hence, finding a suitablearchitecture will not be difficult. The flexure and stiffness modeling methods, which are presented insection “Theoretical Modeling Approaches,” are sufficient for executing S4 and S5 (except the FEanalysis). Therefore, information presented in this section will be useful for designing a parallelcompliant mechanism systematically.

Topological OptimizationA compliant mechanism can also be treated as a continuum structure with either distributed orlumped compliance to deliver specific DOF motion. For example, a compliant gripper shown inFig. 39a was designed based on a four-bar linkage architecture with a slider. On the other hand,topology optimization can also synthesize a continuum structure that not only delivers the samefunction but with better stiffness characteristic as shown in Fig. 39b.

In general, a topological optimization approach is a mathematical approach of finding the optimalway of distributing material within a predefined design domain based on a set of loads, fixedsupports, and boundary conditions such as the performance specifications and task requirements.To conduct a topological optimization, a design domain formed by either discrete number of finiteelements or trusses must be defined. Using finite-element design domain is considered as thehomogenization approach (Bendsoe and Kikuchi 1988), while those with truss structures are termedas the ground structure approach (truss sizing) (Rozvany 1976; Bendsoe et al. 1994). Figure 39bshows a finite-element design domain that was used to synthesize an optimized continuum structure.With this design domain, loads, fixed supports, and output point (motion) were allocated aroundit. Based on an objective function and some boundary conditions, topological optimization wasconducted via an optimization algorithm to determine the state of each element, i.e., either solid orvoid. From Fig. 39b, those elements in black are solid, while those in white are void. The solidelements form the optimized structure topology that meets the objective function. A shape optimi-zation process further removed the unwanted materials and smoothen the edges to form a completecontinuum structure. Lastly, a symmetrical pair of optimized continuum structure will function likea compliant gripper.

Topological optimization approach uses optimization algorithms such as the Generic Algorithm(GA) (Chapman and Jakiela 1996), Solid Isotropic Material with Penalization (SIMP) (Bendsoe1989; Bendsoe and Sigmund 1999), Evolutionary Structural Optimization (ESO) (Xie and Steven1993; Chu et al. 1997), and Optimality Criteria (OC) (Rozvany 1995). Lately, new topological

Fig. 39 (a) A compliant gripper obtained from the kinematic-based design approach versus (b) a new gripper conceptgenerated via the topology optimization approach


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optimization approaches such as the level-set method (Wang et al. 2003b), the morphologicalmethod (Tai and Akhtar 2005), and the mechanism-based seeding approach (Teo et al. 2013) werealso introduced. Over the past 30 years, these research efforts and findings have demonstrated thatthe topological optimization approach is another effective concept of designing the compliantmechanisms.

Using the advantages of modern topology optimization approach, an integrated design approachfor synthesizing an optimized parallel compliant mechanism is introduced in this section. Thisdesign approach is a systematic design methodology that integrates both classical mechanism theoryand modern topology optimization approach. Referring to Fig. 40, the first step is to understand thedesign specifications, e.g., the desired DOF, workspace, and size constraints. Next, appropriateparallel-kinematic architecture will be selected and general kinematic analyses will be conducted. Atsub-chain level, topology optimization will be used to determine the optimized topology of theflexure joint or limb. With these optimized topologies being generated, the compliant matrix of eachflexure joint or limb will be used to determine the overall stiffness of the compliant mechanism viathe classical mechanism theory. At configuration level, the overall stiffness of compliant mechanismwill be optimized based on the desired workspace and size constraints. Subsequently, an optimizedparallel compliant mechanism that meets all desired specifications will be generated.

Fig. 40 A systematic integrated design approach for synthesizing optimized parallel compliant mechanisms

Fig. 41 Schematic representation of 3PPR parallel-kinematic architecture and the stiffness modeling from classicalmechanism theory


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Example. Consider the design of a 3-DOF X-Y-yz planar motion compliant mechanism as anexample to apply the integrated design approach. This compliant mechanism targeteda workspace of 4 mm 2 2 and a footprint of 300 mm2.

To achieve an X-Y-yz motion, 3RRR, 3PRR, and 3PPR (Yang et al. 2008) are possible parallel-kinematic architectures. In this example, 3PPR was chosen because the compliant P-joints are moredeterministic than the compliant R joints. The schematic of 3PPR is shown in Fig. 41 where themoving platform is connected to the fixed base by three identical parallel-kinematic chains. Eachkinematic chain consists of an active P-joint and a passive RP-joint that are connected in series. Bytreating each joint as a spring, the stiffness modeling approach from classical mechanism theory (seesection “Stiffness Modeling”) can be used to determine the overall stiffness of the end effector basedon the compliant joints modules.

After the mechanism synthesis and kinematic analysis, a new topological optimization approachwas used to synthesize individual compliant joint module. Termed as the mechanism-basedapproach (Lum et al. 2013), elementary linkage mechanisms were used as basic genes for thejoint optimization. For example, a generic 4-bar linkage mechanism was used to synthesize the 1-DOF active P-joint and a generic 5-bar linkage mechanism was used to synthesize the 2-DOFpassive PR-joint. Next, the cubic and harmonic curves were added to each link of the mechanism tocreate material as shown in Fig. 42a. Elements that fell within the boundary of the original and thereflected curves became solid. Subsequently, the optimization varied the distribution of the materialby changing the design variables of mechanism (Fig. 42b) and individual link (Fig. 42c) until theobjective functions were met.

For an optimized P-joint, C11, which represents the compliance along the x-axis due toa translation force along the same axis, needs to be as high as possible, while the remainingcomponents of the compliance matrix need to be as low as possible. Hence, the objective functionfor the P-joint optimization was formulated as

Fig. 42 (a) Adding cubic and harmonic curves to create material for each link. (b) Design variables of the mechanismand (c) for each link


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min F xð Þ ¼ P6i¼2P

ij¼1jCijj

C2011

!(81)

For an optimized PR-joint, both C11 and C66, which represents the compliance about the z-axisdue to a moment about the same axis, need to be as high as possible, while the remainingcomponents of the compliance matrix need to be as low as possible. Thus, the objective functionPR-joint optimization was

min F xð Þ ¼ P6i¼2P

ij¼1C

2ij

C1911C

2066

!(82)

In this example, GAwas used as the optimization algorithm. The evolutions from the basic genes(linkage mechanisms) to optimal joint designs in both topology and structural forms are shown inFig. 43. Subsequently, the stiffness matrix of the optimized topology of each joint was used todetermine the stiffness matrix of the end effector based the stiffness modeling approach (see section“Stiffness Modeling”).

At configuration level, stiffness optimization was conducted to optimize the end effector based onthe workspace and size constraints. As mentioned in the beginning of this section, the mostimportant design criteria of a compliant mechanism is to maximize the stiffness ratio, i.e., betweenthe off-axis stiffness and natural stiffness. Hence, the objective function of the stiffness optimizationwas formulated as

Fig. 43 Concurrent evolution of topology and structure for both P- and PR-joints

Fig. 44 3PPR parallel compliant mechanisms articulated by compliant joints with (a) optimized topologies versus (b)conventional topologies


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max ¼ K33K44K55

K11K22K66

(83)

where K22 represents the stiffness along the y-axis due to a translation force along the same axiswhile K33, K44, and K55 are the off-axis components.

Another 3PPS parallel compliant mechanism shown in Fig. 44a, which was articulated by flexurejoints with traditional topologies, was used to evaluate the stiffness characteristic of the optimizeddesign. Termed as the conventional design (Fig. 44b), its PR flexure joint was formed by a cantileverbeam with both ends being fixed to the translation portion of the P-joint, which was formed bya conventional parallel linear spring configuration. The compliance matrix of the optimized design,Ceeopt, and the conventional design, Cee

con, was obtained as

Coptee ¼

3:55E-51:1E-17 3:55E-5 SYM4:7E-25 2:3E-24 1:12E-61:8E-15 3:4E-15 1:7E-21 4:06E-43:4E-15 1:78E-15 1:7E-21 9:7E-16 4:06E-47:3E-20 2:3E-20 1:6E-15 1:7E-21 4:4E-21 2:42E-2

26666664

37777775

(84)

Cconee ¼

1:86E-56:6E-18 1:86E-5 SYM1:7E-25 5:2E-25 1:97E-62:0E-16 3:0E-16 3:9E-21 5:41E-43:0E-16 2:0E-16 4:0E-22 4:6E-16 5:41E-43:8E-20 6:2E-20 6:8E-15 7:9E-21 4:5E-20 2:42E-2

26666664

37777775

(85)

Subsequently, the ratio between Eqs. 84 and 85 is represented in a diagonal matrix form, which is

Fig. 45 Prototype of the 3PPR parallel compliant manipulator that is articulated by flexure joints with optimizedtopologies and driven by three voice-coil actuators


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Rc ¼ Coptee

Cconee

¼ diag 1:91 1:91 0:57 0:75 0:75 1½ (86)

From Eq. 86, results show that the compliance along the x- and y-axis of the optimized design isalmost twice as compared to the conventional design. In addition, the off-axis stiffness is higher forthe optimized design. Hence, this section shows that the integrated design approach is an effectivedesign methodology for using topological optimization approach to synthesize an optimized parallelcompliant mechanism with high stiffness ratio characteristic. Finally, a prototype was developedfrom this approach. Each of the active P-joint is driven by a voice-coil actuator and its position ismeasured via a linear optical encoder. Based on a PID controller, it achieved a high positioningresolution of 50 nm over a workspace of 5 mm2 5 (Fig. 45).

Actuation and Sensing

Compliant manipulators are high-precision mechatronic systems that consist of the compliantmechanisms, high-resolution positioning actuators and sensors, and good control schemes. Withthe main part of this chapter focusing on the strength, limitations, modeling, and design methodol-ogies of the compliant mechanisms, this section reviews the state-of-the-art actuators and sensorsthat can be used to achieve high-precision manipulation where the advantages, limitations, andpotential performance trade-offs for each kind of actuator or sensor will be discussed.

High-Resolution Positioning ActuatorsAn actuator with high positioning resolution is an essential subsystem of the compliant manipulatorthat could ultimately decide the traveling range, output force, stiffness, and even size or footprint ofthe manipulator. These actuators can be classified into two categories, i.e., the solid state and the fieldbased. The solid-state actuators are transducers that convert the electrical energy into the mechanicalenergy via strain in the materials. Piezoelectric (PZT) actuator, shape-memory alloy (SMA), and thethermal actuator are the solid-state actuators, which are commonly used by the compliant manipu-lators. On the other hand, the field-based actuators are transducers that convert the electrical energyinto the mechanical energy via the presence of fields. Field-based actuators that are commonly foundin the compliant manipulators include the electrostatic and the electromagnetic actuators.

Piezoelectric ActuatorsMost conventional compliant manipulators are driven by the PZT actuators due to the nanometricresolution and large actuating force characteristics (Mamin et al. 1985; Fite and Goldfarb 1999;Dong et al. 2000; Kimball et al. 2000; Sun et al. 2002; Zhang and Zhu 1997; Tan et al. 2001). Suchan actuator is made up of ceramics that convert an applied voltage or charge into a mechanicaldisplacement directly through the physical elongation of the material. This small-dimensionalchange leads to extremely small displacement of up to 1 1012 m. Such positioning resolutioncan be adjusted since the dimensional changes are proportional to the applied voltage (Ouyanget al. 2008). PZTactuators can operate at extremely high bandwidth of up to few hundreds kilo-Hertz(kHz). They can also operate up to millions of cycle without deterioration. Due to the nature ofproducing mechanical displacements through dimensional changes, PZT actuators can producelarge force of up to few kilo-Newton (kN) and no wear-and-tear issue and can operate in vacuumand clear room environment. Figure 46a shows some commercial stacked PZT actuators developedby Physik Instrumente (PI).


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The major disadvantages of PZT actuators include the small and nonlinear displacement charac-teristics. The nonlinearity of PZT actuator is contributed by the hysteresis and creep behavior of theceramic materials during the dimensional change. As a result, the forward and backward paths ofa cyclic PZT actuator are different and require mechanical means or advanced control schemes tominimize such nonlinearities. One approach is to preload the moving stage with springs in thedirection that opposes the PZTactuation as shown in Fig. 46b. However, preloading with springs canonly linearize a portion of the stroke. Hence, the maximum displacement of the stage is limited to250 mm (Ho et al. 2004). Another approach is to model the hysteresis and creep of the PZTactuationand linearize the displacement through advanced control schemes (Liu et al. 2013). The biggestlimitation is the limited displacement of a PZT actuator. By stacking the PZT ceramic disks, themaximum achievable stroke is a few hundred micrometers. In addition, this stacking approachgenerates accumulative errors at the end of the stack and increases the stress within each PZTceramic disks. Most importantly, this approach increases the internal resistance which in turnincreases the applied voltage requirements, e.g., >200 VDC.

Shape-Memory AlloySMA material has a unique memory (known as the “shape memory effect”) of its pre-deformedshape. At low temperature, the SMA returns back to its original shape when the temperatureincreases. This behavior makes SMA as a form of solid-state actuation. At high temperature, anapplied force can cause a large deformation, which can also easily recover by releasing the appliedforce. This effect is termed as the “superelasticity.” The use of SMA on the compliant manipulatorsis unique too. Instead of using conventional materials to develop the flexure joints, SMA is used asthe flexure material creating interesting flexible members that double up as both the limbs andactuators of the compliant manipulators (Reynaerts et al. 1995; Hesselbach et al. 1997; Bellouardand Clavel 2004). This is because the SMA flexure joint exhibits higher flexibility as compared tothe conventional flexure joints. Using SMA flexure joints reduced the amount of flexure joints,hence lowering the stiffness of the compliant manipulator in the driving direction. In addition, SMAcan also be considered as a high damping metal, which is a favorable characteristic for spring-dominated systems, i.e., compliant mechanisms.

The main limitation of SMA is its slow rate of cooling, which is usually limited to a few hertz. Asa result, the difference between heating and cooling transition creates a hysteresis effect during theforward and reverse motions as shown in Fig. 47. The slow cooling rate also limits the bandwidth ofany SMA-based compliant manipulator. Hence, these manipulators are only suitable for tasks orapplications that require slow yet precise motion delivery.

Fig. 46 (a) Stacked PZT actuators from PI (Physik Instrumente 2014). (b) A 1-DOF PZT-driven compliant stage withan amplification level and preloaded springs and its schematic representation (Ho et al. 2004)


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Thermal ActuatorsThermal actuation is one of the popular schemes in driving the MEMS-based compliant manipula-tors (Chen and Culpepper 2006; Comtois and Bright 1996, 1997; Qiu et al. 2003; Zhu et al. 2006).This is because the thermal actuators can be easily fabricated and integrated within such microscaledfunctional systems. The basic working principle of the thermal actuation is to generate a smallamount of thermal expansion in a material through joule heating. Simultaneously, this small thermalexpansion will be amplified to produce a deflection motion. Such amplifications are realized throughthe bimorph (asymmetric) or chevron (symmetric) actuator design as shown in Fig. 48.

The bimorph design is a cantilever beam formed by two equal length parallel segments beenjoined together. During operation, one segment will be heated while the other remains cool. Thetemperature difference between two segments causes both segments to expand differently. Asa result, a bending deflection occurs since both segments are fixed at one end. In addition, two ormore thermal actuators can be connected together in parallel to enhance the force output or generatea linear motion (Comtois and Bright 1997). The chevron design can be articulated by one or morepairs of “V”-shaped cantilever beams. For each pair of beams, the applied current passes throughfrom one beam to the other, thus causing them to expand, buckle, and create a linear motion. Ingeneral, the bimorph thermal actuator provides a bending motion, while the chevron thermalactuator provides a translation motion.

Figure 49a shows a 6-DOF compliant manipulator, mHexFlex, that is driven by three pairs of 2-axes bimorph thermal actuators to achieve a workspace of 8.4 12.8 8.8 mm3 and 1.1 1.0 1.9 (Chen and Culpepper 2006). Another MEMS-based compliant stage shown in Fig. 49b isdriven by a single-axis chevron thermal actuator. When the actuator was tested without a specimen,a maximum displacement of approximately 800 nm was generated at a current of 15 mA (Zhuet al. 2006). The main advantage of the thermal actuation is having higher force generation as

0.00

100

200

300

0.2 0.4 0.6 0.8

Range motion (mm)

For

ce (

mN

)

1.0 1.2 1.4 1.6

Binary Ni-TiAnnealed @550°C15 min / Water Quenched

Steel X220CrVMO13–4Max. range 0.34 mm

a b

Fig. 47 A monolithic SMA compliant linear spring and its force-displacement characteristic (Bellouard and Clavel2004)

Fig. 48 Illustrations of the (a) bimorph and (b) chevron thermal actuator designs


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compared to the electrostatic actuation. However, it has lower bandwidth and requires good heatdissipation for continuous operations.

Electrostatic ActuatorsElectrostatic actuation is perhaps the most popular choice for driving the MEMS devices(Toshiyoshi and Fujita 1996; Rosa et al. 1998; Hung and Senturia 1999; Tsou et al. 2005; Chiouand Lin 2005; Borovic et al. 2006). Unlike the thermal actuation, it consumes very small amount ofpower and can operate in high bandwidth. Electrostatic actuator can also be easily fabricated andintegrated as part of the MEMS devices. Governed by Coulomb’s law, the electrostatic actuationuses the attraction force between two point charges to generate displacement or exert force.Assuming that two surfaces are extremely close, the actuation force, Fe, can be expressed as

Fe ¼ 1

2V 2eair

A

d2

(87)

where V represents the voltage and eair represents the permeability of air, while E¼ A/d2 where A isthe area, d is the gap between two areas, and E is the electric field.

Equation 87 suggests that the force generation is proportional to the surface area; the “comb”configuration is commonly used to enhance the force generation. A comb architecture consists oftwo sets of capacitor banks where each bank comprises of a parallel array of capacitors. Eachcapacitor has a pair of parallel surface. However, there can be many variations in the geometry ofthese comb-drives such as the typical parallel configuration shown in Fig. 50b, the spiral (Tanget al. 1989) and the T-bar configurations (Brennen et al. 1990). Figure 50a shows an example of anelectrostatic-driven MEMS-based X-Y-yz micro-/nanopositioning stage articulated by a parallel-

Fig. 49 (a) A 6-DOF microscale nanopositioner, mHexFlex, driven by six 2-axes bimorph thermal actuators (Chen andCulpepper 2006). (b) A MEMS-based compliant stage driven by a chevron thermal actuator (Zhu et al. 2006)


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kinematic architecture. Each electrostatic achieved a travel of 27 mm at 85 V, while the overallworkspace of the end effector is 18 mm2 1.72 when all three actuators are energized.

Electromagnetic ActuatorsElectromagnetic (EM) actuation is a driving scheme that offers noncontact, frictionless, and longtravel characteristics. Over the past two decades, compliant manipulators driven by EM actuationhave achieved millimeters of displacement and subnanometer positioning resolutions with highaccelerations and speed responses. To realize an EM-driving actuation, two types of techniques canbe employed, i.e., the solenoid actuation and the Lorentz-force actuation.

Solenoid Actuation Solenoid actuation is an EM technique based on the attraction ofa ferromagnetic moving part. Based on Fig. 51a, the fixed stator generates an attraction force topropel the ferromagnetic moving part towards it. This force is expressed as

Fig. 50 (a) Schematic of an electrostatic force actuation. (b) An X-Y-yz MEMS-based 3PRR parallel compliantmanipulator driven by electrostatic comb-drives (Deepkishore et. al 2008)

Fig. 51 (a) A schematic representation of an EM solenoid actuation and (b) an EM-driven 3-DOF planar motioncompliant manipulator and its schematic plot of all three motion (Chen et al. 2002)


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F ¼ mN2I2A

2g2(88)

where m is the magnetic permeability of the air, 4p 107 (N/A2), N represents the number of turns,I represents the applied current, A represents the area, and g represents the air gap between the statorand the moving part. The main advantage of this technique is that it generates higher force ascompared to the Lorentz-force actuation. However, it exhibits nonlinear characteristic due to thesquare term in g. Hence, the force increases significantly as the gap reduces.

Figure 51b shows an example of an X-Y-yz compliant manipulator driven by the solenoidactuation. Three pairs of solenoid actuators are used to produce 3-DOF decoupled planar motionwith a minimum positioning resolution of 50 nm over a workspace of80 mm23.52 mrad. Eachsolenoid actuator generates a driving force of 50 N with an input current of 0.5 Amp at an air gap of250 mm. This example also demonstrated that the solenoid actuation is not suitable for largetraveling range and has a nonlinear force-displacement and current-force relationship.

Lorentz-Force Actuation Lorentz-force actuation has a contrast characteristic as compared to thesolenoid actuation. This EM technique offers a linear current-force relationship and can be config-ured to deliver large traveling range with linear force-displacement relationship. Governed byLorentz’s law, the force generation is expressed as

F BILN (89)

where B represents the magnetic flux density from the permanent magnet (PM), I represents theapplied current, N represents the number of coil turns within the effective air, and L represents thecoil length per turn.

Commercially available Lorentz-force actuator shown in Fig. 52a is also known as the voice-coil(VC) actuator. With the VC actuator, the effective air gap represents the region with the presence ofmagnetic flux density emanates from a permanent magnetic source, i.e., PM. In this air gap, themagnetic flux density should be well distributed. Due to the low permeability of air, the magnitude ofthe magnetic flux density is usually lower than the remanence magnetic flux density of a PM. Thebest approach to overcome this limitation is to increase the size of a PM while maintaining a smalleffective air gap. Examples can be found in the VC actuators from H2W Technologies (Stroman2006) show in Fig. 52b and BEI Technologies (Speich and Goldfarb 2000) shown in Fig. 52c. Thesemagnetic circuits increase the magnetic flux density through larger PMs. Nevertheless, they stillrequire very small effective air gaps because the magnitude and uniformity of the magnetic fluxdensity decreases with respect to the increment in distance from the magnet-polarized surface.

Fig. 52 (a) Conventional VC actuators and other variations of VC actuators from (b) H2W Technologies (H2WTechnologies 2014) and (c) BEI Technologies (BEI Kimco Magnetics 2014)


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Due to its linearity and large displacement characteristics, there is a growing trend of usingLorentz-force actuation to drive the compliant manipulators (Teo et al. 2010a; Yang et al. 2008; Teoet al. 2013; Fukada and Nishimura 2007; Bacher 2003; Helmer 2006; Culpepper and Anderson2004). These multi-DOF compliant manipulators have demonstrated nanometric positioning reso-lution capability over a few millimeters and degree workspace. Figure 53a shows an example of a 3-DOF parallel compliant manipulator driven by three VC actuators fromBEI Technologies. Using theVC actuators, it achieved a positioning resolution of 50 nm over a workspace of 5 mm2 5.Customized Lorentz-force actuators for large force generation were also well demonstrated by the 3-DOF out-of-plane motion compliant manipulator shown in Fig. 53b. Using customized Lorentz-force actuators, it achieved 20 nm positioning resolution and 0.05 arcsec angular resolution overa workspace of 5 mm 5 5. With the customized actuators, the manipulator produces 150 N/Amp of output force.

Performance Trade-OffsElectrostatic and thermal actuators are commonly used in theMEMS-based compliant manipulators.The trade-off of using the electrostatic actuators is poor force generation. Switching to the thermalactuator may provide larger force generation but at the expense of lower-frequency responses. Formicro- to macroscale compliant manipulators, the PZTactuators and SMAmaterials are the potentialsolutions for driving them. Both offer limited travel range, but PZT actuators can deliver very highdriving force and bandwidth. On the other hand, SMA materials can be made into a monolithiccompliant manipulator. It is an attractive solution for minimizing moving masses and stiffness inactuating direction. Frommacroscale onwards, EM actuation can be an alternate solution to the PZTactuators. Solenoid actuators may have the advantage of large force-to-size ratio, but the Lorentz-force actuator has linear characteristic and large traveling range. Although PZT actuators havelimited stroke, high force generation and bandwidth are desirable characteristic for driving thecompliant manipulators. Being a solid-state actuator, PZT can still provide additional non-actuationstiffness (even when power-off) to the compliant manipulators. This benefit can never be providedby the EM actuation. To summarize the review of high-resolution positioning actuators, theperformance and trade-offs of each actuator are listed in Table 8.

High-Resolution SensorsA sensor is a transducer that converts one form of energy to another form, for example, a device thatresponds to or detects a physical quantity and transmits the resulting signal to a controller (Slocum1992). Hence, sensors are also considered as the “eyes” or “ears” of a control system that controls

Fig. 53 (a) A 3-DOF planar motion parallel compliant manipulator driven by commercial VC actuators and(b) a 3-DOF out-of-plane motion parallel compliant manipulator driven by customized Lorentz-force actuators


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a manipulator. Hence, it defines the achievable positioning resolution of a compliant manipulatoreven if the actuators have infinite positioning resolution. Between different sensing technologies,there are certain performance specifications that are crucial and are stated as follows (Slocum 1992):Resolution The smallest detectable change to the physical quantity.Accuracy An error in output causes by external disturbances such as the variations of temperature,

humidity, and atmospheric pressure.Noise The magnitude of the output which is not part of the change in the physical quantity.Linearity The percentage of variations in the constant of proportionality between the output signal

and the measured physical quantity.Frequency response The rate of change in output signal due to a change in physical quantity.

In general, a sensor can be classified into two categories: the nonoptical and optical sensors.Optical sensors provide analog or digital signals that are corresponding to the physical quantitychange by optical means, while the nonoptical sensors use other means of measurement. Such kindof sensors include capacitive sensor, hall effect sensor, inductive displacement sensor, variabledifferential sensor, inclinometers, magnetic scales, magnetostrictive sensor, and PZT-based sensor.Among these nonoptical sensors, the capacitive sensor is commonly used in the compliant manip-ulators due to its high-resolution nature. For the same reason, the optical sensors, which include theoptical encoders and interferometric sensors, are the popular choices for the compliant manipulators.

Capacitive SensorsA capacitive sensor determines the gap between a probe and a target by measuring the amount ofcapacitance formed between the two parallel surfaces, i.e., the face of the probe and the target, asshown in Fig. 54a. By applying voltage to one surface, an electric field will exist between the twosurfaces. Electric field is the result of the difference between the electric charges that are stored onthe surfaces. Hence, the capacitance is the “capacity,” which is formed between the two surfaces, to

Table 8 Performances of various actuators that are suitable for driving the compliant manipulator

Actuator Max. range (mm) Stiffness Linearity Accuracy (nm) Bandwidth Cost

Piezoelectric ~500 High Poor <10 High Medium

Electrostatic ~100 0 Square law <1 High Low

Thermal ~20 High Poor <10 Low Low

SMA ~10 Medium Poor <100 Low Medium

Solenoid ~300 0 Poor <1 Low Low

Lorentz-forcea ~50 0 Good <1 Medium MediumaSingle-phase non-commutation

Fig. 54 (a) Illustration of capacitance probe measurement. (b) A commercial capacitance probe from Lion Precision(Lion Precision 2014) and (c) its configuration


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hold the charges. The effect of the gap, g, and the surface area, A, on the capacitance,C, between twoparallel surfaces is expressed as

C ¼ eAg

(90)

where e is the dielectric constant of the material in the gap.Capacitive sensor offers a noncontact measurement with extremely high resolution of ~25 Å (2.5

109) and a typical accuracy of ~0.10–0.20 % over the full-scale range. However, it can onlyprovide a measuring range of up to 0.13 mm or subnanometer resolution and the frequencyresponse is up to 20–40 Hz. The measuring range can be increased but at the expense of losing thenanometric resolution. A unique characteristic of the capacitive sensor is the ability to detect a widerange of materials, e.g., metals, dielectric, and semiconductors. The sensor output is only affected bydifferent types of material surfaces but will not be affected with different contents with the samematerial. For example, a capacitive sensor calibrated over a stainless steel target can also be used tomeasure brass or aluminum target. Apart from the optical means, the capacitive field sensing is theonly nonoptical means of providing subnanometer resolution measurement capability. Hence, it iswidely used in high-precision compliant manipulators.

There are a couple of stringent requirements when using the capacitive sensors. First, there will bestray capacitance that affects the accuracy of the measurement. However, the stray capacitance canbe easily minimize by adding a guard around the sensing electrode and collimate the electric fieldlines between the sensor and the target as shown in Fig. 54c. The trade-off is that the size of the probewill increase due to the presence of the guard. Second, it is important to keep the surfaces of thesensor and the target parallel. Misalignment in parallelism of two surfaces will affect the accuracy ofthe measurement since the capacitance is proportional to the sensing area and gap between the probeand the target.

For the same reason, the third requirement is to ensure the high ratio of the sensing area to the gap.Having high ratio, i.e., huge sensing area with very small gap, means greater accuracy andresolution. Other benefits include minimizing the effect of electromagnetic waves on the accuracyand providing an averaging effect to the output. The forth requirement is to minimize environmentaldisturbances because the dielectric constant can be affected by the temperature, barometric pressure,humidity, and media type. Other than maintaining the environment, a second probe can be used tomeasure a fixed object to record a reference signal that captures the noise or drift cause by the changein barometric pressure, humidity, and temperature. Subtracting the measurement signal with thisreference signal concurrently will produce the actual displacement signal.

Fig. 55 (a) Illustration of an optical measurement principle. (b) A commercial optical linear encoder fromHEIDENHAIN Corporation (Heidenhain 2014)


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Optical EncodersOptical encoder operates based on the principle of counting the slits or windows within a scalethrough a light source and a photodiode as shown in Fig. 55a. Simply imagine a LED that shineslight on a scale, which has an array of equally spaced windows. Behind the scale and directlyopposite the LED is a photodiode. If the LED light transmits through these windows, the photodiodewill receive the transmitted light and generate signals or pulses. On the other hand, no signal will begenerated by the photodiode if the light is block by the material in-between two windows. Thismeasuring scheme can be configured to linear or angular optical encoder that is immune to electricnoise. Most optical encoders mainly produce digital output due to the working principle.

The resolution of optical encoder is dependent to the types of scale gratings. For example, a scalethat can pack more equally spaced slits (gratings) will have higher resolution. Based on thecommercial available optical encoders from HEIDENHAIN Corporation, a scale with the absolutegrating can deliver approximately 10- to 12-bits positioning resolution, a scale with the incrementalgrating can deliver 10- to 16-bits positioning resolution, and a diffraction grating can deliver up to21-bits positioning resolution. Like a ruler with equally spaced numbered markings, the absolutegrating has a fixed and low resolution. For the incremental grating, the quadrature and interpolationmethods are used to enhance the resolution of the encoder. For example, an incremental grating with20 mm pitch can deliver 5 nm per encoder count resolution based on 4,000 steps interpolation. Withsuch interpolation approach, the diffraction grating provides even higher positioning resolution dueto its smaller pitch size. The scale also comes in different materials, i.e., stainless steel tape, glassscale, and zerodur scale. The glass scale has a thermal expansion coefficient of 5.9 106/K, whilethe zerodur material has a thermal expansion coefficient of 0 0.007 106/K from 0 C to50 C. The selection of the material depends on the applications and desired specifications. For high-precision positioning tasks, the zerodur scale may be the best option, but it is the most expensiveamong the rest.

Laser Interferometer SensorsLaser interferometer sensor is the most accurate measurement system that can be applied to today’smechatronics and robotic systems. Hence, it can be used as working standards for machinecalibrations, measurement, and feedback control. Based on the typical size of a photon, theresolution of a laser interferometer sensor is up to 0.15 nm. It also has a measuring range that isbeyond 10 m and a measuring speed of up to 4.2 m/s. Most laser interferometer sensors use Helium-Neon as the laser source (wavelength, l, ¼ 633 nm). The accuracy of laser interferometer sensor isdependent on very stringent metrology conditions:

Fig. 56 Illustration of (a) a 1-DOF measurement setup and (b) a 3-DOF planar motion measurement setup using thelaser interferometer sensor


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1. Temperature stability of 1 C2. Humidity variation of less than 10 %3. Pressure variation of less than 0.25 mmHg4. Target mirror flatness of l/10 PV

Temperature and humidity can be actively controlled in an enclosed environment, e.g., a cleanroom, while the mirror flatness can be achieved through high-precision polishing technology. Asbarometric (environmental) pressure is more difficult to control, the usual approach is to setupa wavelength tracker or a metrology station to monitor the variations in the laser.

A laser interferometer sensor is a noncontact and relative displacement sensor. Hence, it will notgive an absolute displacement value but rather a value relative to the previous value. In addition, itcan only measure relative displacement change beyond a dead path, which represents the minimumlength requirement between laser interferometer sensor and the target. Setting up the laser interfer-ometer sensor requires optics components such as the beam splitters (BS) and retroreflectors toorientate the laser beam from the laser source to the moving stage. Figure 56a illustrates a setup tomeasure a 1-DOF moving stage via the laser interferometer sensor. A BS is used to split the beaminto two paths where one path accounts for the fixed reference position via the fixed retroreflector,while the other path accounts for the position of the moving stage. The displacement is given by the

Fig. 57 Measurement setups using (a) two single-axis interferometric encoders from Renishaw and (b) a three-axesinterferometric sensor from SIOS (SIOS 2014)


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position of the moving stage relative to the fixed reference position. Figure 56b illustrates anothersetup that measures a 3-DOFmoving stage. After the first BS from the laser source, 50% of the beamgoes to another BS that further splits it into two beams (25 % each) where one goes to a wavelengthtracker and the other measures the position along the x-axis. The other 50 % of the beam is also splitinto two beams (25 % each) that measure the position along the y-axis. All three 25 % beams alsomeasure the angular change about the z-axis. The wavelength tracker is used to monitor the accuracyof the beam, which could easily be affected by changing metrology conditions such as thetemperature, humidity, and pressure.

Fiber optics can be used to deliver the laser beam directly to the remote units with built-in opticsand detector. This technology minimizes the needs of using optic components and the setup time toorientate the beam from the laser source and can channel the beam from one laser source to multipleremote units. These units, termed the interferometric encoders, can be placed near the end effectorfor direct measurements. Figure 57a shows a pair of single-axis interferometric encoders fromRenishaw that were used to measure the displacement of the end effector of a compliant manipulator.Using the fiber optics technology, the laser beam from a laser source was channeled into twoseparated interferometric encoders without any loss in the resolution, i.e., 10 nm per count. Thetransmitter, optics, and detector are all built inside each encoder to simplify the setup, while the endeffector carried the metrology mirrors to reflect the transmitted laser beams back to individualencoder. Another example is the triple-beam interferometric encoder from SIOS shown in Fig. 57b.Fiber optics cable channels the beam from a laser source to a sensor head that provides threeindividual measuring beams. These beams are equally spaced in a predefined arrangement that iscapable of measuring 3-axes out-of-plane motion and provides a resolution of 1 nm per count fromeach beam. This encoder system also provides a metrology station that has a similar function asa wavelength tracker.

Performance Trade-OffsLaser interferometer sensor offers the most accurate measurement in modern world and has theability to provide a positioning resolution of up to 0.15 nm. It is a noncontact and relativedisplacement sensor that is commonly used to feedback to position of the end effector rather thanthe joints of a manipulator. The main limitation of laser interferometer sensor is that its accuracy canbe easily affected by changing metrology conditions. On the other hand, these changing metrologyconditions have less effects on an optical encoder since the scale and encoder head are placed inclose proximity. However, this arrangement limits the use of the optical encoders to single-axis ortwo-axes planar motion measurement of the joints rather than the end effector of a manipulator(except for 1-DOF and 2-DOF translational motion stages). The measuring range is limited by thelength of the scale, which typically range from 10 cm to 3 m. Although interpolation offers highresolution, the accuracy is always limited to 1 % of the grating pitch size. One method to overcomethis limitation is to directly relate the change in analog signal with the change in displacement. Suchcalibrations are also commonly done in motion systems that use the analog capacitance sensors. Justlike the laser interferometer sensor, a capacitance sensor also has very high positioning resolution.However, high resolution comes with the expense of small measuring range. Likewise, a capacitancesensor can be calibrated for larger measuring range but sacrifices on the resolution due to the effectsof stray capacitance and the instability of the dielectric constant within a large air gap. Nevertheless,all these technologies are effective measuring means if properly setup via controlled environmentand methodology conditions.


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Future Advancement

The first generation of compliant manipulators offered single-axis translational motion, while thesecond generation delivered multi-DOF planar or spatial motion through the aid of parallel-kinematic architectures. In both generations, the workspace of the compliant manipulators is limitedby the notch-hinge flexure joints and PZT actuators. By exploring the EM actuation and moreflexible beam-based flexure joints, the third generation of compliant manipulators has successfullybroken through the millimeter-range barrier encountered by the previous generations but limited tolow payload positioning applications. More recent research work has introduced the latest genera-tion of compliant manipulator, which offers high payload with large displacement capability,suitable for direct contact applications such as the UV-NIL process (Teo et al. 2010a). The nextgeneration of compliant manipulators is projected to be an integrated system fabricated through 3Dprinting technology. Termed as the Flextronics, these single monolithic compliant manipulators willhave actuators and sensors all printed and embedded into the compliant limbs or flexure joints.Synthesizing the Flextronics requires the advancement in the topological and dynamics optimizationof compliant manipulators. Using the 3D printing technology, the structure arrangement of thecompliant limbs or flexure joints will be isotopic rather than homogenous. Hence, new theoreticalmodels and modeling approach will be needed to predict the stiffness characteristic of theFlextronics. The advancement in material science will be the key to the realization of Flextronicssince this advanced technology requires newmaterial that can be actuated and sense while providingnecessary strength and stiffness to the overall structural integrity. Most importantly, the newengineered material must have high endurance limit and predictable fatigue life. Potential applica-tions of Flextronics can be advanced surgical tools and miniature robotic systems or even providingmacro- to nanoscale positioning tasks in vacuum and harsh environment.

Summary

Development of a compliant manipulator, which is articulated by the flexure joints, involves goodunderstandings on several fundamental topics associated with this advanced robotic system. Mate-rial science plays an important role in developing a reliable compliant manipulator. Accurateidentification of the fatigue life of those flexure joints is crucial as it varies according to the typesof materials, surface finishing, geometries, and loadings. Material properties also affect the stresslevel that each flexure joint can sustain under loading, bending, and torsional bending. Althoughlower Young’s module properties will lessen bending stiffness and thus the stresses, the compliantmanipulators may lose desirable stiffness in the non-actuating directions. Hence, the major chal-lenge in designing a compliant manipulator is to fulfill its desired functions and deflectionworkspace while maintaining the stresses well below the yield strength of the material. Solidmechanics provide important analytical tools for modeling the deflection stiffness of the flexurejoints. Classical bending-moment-curvature equation brought several key formulations in predictingthe stiffness and stresses of functional compliant mechanisms, while the large deflection theoremcovers the nonlinear characteristics of the flexure joints. Advancement in the theoretical studies onlarge deflection characteristic gave birth to the pseudo-rigid-body approximation model, which alsobecame the bridge between the classical rigid-body linkage mechanism and the compliant mecha-nism. This model was further enhanced by the semi-analytic approximation model, which providesa simple, quick, and generic solution for any form of flexure joint configuration. With theseapproximation models, the knowledge of rigid-body linkage mechanism, particularly the parallel-


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kinematic architectures, can be applied in synthesizing the compliant mechanism. Synthesizing ofthe compliant mechanism can also be conducted through the classical exact-constraint designapproach and the state-of-the-art topological optimization techniques. The synthesized compliantmechanism has to be driven by actuators to form a compliant manipulator. The actuators willultimately decide the traveling range, output force, stiffness, and even size or footprint of themanipulator. Yet, the high-resolution sensors will define the step resolution of the manipulator.With the relevant knowledge of each fundamental topic being covered in this chapter, it serves asa guide and reference for designing, analyzing, and developing a compliant manipulator.

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Index Terms:

3-legged Prismatic-Prismatic-Revolute (3PPR) parallel-kinematic architecture 353PPS parallel compliant mechanism 39Capacitive sensor 52Circular-shaped notch hinges 17École Polytechnique Fédérale de Lausanne (EPFL) 7Compliant manipulator 1–3, 9–11, 14, 18, 21, 25, 32, 34, 40, 44, 52, 57

advantages 1benefits 3classification 3compliant mechanism 1DOF compliant manipulator 2exact constraint design approach 34fatigue failure 14flexure-based compliant bearing 3future advancement 57high positioning resolution actuators 52high resolution sensors 52history 9non-contact bearings 3nonlinear large deflection theorems 18parallel-kinematic architecture 40pseudo-rigid-body model 21semi-analytical modeling 25small deflection theorems 18stiffness modeling 32strength and stiffness 10stress failure 11topology optimization approach 44

Compound linear spring mechanism 16Corner-filleted notch hinge 18Degrees-of-Freedom (DOF) compliant manipulator 2Delta3 compliant manipulator 8Double compound linear spring mechanism 16Electromagnetic (EM) actuation 49Electrostatic actuator 48Elliptical notch hinge 18Exact constraint design approach 32Fiber optics 56Finite Element Modeling (FEM) platform 36Flexure joints 3Flexure-based Electromagnetic Linear Actuator (FELA) 39Forward kinematic modeling 37Fully compliant manipulators 2High positioning resolution actuators 45–49, 51

electromagnetic actuators 49


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electrostatic actuators 48performance trade-offs 51piezoelectric (PZT) actuator 45shape memory alloy 46thermal actuators 47

High resolution sensors 52–54, 56capacitive sensor 53laser interferometer sensor 54optical encoder 54performance specifications 52performance trade-offs 56

Laser interferometer sensor 54Leaf-spring compound linear spring mechanism 5Lorentz-force actuation 50MEMS-based micro-actuator 9Michelson interferometer mirror 5Monolithic compound linear spring mechanism 5Nonlinear large deflection theorems 18Notch flexure joints 17Optical encoder 54Parallel linear spring mechanism 15Partially compliant manipulator 2Piezoelectric (PZT) actuators 45Planar motion compliant manipulators 7Pseudo-Rigid-Body (PRB) model 21Semi-analytical modeling 25Shape memory effect 46Slender strips 4Solenoid actuation 49Spatial joint compliant module 2Spherical-Prismatic-Spherical (SPS) serially-connected compliant limbs 7Stiffness modeling 32Superelasticity 46Thermal actuators 47


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