Handbook of Manufacturing Engineering and Technology
DOI 10.1007/978-1-4471-4976-7_102-1
# Springer-Verlag London 2014
Compliant Manipulators
Tat Joo Teoa*, Guilin Yangb and I-Ming Chenc
a
Mechatronics Group, Singapore Institute of Manufacturing Technology, Singapore, Singapore
b
Institute of Advanced Manufacturing, Ningbo Institute of Materials Technology and Engineering of the Chinese
Academy of Sciences, Zhenhai District, Ningbo, Zhejiang, People’s Republic of China
c
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, Singapore
Abstract
Compliant manipulators are advanced robotic systems articulated by the flexure joints to deliver
highly repeatable motion. Using the advantage of elastic deflection, these flexure joints overcome
the 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 are
suitable ideal candidates for micro-/nanoscale positioning tasks. This chapter presents the relevant
knowledge of several fundamental topics associated with this advanced technology. After reviewing
its evolution and applications, the principal of mechanics is used to explain the limitations of these
manipulators. Subsequent topic covers various theoretical modeling approaches that are generally
used to predict the deflection stiffness of flexure joints and stiffness characteristics of compliant
manipulators. Next, various fundamental design concepts for synthesizing the compliant mechanism will be introduced and several examples are used to demonstrate the effectiveness of these
concepts. The topic on actuation, sensing, and control summarizes the types of high-resolution
actuators and sensors which the compliant manipulators use to achieve high-precision positioning
performance. Performance trade-offs between various actuators and among different sensors are
discussed in detail. With this relevant knowledge, this chapter serves as a guide and reference for
designing, analyzing, and developing a compliant manipulator.
Introduction
A compliant manipulator is a motion system that consists of a compliant mechanism driven by highresolution positioning actuators. Unlike traditional rigid-link mechanisms, a compliant mechanism
gains its mobility from the deflection of flexible members (Howell 2001), which is termed as the
flexure joints. Using the advantages of elastic deflection, a flexure joint overcomes the limitations of
a 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 characteristics of these flexure joints. Driven by high-resolution positioning actuators such as the piezoelectric (PZT) solid-state actuators and the electromagnetic (EM) actuators, the compliant
mechanisms become high-precision manipulators that are ideal candidates for micro-/nanoscale
positioning tasks (Teo et al. 2010a; Yang et al. 2008; Ryu et al. 1997; Lee and Kim 1997; Jywe
et al. 2008). Figure 1a shows an example of a compliant manipulator developed by the Singapore
*Email: [email protected]
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Handbook of Manufacturing Engineering and Technology
DOI 10.1007/978-1-4471-4976-7_102-1
# Springer-Verlag London 2014
Fig. 1 (a) A 3-DOF out-of-plane motion compliant manipulator and (b) a 3-DOF in-plane motion compliant mechanism developed by SIMTech
Fig. 2 (a) A spatial joint compliant module developed by SIMTech and (b) a bearing-based spherical joint from
HEPHAIST SEIKO (S. HEPHAIST 2014)
Institute of Manufacturing Technology (SIMTech) to automate the out-of-plane alignment and
imprinting tasks for a Nanoimprint Lithography (NIL) process. This three degree-of-freedom
(DOF) compliant manipulator is driven by Lorentz-force EM actuators and has achieved
a positioning and angular resolution of 10 nm and 0.05 arcsec (0.242 106 rad), respectively, throughout a workspace of 5 5 5 mm.
Compliant manipulator can be classified as a partially or fully compliant manipulator. A partially
compliant manipulator has limbs that consist of both rigid bodies and flexure joints, while a fully
compliant manipulator has continuous flexible limbs. For example, the manipulator shown in Fig. 1a
is considered as a partially compliant manipulator since each limb is articulated by a combination of
rigid bodies and flexure joints. On the other hand, the mechanism shown in Fig. 1b is considered as
a fully compliant manipulator because each limb is formed by a continuous flexure joint without the
need of rigid bodies. Other than gaining performance, these compliant manipulators also benefit
from 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 monolithiccut joint that has significantly less parts as compared to a bearing-based spherical joint as shown in
Fig. 2b. Having less parts may shorten the manufacturing and assembly time and eventually reduce
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Handbook of Manufacturing Engineering and Technology
DOI 10.1007/978-1-4471-4976-7_102-1
# Springer-Verlag London 2014
cost. In addition, parts reduction may also reduce assembly errors, which often affect the accuracy
and 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 and
maintenance-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 is
a significant merit because the compliant manipulators could operate in harsh environments that may
damage or degrade the joints. For example, flexure joints made of Teflon could be used in chemical
solutions and even space exploration systems since no lubrication is required. The frictionless
characteristics of a compliant manipulator also suit the clean vacuum environment since no particles
will be generated through friction. Most importantly, reduction in part counts coupled with the
simple constructions of flexure joints becomes an attractive solution for developing microscale
compliant manipulators such as Micro-Electro-Mechanical Systems (MEMS) and macro-/
microscale nanopositioner (Chen and Culpepper 2006). Figure 1b shows a 3-DOF macroscale
fully 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 deflection
from the flexure joints is also accompanied by limitations. As all flexure joints are required to deflect
or bend within the elastic region of the materials, the deflection of these flexible members limits the
motion of the compliant manipulator. For example, a flexure joint cannot produce the continuous
rotation motion of a ball-bearing rotary joint. In general, the workspace of a typical compliant
manipulator is limited to a few millimeters and degrees as the stress concentration of each flexure
joint must not exceed the yield strength of the material. Other than stress–strain characteristics of the
materials, the force-displacement (or stiffness) characteristics of a flexure joint are also crucial to the
development of a compliant manipulator. Accurate prediction of the stiffness characteristics requires
in-depth knowledge of the principal of mechanics, deflection theory, mechanism synthesis, and
synthesis methods. Yet, from the recent advancements in compliant manipulator, flexure joints are
tasked to produce larger deflections, which exhibit nonlinear behavior. Hence, classical beam
equations derived from the small deflection theory are no longer valid. Even with sufficient
knowledge, manufacturing tolerances can easily result in uncertainties to the actual performance
of the compliant manipulator.
Dealing with flexure-based (or compliant) bearing involves the transfer or transformation of
stored energy from input to output. To overcome the stored energy, the energy used to create
a displacement tends to be higher as compared to other noncontact bearings. Under significant
stress, prolonged stress, or constant exposure to high temperature, the stored energy will also cause
a certain degree of hysteresis in the stress–strain characteristics resulting in creep effects. Although
stored energy can be reduced by lowering the displacement stiffness, this approach further reduces
the off-axis stiffness of the compliant manipulator, which is relatively lower as compared to motion
systems using other noncontact bearings. Thus, the compliant manipulators are not suitable for
carrying high payloads and any accidental overload may lead to instabilities, e.g., buckling. Fatigue
is another crucial factor that determines the reliability of a compliant manipulator since the flexure
joints are deflecting cyclically with constant load. Unfortunately, existing theoretical model, which
predicts the fatigue life, is only applicable to standard specimen shapes, e.g., rectangular and
circular. In addition, the material properties, geometries/dimensions of the flexure joints, and
types/amount of loadings have significant effects on the fatigue life. As a result, tedious and
meticulous evaluations are required to determine the life span of any compliant bearing in
performing its prescribed tasks.
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Fig. 3 H. A. Roland with his ruling engine for diffraction gratings (Jones 1988)
This chapter introduces the essence of compliant manipulators by presenting several fundamental
topics that are associated with this advanced technology. After the brief history on its evolution and
applications, the first topic covers the principal of mechanics, which is used to explain the limitations
of these manipulators. The next topic introduces the fundamental design concepts that are typically
used to synthesize the compliant mechanism. Subsequent topic covers various theoretical modeling
approaches that are generally used to predict the deflection stiffness of flexure joints and the stiffness
characteristics of the compliant manipulators. Following these fundamental topics, the material
properties, the types of fabrications, and the manufacturing limitations are discussed. The topic on
actuation, sensing, and control summarizes the types of high-resolution actuators and sensors which
the compliant manipulators use to achieve high-precision positioning performance. Lastly, the future
advancement of the compliant manipulator technology is presented.
Brief History
Implementation of flexure joints into precision machines can be dated as early as 1826, when metal
strips were first used to replace torsional members of a classical torsion balance to increase its
precision 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 with
meaningful observations of 109 rad change in orientations. A cross-strip hinge, which was later
introduced to increase the stiffness of non-actuating directions, was well adopted by subsequent
torsion balances. In 1902, such slender strips were used to support the ruling engine for grating
diffraction (Fig. 3) so as to avoid the effects of friction (Jones 1988). This slender strip, which is
termed as a leaf spring, is considered as the earliest form of a flexure joint. By the dawn of World War
II, these shock-proof torsional leaf springs have been increasingly used in electrical instruments to
replace jeweled pivots (Jones 1962). During the war, the flexure joints became widely used in
developing highly sensitive measurement instruments such as highly accurate load cells for force
measurement and the pendulum pivots of miniaturized force-balance accelerometers (Motsinger
1964; Tuttle 1967).
After the war, the compliant mechanisms were widely used to develop metrology systems when
precise positioning of optical lens or mirrors was needed (Jones 1951, 1952, 1955, 1956). For
example, the optical slit mechanism of an infrared spectrometer shown in Fig. 4a is a compliant
mechanism. Using leaf springs to support the slit jaws, the resolution of the micrometer is directly
translated onto the optical slit mechanism since there is no backlash between the micrometer and the
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# Springer-Verlag London 2014
Fig. 4 (a) An optical slit mechanism, (b) a parallel spring mechanism for positioning the Michelson interferometer
mirror, 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)
jaws. Due to their ability to provide direct transmission between the input and the output, leaf springs
were 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 leaf
springs and Fig. 4c shows a seismograph that used these compliant bearings to achieve direct
measurement of the amplitude of earthquakes without losses through friction. Using leaf springs as
flexure joints requires additional assemblies that generally affect the precision of a mechanism. In
addition, the leaf springs have poor stiffness in other non-actuation directions, which will further
deteriorate precision when subjected to off-axis external loading.
To address these issues, the notch hinge was introduced as an alternative solution for creating the
compliant mechanism. A monolithic-cut approach was used to produce such notch hinges and the
entire mechanism could be fabricated from a single piece of workpiece without involving any
assembly. Figure 5 illustrates two compound linear spring compliant mechanisms constructed by the
leaf springs and notch hinges, respectively. The notch hinges were localized flexure joints that
formed only a small portion of the entire mechanism. As a result, the off-axis (or non-driving
directions) stiffness of the monolithic compound linear spring compliant mechanism is more
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Fig. 6 (a) A 1-DOF compliant stage developed by SIMTech (Ho et al. 2004) and (b) a 2-DOF nested compliant stage
developed by NIST (Boone et al. 2002)
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 predict
the deflection stiffness of these notch hinges. Monolithic-cut fabrication also reduced the accumulative assembly errors that could potentially cause the motion of the compliant mechanism to be
indeterministic. Thus, compliant mechanisms that used notch hinges were more predictable than
those with leaf springs.
Between the late 1960s and the early 1990s, compliant mechanisms were mainly developed via
the 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 parallel
linear spring concept that resembles the classical four-bar linkage mechanism. To double the
displacement range, a compound linear spring concept, which is formed by connecting a pair of
linear spring in series, was introduced. Examples of the compound linear spring compliant mechanisms were illustrated in Fig. 5. To further enhance the robustness of these stages towards external
disturbance, a symmetrical double compound linear spring concept was introduced. Figure 6a shows
an example of a single DOF compliant stage that was developed via a double compound linear
spring concept and notch hinges. Driven by a PZT actuator, it has a positioning resolution of 50 nm
over 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 of
another in series. Another approach was to nest a 1-DOF compliant stage within another stage. This
approach was used to develop several multi-DOF nanopositioning stages (Her and Chang 1994; Gao
et al. 1999). Figure 6b shows an example of an X-Y translational optical lens steering stage
developed by the US National Institute of Standard and Technology (NIST) for space communication purposes (Boone et al. 2002). This 2-DOF PZT-driven compliant stage was developed based on
a nested approach where the y-axis stage is embedded inside the x-axis stage. Precisions of these
stacked and nested compliant stages are usually affected by the accumulative positioning errors. The
responses of these stages are also slow because the lower stage needs to carry the inertial mass of the
upper stage via the stacked approach and the outer stage carries the inertial mass of embedded stage
for the nested approach. Inertial masses of the upper and embedded stages from both approaches also
caused these 2-DOF compliant stages to have nonsymmetric natural frequencies.
Towards the late 1990s, the parallel-kinematic architectures were widely adopted by modern
compliant manipulators to achieve a higher precision and better performance in multiple DOF
manipulations. This architecture plays an important role in the success of these manipulators due to
its advantages of a lower inertia, programmable centers of rotations, superior dynamic behavior, and
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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)
insensitivity to external disturbances, e.g., thermal expansion. Most importantly, the limited deflections of flexure joints also suit the limited workspace characteristics of a parallel-kinematic architecture. Figure 7a shows an example of a planar motion compliant manipulator developed based on
a parallel-kinematic architecture. Being used as an X-Y-yz wafer positioning stage, this compliant
manipulator delivers a positioning resolution of 8 nm along the x- and y-axes and a rotational
resolution 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 developed
based 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 an
X-Y translational workspace of 100 mm2 with a rotational range of 0.1 . A similar concept was found
in another X-Y-yz compliant manipulator as shown in Fig. 7b, which was used as a precision stage for
positioning the samples within a Scanning-Electron-Microscope (SEM) machine (Yong and Lu
2009).
With the advancement of electrical discharged machine (EDM) in the 1990s, the dimensions of
flexure joints can be fabricated more precisely. Hence, the modern flexure joints played an important
role in realizing various types of spatial motion compliant manipulators (Han et al. 1991; Hudgens
and 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-Z
spatial motion compliant manipulator was constructed based on the kinematics of a Delta robot
where 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 a
positioning repeatability of 100 nm within a workspace of 1 cm3. Figure 8b shows another
interesting 3-axes translational motion compliant manipulator that was developed by Nanyang
Technological University (NTU). Other than using a three-limbed parallel configuration to
construct this compliant manipulator, each macroscale compliant limb is articulated by a 3RRR
parallel-kinematic architecture. As a result, all three translational axes are kinematically decoupled
and 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
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Fig. 8 (a) Delta3 compliant manipulator developed by EPFL (Henein 2000) and (b) a 3-axes translational motion
compliant 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 compliant
manipulator developed by MIT (Zuo et al. 2003)
accuracy of 2 m rad (Oiwa and Hirano 1999). Due to the limited displacement stroke of PZT
actuators, it could only achieve 10 mm in all translational directions, 200 mrad about the x- and
y-axes, and 100 mrad about the z-axis. Figure 9b shows another example of a 6-DOF compliant
manipulator developed by Massachusetts Institute of Technology (MIT). It was constructed based
on a hybrid concept by stacking an X-Y-yz compliant mechanism on top of a Z-yx-yy compliant
mechanism (Zuo et al. 2003). As a result, the planar motion is decoupled with the out-of-plane
motion. 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 achieved
a workspace of 5 mm 2.4 2.4 .
Today, the compliant manipulators can be fabricated in the micron level through the semiconductor photolithography process. Driven by electrostatic actuation, thermal actuation, and even
microscale gears, these compliant manipulators become MEMS devices such as microsensors and
micro-actuators. Figure 10a shows a MEMS-based micro-actuator developed by MEMS and
Nanotechnology Exchange. The compliant mechanism adopts a double compound linear spring
concept with slender hinges with the translator driven by a pair of electrostatic actuators. The latest
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Fig. 10 (a) A MEMS-based micro-actuator from MEMS and Nanotechnology Exchange and (b) a 6-axes microscale
nanopositioner developed by MIT (Zuo et al. 2003)
and most noticeable development of microscale compliant manipulator comes from MIT, Precision
Systems Laboratory. The research group presented a 6-axes microscale nanopositioner, termed
mHexFlex, based on a fully compliant design concept as shown in Fig. 10b. The mHexFlex consists
of a central stage that is connected to the base via three parallel flexure joints where each joint is
driven by two microscale thermal mechanical actuators. As each thermal actuator has 2-DOF
actuation, i.e., in-plane and out-of-plane deflections, a combination of six actuators delivers a
6-DOF motion to the system. mHexFlex registered a positioning error of 10 nm over a workspace
volume 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” with
high stiffness ratios. The basic working principles of these flexure joints are elastic bending and
torsion. The advantages of elastic bending or torsional motion include frictionless, contactless, and
non-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 section
reviews the characteristics of the elastic bending and torsional motion based on the principal of solid
mechanics.
Strength and Stiffness
The strength and stiffness offer different insights to the deflection of the flexure joint. Strength
determines the stress a deflected flexure joint can withstand before failure and is associated with the
property of the material. On the other hand, stiffness determines how much a flexure joint deflects
due to a load. Based on the Bernoulli-Euler law, the bending moment is proportional to the beam
curvature
M ¼ EI
dy
ds
(1)
where M represents the bending moment, dy/ds is the rate of change in the deflection angle along the
curvature of the beam, and EI represents the bending rigidity with E representing the Young’s
modulus (modulus of elasticity) of the material and I is the cross-sectional moment of inertia (the
second moment of area).
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Fig. 11 A cantilever beam subjected to two independent loads along the x- and y-axes
The bending rigidity is a function of its material properties and the geometries due to the presence
of the Young’s modulus and the cross-sectional moment of inertia. Figure 11 illustrates a cantilever
that 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 governed
by 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 rigidity
along the y-axis. This example explains that even with similar Young’s modulus and strength in both
directions, the stiffness between both directions may not necessarily be the same. It also highlights
that 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. For
example, an aluminum (E ¼ 71 GPa) beam will be approximately 3 times more flexible than
a steel (E ¼ 210 GPa) beam, while the flexibility of a Teflon (E ¼ 0.5 GPa) beam will be 142 times
higher than the aluminum beam. These comparisons also highlight the importance of material
selection in the flexure joint design.
Stress Failure
In general, the deflection of a flexure joint is limited by the bending or torsional stress. As these joints
only operate within the elastic region, the yield strength of the material becomes the maximum
allowable stress and the maximum stress generated via the deflection must be kept below it. In
theory, 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 into
Eq. 1 and let c ¼ h/2, the maximum stress due to bending moment about the x-axis is expressed as
h
dyx
smax ¼ EI xx
(3)
2l
ds
By referring to Eqs. 2 and 3, the maximum bending stress is proportional to the maximum bending
moment and geometry also plays an important role in reducing the stress. In addition, selecting
materials with low Young’s modulus could also reduce the stress. Yet, materials with low Young’s
modulus would have lower yield strength. Based on a stress–strain curve shown in Fig. 12, the
Young’s modulus of a material is calculated based on the linear stress–strain relationship, while the
yield strength is beyond the proportional limit. Hence, choosing a material with lower Young’s
modulus will effectively lower the maximum allowable stress. Yet in some cases, higher Young’s
modulus is desirable for achieving higher stiffness characteristics. Consequently, the major
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Fig. 12 Stress–strain curve of a ductile material
Fig. 13 S-N curve of steel material
challenge in designing a flexure joint is to obtain the desired deflection that could fulfill the function
of the compliant mechanism while maintaining the stresses well below the yield strength of
the material. Table 1 summarizes this section by listing some materials that are commonly used to
develop the compliant mechanism.
Fatigue Failure
Any flexure joint that flexes to deliver motion is subjected to fatigue failure. The fatigue life of any
material 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, high
cycle, and infinite life (Howell 2001). For the low cycle category, the fatigue failure usually occurs
between 1 and 1,000 cycles. As for the high cycle category, the fatigue failure typically occurs
beyond 1,000 cycles. The infinite life region is for the flexure joints that required to flex constantly
and 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 of
the material. The first limit that bounds the low cycle region is represented by SL, while the second
limit that bounds the finite life is known as the endurance limit, Se. This limit is common in many
low-strength carbon and alloy steels, some stainless steels, irons, molybdenum alloys and titanium
alloys, and certain polymers (Dowling 1993). If the stress is kept below the endurance limit,
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0
0
Table 1 Initial approximation values of Se and Sf for some materials
Material
Steel
Classification
Endurance limit
Iron
Endurance limit
Aluminum
No endurance limit, fatigue estimation for 5 108 cycles
Values
0
Se 0.5Sut
0
Se 700 MPa
0
Se 0.4Sut
0
Se 160 MPa
0
Sf 0.4Sut
0
Sf 130 MPa
Conditions
Sut < 1,400 MPa
Sut 1,400 MPa
Sut < 400 MPa
Sut 400 MPa
Sut < 330 MPa
Sut 330 MPa
continuous cycle without fatigue failure is possible and the flexure joint has infinite life. From past
literature (Juvinall 1967), SL can be approximated as
S L ¼ cf S ut
(4)
where
cf ¼
0:9
0:75
bending
axial loading
(5)
For low cycle fatigue estimation, the maximum stress must not exceed SL. For high cycle fatigue
estimation, Sf can be approximated as
S f ¼ af N bf
(6)
where
2
cf S ut
af ¼
Se
cf S ut
1
bf ¼ log
3
Se
(7)
Based on Eq. 6, the number of cycle can be estimated by assuming that smax ¼ Sf. For materials
without endurance limit, af and bf are expressed as
ðlogN 2 Þ logcf S ut 3logS f 2
cf S ut
1
bf ¼
log
(8)
logaf ¼
3 logN 2
logN 2 3
Sf 2
where Sf1 ¼ cfSut, N1 ¼ 1 103, Sf2 ¼ Se, and N2 ¼ 1 106. For materials with endurance
limit, Se
0
obtained through these specimen tests are termed as uncorrected endurance limit, Se. For materials
without endurance
limit, the Sf obtained via the specimen tests are known as uncorrected fatigue
0
0
strength,
Sf. As information is often unavailable, Table 1 listed some initial approximations of Se and
0
Sf for 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
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Table 2 Curve-fitted a and b parameters for surface finishing
Surface finish
Ground
Machined, cold rolled
Hot rolled
As forged
a (MPa)
1.58
4.45
58.1
271
b
0.086
0.265
0.719
0.995
Table 3 Marin correction factors for size
a
Csize
1
d 0:1133
Conditions dependent on the diameter of the sample, d
For d < 2.79 mm
If d is in inches and 0.11 d 2 in.
0:3
d 0:1133
7:62
If d is in millimeters and 2.79 d 51 mm
For d > 51 mm
pffiffiffiffiffi
For rectangular shape subjected to zero-torsional bending, d ¼ 0:808 bh
0.6
a
S e ¼ csur f csize cload creliab S 0e
(9)
S f ¼ csur f csize cload creliab S 0f
(10)
where csur f represents the Marin correction factor for surface finishing, csize for size, cload for
loading, and creliab for reliability. csur f can be approximated as
b
aS ut if aS but < 1
csur f ¼
(11)
1
if aS but 1
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), cload ¼ 1 for bending load, cload ¼ 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 the
continuous evolution of the flexure joints and the applications of the compliant manipulators. From
the very beginning, modeling of the flexure joints from classical bending-moment equation was
sufficient when the flexure joints were expected to deliver small deflection motions. As the desire for
large deflection increased, analytical models focusing on nonlinear deflection behavior of the flexure
joints were introduced by the late 1960s. When high-precision compliant manipulators were needed
in the semiconductor industry in the 1990s, the evolution of notch-hinge flexure joints spurred the
efforts in finding more accurate modeling approaches. These efforts continue till today due to new
beam-based flexure joints that were introduced for delivering large deflection motions. After
understanding the limitations of the flexure joints and the design constraints, this section presents
a comprehensive library of theoretical models and modeling methods that can be useful for modeling
different types of flexure joints.
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Table 4 Marin correction factors for reliability for steel material
Reliability (%)
creliaba
a
50
1.000
90
0.897
99
0.814
99.9
0.753
99.99
0.702
99.9999999
0.520
Assuming a standard deviation of 8 %
Small Deflection Theorems
Leaf spring is considered as the earliest form of a flexure joint. It can be modeled as a cantilever
beam based on the Bernoulli-Euler law. From Eq. 1, the beam curvature due to a bending moment
can be represented in rectangle coordinates
d 2 y=dx2
M ¼ EI
f ðx; yÞ
(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-momentcurvature equation given as
M ¼ EI
d2y
dx2
(13)
For a cantilever beam subjected to an end load shown in Fig. 14a, the summation of moment gives
M ¼ P(l x). By solving Eq. 13 with M ¼ P(l x), the maximum deflection along the y-axis occurs
at x ¼ l and is expressed as:
dmax ¼
Pl 3
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
3Edmax h
2l 2
(15)
For pure translation motion as shown in Fig. 14b, the summation of moment gives M ¼ P(l s x).
By solving Eq. 13 with M ¼ P(l s x), the maximum translation motion along the y-axis, which
occurs at x ¼ l and s ¼ l/2, is expressed as
dmax
Pl 3
¼
12EI
(16)
Using Eq. 2 with M ¼ Pl and c ¼ h/2, the maximum stress is given as
smax ¼
3E dmax h
l2
(17)
The pure translation motion equation expressed in Eq. 16 is useful for finding the translation
stiffness of a parallel linear spring mechanism shown in Fig. 15a. To avoid parasitic torsional
motion, this compliant mechanism employs two parallel leaf springs to achieve a pure translation
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Fig. 14 (a) A beam that is subjected to an end load and (b) delivers a pure translation motion
Fig. 15 (a) Parallel linear spring mechanism and its (b) front view and (c) side view
(or prismatic) motion along the y-axis. Based on the same notations used to define the geometries of
a cantilever beam shown in Fig. 11, the translation stiffness along the y-axis, K dy Py, which is twice of
Eq. 16, is expressed as
K dy P y ¼
Py 24EI
¼ 3
dy
l
(18)
To achieve maximum translation motion from a given load, Py, the loading point must be located
at l/2 away from the base as illustrated in Fig. 15b. Although the stiffness is doubled due to the
parallel configuration, the amount of deflection remains unchanged and the maximum stress is
similar to a spring leaf-spring configuration, which is given in Eq. 17. Based on Eq. 18 with Iyy, the
translation along the x-axis, K dx Px , is expressed as
K dx P x
3
b
¼ 2Eh
l
(19)
The torsional stiffness about the x- and z-axes is expressed as
K yx M x ¼
K yz M z
Ebhe2
2l
Ehe2 b 3
¼
l
2
(20)
(21)
Any translation motion provided by the linear spring mechanism will be accompanied by
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Fig. 16 (a) Compound linear spring mechanism and (b) double compound linear spring mechanism
a parasitic height variation, l. Figure 16a illustrates a compound linear spring mechanism that can
eliminate or minimize this error. Using two linear spring mechanisms connected in series, the
parasitic height variation of the moving platform can be canceled out by the parasitic height variation
of the intermediate platform. Due to the series connection, the translation stiffness is half of the linear
spring mechanism. Figure 16b shows a double compound linear spring mechanism, which is
a symmetrical concept of a compound linear spring mechanism. It was introduced to obtain superior
rectilinear 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 the
translation stiffness along the y-axis and maximum stress for both types of compound linear spring
mechanisms.
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 or
avoided to make the compliant manipulators more deterministic. The simplest form of a notch hinge
shown Fig. 17a has a circular shape profile, which incorporates a circular cutout on both side of
a blank to form a necked-down section. This necked-down section, which serves as a fixed center of
rotation, exhibits a pure rotational motion within a small dedicated range. In 1965, Paras and
Weisbord (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
K yz M z ¼
M z 2Ebt5=2
¼
yz
9pR1=2
(22)
For circular shapes defined by t < R < 5 t, the angular stiffness is expressed as
K yz M z ¼
Ebt 3
24kR
(23)
where the correction factor, k, is given as
t
k ¼ 0:565 þ 0:166
R
(24)
The maximum stress is determine by
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Table 5 Translation stiffness along the y-axis and maximum stress for different types of compound linear spring
mechanisms
Type
Single
Double
Translation stiffness, K dy Py
Maximum stress, smax
12EI
l3
24EI
l3
3Edmax h
2l 2
3Edmax h
2l 2
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 spring
mechanisms
smax ¼
k t Et
ymax
4kR
(25)
where the stress correction factor, kt, is given as
kt ¼
2:7t þ 5:4R
þ 0:325
8R þ t
(26)
The monolithic translation motion compliant mechanisms shown in Fig. 18 are constructed via
the circular notch-hinge flexure joints. Unlike the leaf-spring versions, each limb is formed by a pair
of notch-hinge joints connected in series. Table 6 lists the translation stiffness and the maximum
stress 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 and
King 1996; Tseytlin 2002; Lobontiu et al. 2002; Lobontiu et al. 2004; Yong et al. 2008). Examples of
these flexure joints include the elliptical shape shown in Fig. 17b and the corner-filleted shape hinge
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Table 6 Translation stiffness and maximum stress for different notch-hinge type mechanisms
Type
Linear spring mechanism
Single compound linear spring mechanism
Double compound linear spring mechanism
Translation stiffness, K dy Py
Maximum stress, smax
Ebt 3
6kℒ2 R
Ebt 3
12kℒ2 R
Ebt 3
6kℒ2 R
k t Etdmax
4kRℒ
k t Etdmax
8kRℒ
k t Etdmax
8kRℒ
shown in Fig. 17c. Based on past literature (Lobontiu 1962), the angular compliance of the elliptical
notch hinge can be expressed as
"
rffiffiffiffiffiffiffiffiffiffiffiffiffi#
2
yz
12l
6c
ð
2c
þ
t
Þ
4c
(27)
¼
6c2 þ 4ct þ t2 ; þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan 1 þ
C yz M z ¼
2
2
2
t
M z Ebt ð2c þ t Þð8c þ t Þ
t ð4c þ t Þ
For the corner-filleted notch hinge, the angular compliance is given as
(
)
12
2r
C yz M z ¼
1 2r þ
Ebt3
ð2r þ t Þð4r þ t Þ3
"
rffiffiffiffiffiffiffiffiffiffiffiffiffi#
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
4r
t ð4r þ t Þ 6r þ 4rt þ t 2 þ 6rð2r þ t Þ2 t ð4r þ t Þ; arctan 1 þ
t
(28)
Many studies were conducted to find the optimal elliptical and corner-filleted shapes. Figure 19
plots the results obtained from one investigation (Henein 2006) that presented one graph that plots
the 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 ellipse
ration, ry/rx, of the elliptical shape hinge must not exceed 0.025. Investigation results also suggest
that 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 % lower
than the elliptical shape hinge of R ¼ 2tmin and 5 times lower than a circular shape hinge. Yet, it is
still 13 % higher than the ideal prismatic beam
Nonlinear Large Deflection Theorems
As compliant manipulators progressed to larger displacement, the ideal beam shape became
a promising solution due to its low stress but large deflection characteristics. However, delivering
large deflection means that these flexure joints will experience parasitic shift in the pivot point and
exhibit nonlinear characteristics. Figure 20a shows that with a fixed pivot point, the ideal deflection
path is a concentric arc about the pivot point. Once there is a shift in the pivot point, the deflection
path is altered causing a variation between the actual and targeted deflected position. This shift is
commonly known as the parasitic shift.
The effects of parasitic shift are accounted by the Bernoulli-Euler law expressed in Eq. 12. In
small deflection theory, the slope dy/dx is assumed to be zero resulting in the derivation of the
classical bending-moment equation. However, this assumption is invalid for large deflection
analysis. Considering the f(x, y) within Eq. 12, Table 7 lists the values from f(x, y) due to the
changes of slope. When the deflection angle is small, the effects of f(x, y) are negligible. At 26.6 , the
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Stress from elliptical shape
Stress from corner-filleted shape
0.06
0.12
0.055
0.1
0.045
σ/(Eα)
σ/(Eα)
0.05
Analytical
FEM
0.04
0.08
0.06
0.035
Analytical (Gross stress)
FEM (Gross stress)
FEM (Local stress)
0.03
0.04
0.025
0
0.05
0.1
0.15
0.02
0.2
0
2
4
6
ry /rx
8
10
r / tmin
l
rx
Circular
Elliptical
Corner-filleted
r
ry
Ideal beam
tmin
R
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
value of f(x, y) increases to 40 %. By 45 , the value of f(x, y) increases up to 2.8 times higher than the
initial value. Consequently, this investigation shows the importance of f(x, y) as it will account for
the nonlinearity behavior of the large deflection.
For large deflection analysis, the angular compliance of a cantilever beam subjected to a moment
loading at free end (Fig. 21c) can be derived directly from Eq. 1 and written as
C yM ¼
l
EI
(29)
By applying the cross-product rule on Eq. 1, the deflection along the x- and y-axes can be
expressed as
dx ¼ l l sin y
y
(30)
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Table 7 Effects of slope on f(x, y)
y (deg)
0.6
5.7
26.6
45
dy/dx
0.01
0.10
0.50
1.00
f(x, y)
1.0001
1.0150
1.3976
2.8281
Fig. 21 (a) A 1-DOF cross-spring pivot that delivers (b) angular rotation via (c) large deflection of cantilever beams
dy ¼
lð1 cos yÞ
y
(31)
Equation 29 can be used to find the angular compliance of a 1-DOF cross-spring pivot shown in
Fig. 21a by considering it as a pair of cantilever beams subjected to an external moment loading at
free 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 EI
dy
¼ P ð l x dx Þ
ds
(32)
By integrating Eq. 32 by s yields
2
Solving Eq. 33 with a2 ¼ Pl
EI
1
dy 2P
¼
ð sin y0 sin yÞ2
ds EI
Ð
and an assumption that y0ds ¼ l yields
1
a ¼ pffiffiffi
2
ð y0
0
dy
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sin y0 sin y
(33)
(34)
Elliptic integrals or numerical integration can be used to solve Eq. 34. As there are many examples
and sources (Howell 2001; Frisch 1962; Byrd and Fredman 1954), this section will not go further to
obtain the closed-form solution.
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Fig. 22 (a) A cantilever beam subjected to a perpendicular point loading at free end and (b) its corresponding
representation based on PRB modeling
Pseudo-Rigid-Body Model
For large deflection analysis, the elliptic integrals or numerical integration can offer closed-form
accurate solutions. However, it is observed that these methods are cumbersome during the design
stage of compliant mechanism. To make analysis of large nonlinear deflection easy to predict yet
accurate, 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 to
a pivot point. In PRB modeling, this pivot point is predefined based on the types of loading and the
nature of the flexure joint. Another uniqueness of this method is a torsional spring, K, which is
attached to each pivot. This spring governs the torsional stiffness of the flexure joint. Revisit the
problem of a cantilever beam subjected to a perpendicular point loading at free end. Figure 22b
shows the equivalent PRB model representation with a characteristic pivot and a characteristic
radius that defines the deflection path. Based on PRB modeling (Howell 2001), the deflection along
the x- and y-axes can be approximated by
c ¼ l ½1 gð1 cos YÞ
(35)
dy ¼ gl sin Y
(36)
where g ¼ 0.85 for this specific case. For accurate prediction using these equations, the deflection
angle must maintain below 64.3 . With the torsional spring governing the angular stiffness, the force
and angle are related by an applied torque about the characteristic pivot. Hence, the relationship
between the load and angle is given as
P¼
KY
gl sin ðf YÞ
(37)
where K is the spring constant and f represents the angle of the load (e.g., vertical load gives f ¼ p2).
The spring constant is expressed as
K¼b
EI
l
(38)
where b ¼ 2.25 for this specific case. For other cases of configurations or loading conditions, the
PRB method offers specific representation, modeling approach, and parameters for each case. Some
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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
examples are shown in Fig. 23 where each individual configuration or loading condition has
a specific PRB modeling representation.
In the case of a small length flexure pivot configuration shown in Fig. 23a, the flexure joint is
coupled with a rigid link to amplify the deflection. Based on the PRB modeling (Howell 2001), the
deflection along the x- and y-axis can be approximated by
l
l
cos Y
(39)
c¼ þ Lþ
2
2
dy ¼
l
sin Y
Lþ
2
(40)
where l represents the flexure joint length and L represents the rigid-link length. The relationship
between the load and angle is given as
P¼
Lþ
l
2
KY
sin ðf YÞ
(41)
where K is expressed in Eq. 38 with b ¼ 1. For accurate analysis, L l must be satisfied, e.g., L must
be at least 10 times greater l.
For a cantilever beam with pure translation motion, this case can be modeled as a fixed-guided
cantilever beam shown in Fig. 23b. Based on the PRB modeling (Howell 2001), the deflection along
the x- and y-axis can be approximated by
c ¼ l ½1 gð1 cos YÞ
(42)
dy ¼ gl sin Y
(43)
where g ¼ 0.8517 for this specific case with a constant vertical load and reaction moment. The spring
constant for each torsional springs is given as
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Fig. 24 Different flexure configurations for different types of flexure joints found within one compliant manipulator
K ¼ 2gK Y
EI
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, with
g ¼ 0.7346. In this case, the spring constant is expressed in Eq. 38 with b ¼ 1.5164. There are other
cases that have been presented in past literature (Howell 2001; Howell et al. 1996; Howell and
Midha 1994). This section only provides a few cases and the remaining cases can be found from
these sources. One important note is that the PRB modeling method offers simplistic and accurate
solutions for analyzing various flexure configurations and loading conditions. Yet, the most significant contribution is that the PRB modeling method has linked the classical linkage mechanism and
the compliant mechanism together. Using the unique concept of adding a torsional spring to each
pivot point to describe the angular stiffness, the knowledge of rigid-body mechanism can be used to
design a compliant mechanism.
Semi-Analytic Model
The PRB modeling method has specific locations to place the pivot points, specific values for the
torsional spring constant, and specific representations for each flexure configuration and loading
conditions. As a result, any misjudgment and inappropriate selection of these PRB models often lead
to inaccurate results, especially for large deflection analyses. Figure 24 shows different types of
flexure joints found within a compliant manipulator. Analyzing each flexure joint with specific
flexure configuration requires a person who is well verse in PRB modeling method. In addition,
pairing a PRB model with a flexure configuration during the design stage is extremely restrictive and
could lead to inaccurate analysis. This is because designing a flexure joint often goes through an
iterative process of changing the geometries of the rigid links or the flexure joints to achieve desired
stiffness and off-axis stiffness within a given size constrain. Considering the flexure joints with
flexure configuration of L l shown in Fig. 24, the configuration could have change to L < l or L ¼ l
during 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 all
forms of flexure configurations.
A semi-analytic modeling method offers a generic, simple, and quick solution for analyzing the
nonlinear characteristic of large deflection motion produce from any flexure configuration (Teo
et al. 2010b). The term flexure configuration represents a flexure joint coupled with a rigid link
shown in Fig. 25a. The force, F, applied at the rigid-link end becomes a moment load at the end of the
flexure joint shown in Fig. 25b. Based on the derivations presented in past literature (Teo
et al. 2010b), the deflection of a generic flexure configuration along the driving direction can be
approximated as
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Fig. 25 (a) A generic flexure configuration subjected to a loading at the rigid-link end and (b) a close-up on the large
deflected 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
D¼
l
sin y
Lþ
2ˆ
(45)
where a represents the deflection angle and ˆ is a Sinc function, i.e., ˆ ¼ sin y/y. Referring to
Fig. 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
l
l
Lþ
cos y
(46)
Dp ¼ L þ
2
2ˆ
The angular stiffness is derived based on the hypothesis that the relationship between the applied
torque and the deflection angle of the torsional spring is governed by a moment arm. This moment
arm is formed by the rigid link and a portion of the flexure joint as shown in Fig. 26a. This portion of
the flexure joint represents the distance from the center of the torsional spring to the coupling point
between the rigid link and the flexure joint. This portion of the flexure joint is termed as the changing
arm, S, and can be expressed as
S¼r
l
2ˆ
(47)
where r is introduced as
pffiffiffiffiffiffiffi
l 1:8 þ L
r¼
lþL
(48)
From Fig. 26b, the changing arm varies according to different flexure configurations even with
similar flexure joint lengths. To address this issue, r is an empirical factor that is used to determine
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the changing arm with respect to any flexure configuration. Subsequently, the changing torque, Ty, is
recognized as a tangential force, Ft, applied to the moment arm and is expressed as
rl
Ty ¼ F T L þ
(49)
2ˆ
The changing angular stiffness of a torsional spring is given as
Ky ¼
Ty
y
(50)
By substituting Eqs. 49 and 29 into Eq. 50, the relationship between the force and angle is
expressed as
EIy
F¼ p
rl
sin 2 y
l L þ 2ˆ
(51)
In semi-analytical modeling,
p
the changing angular stiffness of a torsional spring is expressed in
Eq. 51. With F T ¼ F sin 2 y , the vertical force, F, applied on the moment arm can be determined
directly based on a known deflection angle shown in Fig. 26a. Last but not least, the maximum
bending stress, smax, is given as
F l
l
h
þ Lþ
smax ¼
cos a
(52)
I 2
2ˆ
2
The uniqueness of the semi-analytic modeling is that it is generic for all flexure configurations. In
cases where there is no rigid link, L will be zero. For any other case, the presence of r accounts for
the change in length of the moment arm. Together with Sinc function, ˆ, which governs the parasitic
shift of pivot point, this method is a generic, simple, and quick tool for analyzing any flexure
configuration.
Stiffness Modeling
Flexure joints are considered as spring members within a compliant mechanism. Hence, the moving
platform of a compliant mechanism can be connected to a fixed base by a series of springs and
parallel springs. When the springs are connected in series, the overall stiffness is expressed as
1
K total
¼
1
1
1
þ
þ þ
K1 K2
Kn
(53)
or
C total ¼
n
X
Ci
(54)
i¼1
where the compliance C ¼ K1. When the springs are connected in parallel, the overall stiffness is
expressed as
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Fig. 27 (a) A 5-DOF spatial compliant joint module formed by (b) a pair of parallel limbs connected orthogonally in
series. (c) Representation of each flexure joint in a translational motion
K total ¼
n
X
Ki
(55)
i¼1
Example. Apply this concept to model the translation stiffness of a 5-DOF spatial compliant joint
module shown in Fig. 27a. This module is formed by two identical segments where each segment
consists 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 spatial
motions, 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 form
and can be represented as two identical flexure joints with individual length being l/2 and the
deflection of each flexure joint is D/2 as shown in Fig. 27c. Hence, the translation stiffness of
a flexure joint, Kl, is expressed as
Kl ¼ 2
F
D
(56)
Based on semi-analytic modeling, the driving force, F, can be obtained from Eq. 51 with the
deflection angle, y, deriving from Eq. 45 based on D ¼ D/2, l ! l/2, and L ¼ 0. With each limb being
formed by two identical flexure joints connected in series, the translation compliance of each limb,
Climb, is
C limb ¼
1
K limb
2
X
1
¼
Kl
i¼1
(57)
As each segment comprises of two parallel limbs, the linear translation stiffness of the spatial joint
module is
K DF
total ¼
2
X
1
i¼1
C limb
(58)
Example. Considering the prismatic compliant joint module shown in Fig. 28a, it is formed by
a compound linear spring module, which comprises of two parallel springs connected in series as
shown in Fig. 28b. Each parallel spring is articulated by two parallel limbs and each limb comprises
of two flexure joints connected together via a rigid link. Hence, each limb can be represented by two
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Fig. 28 (a) A double compound linear spring module and (b) the schematic representation of this compliant joint
module and (c) each parallel limb
identical flexure configurations where each flexure joint is represented by a revolute joint with
a 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, Kzfc, is expressed as
K zfc ¼ 2
Fz
Dz
(59)
Based on semi-analytic modeling, the driving force, Fz, can be obtained from Eq. 51 with the
deflection angle, y, deriving from Eq. 45 based on D ¼ Dz/2 and L ¼ L/2. With two identical flexure
configurations connected in series to form each limb, the deflection compliance of each parallel
spring along the z-axis of each limb, C zlimb , is
C zlimb ¼
2
X
1
K zfc
i¼1
(60)
As each parallel spring is articulated by two parallel limbs, the linear translation stiffness along the
z-axis, Kzps, is
K zps
¼
2
X
1
C zlimb
i¼1
(61)
With two parallel springs connected in series to form a compound linear spring, the translation
stiffness of the compound linear spring along the z-axis, KDzFz
total , is
1
K DzFz
total
¼
2
X
1
K zps
i¼1
(62)
This spring-based modeling concept can be extended to model the stiffness of the entire compliant
manipulator. Considering that a compliant manipulator comprises of a moving platform supported
by j number of parallel and symmetrical limbs, each limb can be formed by a group of compliant
joint modules connected in series by the links as shown in Fig. 29a. For a link in Cartesian space, the
moment vector, m, is a cross-product of the link vector, r, and force vector, f, expressed as
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Fig. 29 (a) A limb, which is constructed from a serially connected compliant joint module, is used to (b) form the
symmetrical limbs of a compliant manipulator
2
3 2 3 2 3 2
0
mx
rx
fx
4 my 5 ¼ 4 r y 5 4 f y 5 ¼ 4 r z
ry
mz
rz
fz
rz
0
rx
32 3
ry
fx
rx 5:4 f y 5
0
fz
(63)
In addition, the linear displacement vector, dd, is a cross-product of the angular displacement
vector, dy, and the link vector expressed as
3 2
2
3 2
3 2 3 2
3
ry
0
rz
dyx
dyx
rx
ddx
4 ddy 5 ¼ 4 dyy 5 4 ry 5 ¼ 4 rz 0
rx 5 4 dyy 5
(64)
ry
rx 0
ddz
dyz
rz
dyz
Within each limb, only the compliant joint module is assumed to be a spring member with
a compliant matrix, Ci, established at the local coordinate frame attached to it. To establish the
compliant matrix of each limb, Climb, a Jacobian matrix, Ji, that maps the local coordinate frame of
each compliant joint module to the local coordinate frame of the tip of each limb is required. Based
on Eq. 64, the displacement vector of each limb, X, can be expressed as
2
3 2
32 3
ryi
ddx
1 0 0 0
rzi
xi
6 ddy 7 6 0 1 0 rzi 0
6
7
7
r
xi 7 6 yi 7
6
7 6
6 ddz 7 6 0 0 1 ryi
6
7
rxi 0 7 6 zi 7
6
7¼6
7
(65)
6 dyx 7 6 0 0 0 1
7:6 yxi 7 ¼ Ji xi
0
0
6
7 6
76 7
4 dyy 5 4 0 0 0 0
1
0 5 4 yyi 5
dyz
yzi
0 0 0 0
0
1
Based on Eq. 63, the force and moment applied to the end-effector, F, can be expressed as
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3 2
1
Fx
6 Fy 7 6 0
6
7 6
6 Fz 7 6 0
6
7 6
6 Mx 7 ¼ 6 0
6
7 6
4 M y 5 4 rzj
ryj
Mz
2
0
1
0
rzj
0
rxj
0
0
1
ryj
rxj
0
0
0
0
1
0
0
32
3
f xj
0
6
7
07
7 6 f yj 7
6
7
0 7 6 f zj 7
7 ¼ JT f j
:6
j
7
07
m
xj
76
7
4
5
5
0
myj
1
mzj
0
0
0
0
1
0
(66)
As each limb is formed by the compliant joint modules connected in series via the links shown in
Fig. 29a, the displacement of the limb is Xlimb ¼ J1x1 + J2x2 + J3x3 + . . . based on Eq. 65,
reexpressed as ClimbF ¼ J1f1 + J2f2 + J3f3 + . . . based on CF ¼ X. As Eq. 66 can be reexpressed as
f ¼ JTF, the compliance matrix of each limb is given as
Climb ¼
n
X
Ji Ci JTi
(67)
i¼1
From Fig. 29b, the end effector is supported by a group of parallel limbs. Based on Eq. 66, the
T
T
T
total force and moment applied to the end effector is F ¼ J
1 f1 + Jz f2 + J3 f3+...., reexpressed as
T
T
T
KX ¼ J
1 K1x1 + J2 K2x2 + J3 K3x3 + . . . based on F ¼ KX. By reexpressing Eq. 65 into
1
x ¼ J X, the stiffness matrix of a compliant mechanism is given as
K¼
n
X
1
JT
j K limb, j Jj
(68)
j¼1
Example. Consider the stiffness modeling of a 3-DOF compliant manipulator shown in Fig. 30a as
an example. This compliant manipulator is formed by three symmetrical parallel limbs and each
limb consists of spatial and prismatic compliant joint modules, as shown in Fig. 30b. All three
parallel limbs are placed 120 apart to support a moving platform with its end effector located at the
center.
In this example, the compliance matrix of the prismatic compliant joint module, CA, and the
spatial compliant joint module, CB, is given as
2
3
1:56e-6
6 1:41e-7 3:33e-7
7
SYM
6
7
6 1:95e-8 0:23e-9
7
3:41e-4
7
CA ¼ 6
(69)
6 0:15e-9 5:00e-9 5:58e-5
7
1:28e-3
6
7
4 0:60e-9 0:25e-9 1:70e-9 3:95e-8
5
4:61e-3
3:03e-5 3:03e-6
1:30e-9 0:01e-9 1:33e-8 6:65e-4
2
1:71e-6
6 1:03e-6
6
6 7:06e-8
CB ¼ 6
6 3:19e-5
6
4 5:24e-5
0
3
1:71e-6
7:06e-8
5:24e-5
3:19e-5
0
SYM
4:80e-9
2:17e-6
2:17e-6
0
0:338
9:81e-4 0:338
0
0
0:031
7
7
7
7
7
7
5
(70)
Next, individual local coordinate frame is attached to the center of the prismatic compliant module
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Fig. 30 (a) A compliant manipulator formed by three symmetrical parallel limbs where (b) each limb consists of
a spatial and prismatic compliant joint module to support (c) a moving platform at three symmetrical corners which are
120 apart
and the tip of the flexure joint within the spatial compliant joint module as shown in Fig. 30b. These
local coordinate frames have the same orientation but different offsets with respect to the local
coordinate frame attached to the tip of the limb. Hence, the Jacobian matrix for each module, Jm, is
expressed as
2
3
1 0
0 0
hm 0
6
07
1
0 hm 0
6
7
6
1 0
0
07
6
7
Jm ¼ 6
(71)
7
1
0
0
6
7
4
zeros
1
05
1
where m represents A or B for respective modules. In this example, hA ¼ 87 mm and hB ¼ 19.5 mm.
The compliance matrices CA and CB are assembled to form
Ctotal ¼ diag ðCA ; CB Þ
(72)
In addition, the Jacobian matrices JA and JB are assembled into
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Jlimb ¼ ½ JA
JB (73)
From Eq. 67, the compliance matrix of each limb is derived as
Climb ¼2 Jlimb Ctotal JTlimb
1:65e-4
6 8:05e-7
1:43e-4
6
6 8:94e-9 4:74e-6
¼6
6 1:28e-5 6:79e-3
6
4 6:97e-3
5:10e-5
3:03e-5
3:10e-6
3
7
7
7
7
7
7
5
SYM
3:41e-4
5:36e-5 0:341
2:17e-6
9:81e-4
0:344
1:30e-9 0:01e-9 1:33e-8
(74)
0:032
Prior to the derivation of stiffness matrix, the Jacobian matrix, Jj, that maps the local coordinate
frame of each limb to this reference coordinate frame must first be established. Figure 30c illustrates
that the local coordinate frame attached to the tip of each limb has different orientations and a linear
displacement 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 same
plane as the local coordinate frame at the tip of each limb. Hence, there is no variation in the z-axis
between both coordinate frames. The orientation of the local coordinate frame at the tip of each limb
w.r.t. the reference coordinate frame in matrix form is expressed as
cos cj cos bj xj
X
(75)
¼
sin cj sin bj
yj
Y
where bj ¼ cj + p/2 and j represents either 1, 2, or 3. Based on Eqs. 75 and 65, the Jacobian matrix
that maps the local coordinate frame of each limb to the reference coordinate frame is
2
3
cos cj cos bj 0 0
0
r sin yj
6 sin cj sin bj 0 0
0
r cos yj 7
6
7
60
7
0
1
r
sin
y
r
cos
y
0
j
i
7
Jj ¼ 6
(76)
60
7
0
0
0 cos cj cos bj
6
7
40
5
0
0
0 sin cj sin bj
0
0
0 0
0
1
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 ¼
64.83 mm. Based on Eqs. 68, 74, and 76, the stiffness matrix of the compliant manipulator is given
as
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3
X
1
1
K¼
JT
j Climb, j Jj
2j¼1
2:57e5
60
2:57e5
6
6 4:3e-12
2:6e-12
¼6
6 1:63e2
5:14e3
6
4 5:14e3 1:63e2
2:6e-11 1:6e-11
3
7
7
7
7
7
7
5
SYM
8:83e3
5:7e-14
1:0e-13
2:71e2
1:30e2
5:9e-14
2:8e-13
1:30e2
5:4e-13
(77)
1:72e3
The obtained stiffness matrix was compared against the experimental values and results have
demonstrated 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 analysis
on 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 the
natural stiffness quantifies the compliance in the desired driving directions while off-axis stiffness
quantifies how stiff the mechanism is in non-driving directions. Hence, a compliant mechanism must
have high stiffness ratio since it directly affects the robustness of the compliant manipulator. This
section highlights the several existing and new design concepts that can be used to synthesize the
compliant mechanisms.
Exact Constraint Design
Exact constraint design approach excludes the principles of kinematics to synthesize a compliant
mechanism. Its basic objective is to achieve desired degrees of motions or no motion through
applying minimum number of constraints to a rigid body. Based on the principles of kinematics, an
unconstraint rigid body has six degrees of motion, i.e., three translational and three rotational
motions. Based on the principles of exact constraint design, a nonrigid body may have one or
more degrees of “flexibility” which serves as additional degrees of motions. Considering a box with
lid as an example, it has six degrees of motions when the lid is on. Without the lid, the open box will
have seven or even eight degrees of motions due to the additional torsional motions. A total of
twelve fundamental ideas are associated with this design approach and can be found in the past
literatures (Blanding 1992; Hale 1999; Slocum 1992). Most of these fundamental ideas are summarized in the matrix table shown in Fig. 31 illustrating how constrains can be added to achieve
desired 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 in
Fig. 32a, it creates an instantaneous center (I.C.) of rotation at the center of the circle. By adding
another constraint that reacts with a moment about the center, the rotation about the center of the
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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 rotation
motion
circle is constrained. However, the circle may not be perfectly constrained because the length of the
additional constraint arm is not equal to the other non-orthogonal constraints. A perfect example that
reflects this fundamental idea is shown in Fig. 32b where all constraint arms are equal in length and
symmetrical. In some conditions, nonsymmetrical arrangement is acceptable, but all constraint arms
must remain equal as shown in Fig. 32c. This fundamental idea is useful for synthesizing compliant
mechanism 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 three
degrees of freedom: Y, yx, and yz.
An Ideal Wire Flexure imposes absolutely rigid constraint along its axial (dx), but it allows five degrees of
freedom: dy, dz, yx, yy, yz.
The fundamental idea of an Ideal Sheet Flexure that offers 3-DOF out-of-plane motions is shown
in Fig. 33a. Using a leaf spring as an example, the in-plane motion is constrained and can be
represented by two vertical constraints and one diagonal constraint. When one set of constraints is
used 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
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Fig. 33 (a) A leaf-spring flexure joint represented by two vertical constraints and one diagonal constraint. (b) This set of
constraints is used to constraint a rigid body that eventually leads to (c) a parallel spring mechanism
degree of motion left is along the y-axis, dy, as shown in Fig. 33c. As each set of constraints
represents 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 shown
in Fig. 27a. The design concept of this spatial compliant joint module is also governed by the
following fundamental idea of exact constraint design (Blanding 1992).
When parts are connected in series (cascaded), add the degrees of freedom. When the connections occur in
parallel, add constraints.
Referring to Fig. 27b, two parallel beam-based flexure joints, which forms one segment, increase
the deflection stiffness of the spatial compliant joint module. Yet, connecting two segments in series
and orthogonal arrangement brings additional degrees of motions. Alternatively, the fundamental
idea of an Ideal Wire Flexure that offers 5-DOF motions is also useful for designing a spatial
compliant 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 (Blanding
1992)
A constraint ℭ properly applied to a body (i.e., without overconstraint) has the effect of removing one of the
body’s rotational degrees of freedom (ℜ's). The ℜ 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 no
constraint 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 underconstrained. An extension of this idea was to generalize it to nonrigid bodies (Hale 1999). By adding
number of DOF with due to the DOF of flexure joints, the mobility equation is rewritten as dof ¼ 6 +
f ℭ. Hence, ℭ must be sufficient to achieve the desired DOF. In addition, it is also important that
there must be no redundant ℭ. If removal of a ℭ does not affect the DOF, the remaining ℭ stays in the
mechanism. Lastly, the mechanism is exactly constrained if the removal of any single ℭ increases the
DOF by one. With this final but most crucial fundamental idea, this section wraps up the review of
exact constraint design approach.
Parallel-Kinematic Architecture
Parallel-kinematic architecture plays an important role in the success of the compliant manipulator
due to its advantages of a lower inertia, programmable centers of rotations, superior dynamic
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Fig. 34 Different parallel-kinematic architectures
behavior, higher stiffness, and less sensitive to external disturbances as compared to its serial
counterpart. In addition, the limited deflection of the flexure joints suits the limited motion range
of the parallel-kinematic architecture. Therefore, the parallel-kinematic architectures are commonly
used to synthesize the mechanisms within the compliant manipulators. In general, a parallel
mechanism is made up of an end effector (located at the center of the moving platform) with
n degrees of motions and of a fixed base that are both linked together by at least two independent
parallel-kinematic chains (Merlet 2000). Each kinematic chain is articulated by a set of elementary
kinematic 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 planar
motion parallel mechanism can deliver out-of-plane motions.
Figure 34a illustrates a 3-DOF planar motion parallel mechanism that can only deliver two
de-coupled X and Y translational motions and a coupled yz rotational motion. This parallel mechanism consists of three kinematic chains. Starting from the fixed base to the moving platform, each
kinematic chain is formed by one P joint, another P joint orthogonal to the first, and one R joint. This
is 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 remaining
P and R joints become the passive joints. To prevent high inertial and moving mass, it is always ideal
to fix the active P joints on the base. Figure 34b illustrates a spatial motion parallel mechanism that
produces 3-DOF out-of-plane motions, i.e., Z, yx, yy. This parallel mechanism is formed by a 3RPS
parallel-kinematic architecture and has an active P joint in each kinematic chain. A 6-DOF spatial
parallel mechanism is illustrated in 34c. This parallel mechanism is formed by a 6UPS parallelkinematic architecture with an active P joint in each kinematic chain. This parallel-kinematic
architecture, which offers 6-DOF motions, is also known as the Steward platform. A compliant
mechanism 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 from
different 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 any
linkage-joint parallel mechanism into a parallel compliant mechanism. Termed as a task-oriented
design approach, the first step (S1) is to identify the task or application, desired DOF, and
workspace. These data become the design criterions, which are used in the second step (S2), to
synthesize the right-type parallel-kinematic architecture, which is suitable to deliver the targeted task
specifications. Upon selection of suitable architecture, the third step (S3) is to perform kinematic
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Fig. 35 Task-oriented design approach; a systematic methodology to convert any linkage-joint parallel mechanism into
a compliant-based parallel mechanism
analysis, i.e., forward and reverse kinematic analyses. The forward kinematic analysis works out the
position and orientation of the end effector (task space) based on given actuator displacements (joint
space). Hence, it is used to determine the required displacements from the actuators and the
compliant joint modules to achieve the targeted workspace. The inverse kinematic analysis provides
the actuator displacements based on given position and orientation of the end-effector. This analysis
is 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 joint
modules. Here, a compliant joint module represents a specific conventional kinematic joint, e.g., the
revolute joint, the prismatic joint, the spherical joint etc., which can be formed by a single flexure
joint, a series or a group of flexure joints. Next, the parametric analyses will be conducted based on
the forward or inverse kinematic solution to provide the estimated displacements required from each
compliant joint module. Subsequently, the design of each module is conducted through analytical
modeling of the required stiffness within those estimated displacements and material’s yield
strength. The last step (S5) is to perform design optimization of the complete compliant mechanism
through numerical simulation using finite-element modeling (FEM) platform, i.e., ANSYS, and
analytical stiffness modeling. S5 evaluates the achievable workspace, the stress concentration of the
flexure joints, and the natural frequency of the FPM. An iterative process between S4 and S5 is
necessary should any of those parameters fall out of the desired specifications. Using this design
approach, 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 an
example to apply the task-oriented design approach. This mechanism with active actuation and
control formed a compliant manipulator targeted to automate the imprinting and out-of-plane
alignment tasks within an Ultraviolet Nanoimprint Lithography (UV-NIL) process. The targeted
workspace 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 task
requirements, was conducted in S2. From past literatures (Merlet 2000; Tsai 1999), four possible
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Fig. 36 (a) A 3PPS parallel-kinematic architecture synthesized from S2. (b) Projection of a rotation vector to represent
the yx-yy-Z motion for kinematic and workspace analyses in S3
architectures were identified, i.e., 3RRS, 3PRS, 3RPS, and 3PPS. To design a high-precision
mechanism, P-joint, which provides stiffer and higher precision guide ways, is always preferred
over the rotation counterpart. Considering the requirements of having high imprinting force and the
contact task, 3PPS parallel-kinematic architecture was selected as shown in Fig. 36a. To reduce the
moving mass and inertia, the active P-joint was placed nearest to the base platform (note: the
underlined P represents active P-joint). To effectively translate the output force of the linear
actuators 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 towards
the center of the equilateral triangular base.
The forward and inverse kinematic models were derived in S3. Forward kinematic modeling is
used to determine the moving platform pose based on the known active P-joint displacements and
the solutions are given as (Yang et al. 2011)
ex ¼
z2 z3
a
pffiffiffi
3ð2z1 z2 z3 Þ
ey ¼
3a
(78)
(79)
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Fig. 37 Initial conceptual design of a 3PPS parallel compliant mechanism
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Fig. 38 (a) FE analysis conducted on the initial proposed mechanism. (b) Final prototype with simplified compliant
limbs
zp ¼
z1 þ z2 þ z3
3
(80)
where a represents the edge length of the equilateral triangular moving platform while z1, z2, and z3
represent the displacement of the active P-joints. Referring to Fig 36b, these forward kinematic
solutions were derived based on the notation that the 3-DOF motions can be represented by
a rotation vector, o, with Z ¼ (0,0,1) and Z0 ¼ (ex, ey, ez) being the unit directional vectors of the
original 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 rotation
angles y and f. These kinematic solutions were used to determine the design parameters of the
mechanism based on the desired workspace. These parameters include the dimensions of the moving
platform, the desired translation, and rotation displacement from each joint. To obtain a vertical
displacement of 5 mm, a long-stroke flexure-based electromagnetic linear actuator (FELA) was
employed (Teo et al. 2008) as the active P-joints. Using the kinematic solutions and size constraint of
the FELA, a was chosen to be 112.29 mm. Based on the selected design parameters, results obtained
from the workspace analysis suggested that the maximum orientation and translation displacements
of the moving platform were 5.1 and 5 mm, respectively (Fig. 36).
Subsequently, an initial conceptual design of a 3PPS parallel compliant mechanism was proposed
in S4. It was articulated by three symmetrical compliant limbs where each limb consisted of a series
of compliant joint modules that mirrored the PPS kinematic chain. The stiffness analysis of each
compliant joint module was conducted via the semi-analytic modeling approach (see section “SemiAnalytic Model”) to determine the geometries of the flexure joints. This stage of design was
conducted 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 workspace
of 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 compliant
module and a segment of the S-joint compliant module were redundant because there was no stress
within the flexure joints. Returning to S4, a simple 5-DOF spatial compliant joint module was
proposed to replace the passive S- and P-joint compliant modules. Reduction of the compliant
modules also simplified the entire mechanism design and increased the off-axis stiffness. Finally, the
improved version was evaluated through the FE simulation before the actual prototype was
developed as shown in Fig. 38b.
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Fig. 39 (a) A compliant gripper obtained from the kinematic-based design approach versus (b) a new gripper concept
generated via the topology optimization approach
This example provides an overview on how the task-oriented design approach can be used to
design a parallel compliant mechanism. For S2 and S3, most of the parallel-kinematic architectures
can be found in many sources and past literatures (Merlet 2000; Tsai 1999). Hence, finding a suitable
architecture will not be difficult. The flexure and stiffness modeling methods, which are presented in
section “Theoretical Modeling Approaches,” are sufficient for executing S4 and S5 (except the FE
analysis). Therefore, information presented in this section will be useful for designing a parallel
compliant mechanism systematically.
Topological Optimization
A compliant mechanism can also be treated as a continuum structure with either distributed or
lumped compliance to deliver specific DOF motion. For example, a compliant gripper shown in
Fig. 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 same
function but with better stiffness characteristic as shown in Fig. 39b.
In general, a topological optimization approach is a mathematical approach of finding the optimal
way of distributing material within a predefined design domain based on a set of loads, fixed
supports, 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 finite
elements or trusses must be defined. Using finite-element design domain is considered as the
homogenization approach (Bendsoe and Kikuchi 1988), while those with truss structures are termed
as the ground structure approach (truss sizing) (Rozvany 1976; Bendsoe et al. 1994). Figure 39b
shows 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 around
it. Based on an objective function and some boundary conditions, topological optimization was
conducted via an optimization algorithm to determine the state of each element, i.e., either solid or
void. From Fig. 39b, those elements in black are solid, while those in white are void. The solid
elements form the optimized structure topology that meets the objective function. A shape optimization process further removed the unwanted materials and smoothen the edges to form a complete
continuum structure. Lastly, a symmetrical pair of optimized continuum structure will function like
a compliant gripper.
Topological optimization approach uses optimization algorithms such as the Generic Algorithm
(GA) (Chapman and Jakiela 1996), Solid Isotropic Material with Penalization (SIMP) (Bendsoe
1989; Bendsoe and Sigmund 1999), Evolutionary Structural Optimization (ESO) (Xie and Steven
1993; Chu et al. 1997), and Optimality Criteria (OC) (Rozvany 1995). Lately, new topological
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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 classical
mechanism theory
optimization approaches such as the level-set method (Wang et al. 2003b), the morphological
method (Tai and Akhtar 2005), and the mechanism-based seeding approach (Teo et al. 2013) were
also introduced. Over the past 30 years, these research efforts and findings have demonstrated that
the topological optimization approach is another effective concept of designing the compliant
mechanisms.
Using the advantages of modern topology optimization approach, an integrated design approach
for synthesizing an optimized parallel compliant mechanism is introduced in this section. This
design approach is a systematic design methodology that integrates both classical mechanism theory
and modern topology optimization approach. Referring to Fig. 40, the first step is to understand the
design specifications, e.g., the desired DOF, workspace, and size constraints. Next, appropriate
parallel-kinematic architecture will be selected and general kinematic analyses will be conducted. At
sub-chain level, topology optimization will be used to determine the optimized topology of the
flexure joint or limb. With these optimized topologies being generated, the compliant matrix of each
flexure joint or limb will be used to determine the overall stiffness of the compliant mechanism via
the classical mechanism theory. At configuration level, the overall stiffness of compliant mechanism
will be optimized based on the desired workspace and size constraints. Subsequently, an optimized
parallel compliant mechanism that meets all desired specifications will be generated.
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Fig. 42 (a) Adding cubic and harmonic curves to create material for each link. (b) Design variables of the mechanism
and (c) for each link
Example. Consider the design of a 3-DOF X-Y-yz planar motion compliant mechanism as an
example to apply the integrated design approach. This compliant mechanism targeted
a 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 parallelkinematic architectures. In this example, 3PPR was chosen because the compliant P-joints are more
deterministic than the compliant R joints. The schematic of 3PPR is shown in Fig. 41 where the
moving platform is connected to the fixed base by three identical parallel-kinematic chains. Each
kinematic chain consists of an active P-joint and a passive RP-joint that are connected in series. By
treating each joint as a spring, the stiffness modeling approach from classical mechanism theory (see
section “Stiffness Modeling”) can be used to determine the overall stiffness of the end effector based
on the compliant joints modules.
After the mechanism synthesis and kinematic analysis, a new topological optimization approach
was used to synthesize individual compliant joint module. Termed as the mechanism-based
approach (Lum et al. 2013), elementary linkage mechanisms were used as basic genes for the
joint optimization. For example, a generic 4-bar linkage mechanism was used to synthesize the 1DOF active P-joint and a generic 5-bar linkage mechanism was used to synthesize the 2-DOF
passive PR-joint. Next, the cubic and harmonic curves were added to each link of the mechanism to
create material as shown in Fig. 42a. Elements that fell within the boundary of the original and the
reflected curves became solid. Subsequently, the optimization varied the distribution of the material
by changing the design variables of mechanism (Fig. 42b) and individual link (Fig. 42c) until the
objective functions were met.
For an optimized P-joint, C11, which represents the compliance along the x-axis due to
a translation force along the same axis, needs to be as high as possible, while the remaining
components of the compliance matrix need to be as low as possible. Hence, the objective function
for the P-joint optimization was formulated as
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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
min F ðxÞ ¼
P6i¼2 Pij¼1 jC ij j
C 20
11
!
(81)
For an optimized PR-joint, both C11 and C66, which represents the compliance about the z-axis
due to a moment about the same axis, need to be as high as possible, while the remaining
components of the compliance matrix need to be as low as possible. Thus, the objective function
PR-joint optimization was
!
P6i¼2 Pij¼1 C 2ij
min F ðxÞ ¼
(82)
20
C 19
11 C 66
In this example, GA was 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 in
Fig. 43. Subsequently, the stiffness matrix of the optimized topology of each joint was used to
determine 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 on
the workspace and size constraints. As mentioned in the beginning of this section, the most
important design criteria of a compliant mechanism is to maximize the stiffness ratio, i.e., between
the off-axis stiffness and natural stiffness. Hence, the objective function of the stiffness optimization
was formulated as
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Fig. 45 Prototype of the 3PPR parallel compliant manipulator that is articulated by flexure joints with optimized
topologies and driven by three voice-coil actuators
max ¼
K 33 K 44 K 55
K 11 K 22 K 66
(83)
where K22 represents the stiffness along the y-axis due to a translation force along the same axis
while K33, K44, and K55 are the off-axis components.
Another 3PPS parallel compliant mechanism shown in Fig. 44a, which was articulated by flexure
joints with traditional topologies, was used to evaluate the stiffness characteristic of the optimized
design. Termed as the conventional design (Fig. 44b), its PR flexure joint was formed by a cantilever
beam with both ends being fixed to the translation portion of the P-joint, which was formed by
a conventional parallel linear spring configuration. The compliance matrix of the optimized design,
con
Copt
ee , and the conventional design, Cee , was obtained as
2
3
3:55E-5
6 1:1E-17 3:55E-5
7
SYM
6
7
6
7
4:7E-25
2:3E-24
1:12E-6
opt
7
Cee ¼ 6
(84)
6 1:8E-15
7
3:4E-15
1:7E-21
4:06E-4
6
7
4 3:4E-15
5
1:78E-15 1:7E-21
9:7E-16
4:06E-4
7:3E-20 2:3E-20 1:6E-15 1:7E-21 4:4E-21 2:42E-2
2
Ccon
ee
1:86E-5
6 6:6E-18
6
6 1:7E-25
¼6
6 2:0E-16
6
4 3:0E-16
3:8E-20
3
1:86E-5
5:2E-25
3:0E-16
2:0E-16
6:2E-20
7
7
7
7
7
7
5
SYM
1:97E-6
3:9E-21
4:0E-22
6:8E-15
5:41E-4
4:6E-16
7:9E-21
5:41E-4
4:5E-20
(85)
2:42E-2
Subsequently, the ratio between Eqs. 84 and 85 is represented in a diagonal matrix form, which is
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Rc ¼
Copt
ee
con ¼ diag½ 1:91 1:91
Cee
0:57
0:75
0:75
1
(86)
From Eq. 86, results show that the compliance along the x- and y-axis of the optimized design is
almost twice as compared to the conventional design. In addition, the off-axis stiffness is higher for
the optimized design. Hence, this section shows that the integrated design approach is an effective
design methodology for using topological optimization approach to synthesize an optimized parallel
compliant mechanism with high stiffness ratio characteristic. Finally, a prototype was developed
from this approach. Each of the active P-joint is driven by a voice-coil actuator and its position is
measured via a linear optical encoder. Based on a PID controller, it achieved a high positioning
resolution 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 compliant
mechanisms, high-resolution positioning actuators and sensors, and good control schemes. With
the main part of this chapter focusing on the strength, limitations, modeling, and design methodologies of the compliant mechanisms, this section reviews the state-of-the-art actuators and sensors
that can be used to achieve high-precision manipulation where the advantages, limitations, and
potential performance trade-offs for each kind of actuator or sensor will be discussed.
High-Resolution Positioning Actuators
An actuator with high positioning resolution is an essential subsystem of the compliant manipulator
that could ultimately decide the traveling range, output force, stiffness, and even size or footprint of
the manipulator. These actuators can be classified into two categories, i.e., the solid state and the field
based. The solid-state actuators are transducers that convert the electrical energy into the mechanical
energy via strain in the materials. Piezoelectric (PZT) actuator, shape-memory alloy (SMA), and the
thermal actuator are the solid-state actuators, which are commonly used by the compliant manipulators. On the other hand, the field-based actuators are transducers that convert the electrical energy
into the mechanical energy via the presence of fields. Field-based actuators that are commonly found
in the compliant manipulators include the electrostatic and the electromagnetic actuators.
Piezoelectric Actuators
Most conventional compliant manipulators are driven by the PZT actuators due to the nanometric
resolution 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). Such
an actuator is made up of ceramics that convert an applied voltage or charge into a mechanical
displacement directly through the physical elongation of the material. This small-dimensional
change leads to extremely small displacement of up to 1 1012 m. Such positioning resolution
can be adjusted since the dimensional changes are proportional to the applied voltage (Ouyang
et al. 2008). PZT actuators 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 of
producing mechanical displacements through dimensional changes, PZT actuators can produce
large force of up to few kilo-Newton (kN) and no wear-and-tear issue and can operate in vacuum
and clear room environment. Figure 46a shows some commercial stacked PZT actuators developed
by Physik Instrumente (PI).
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Fig. 46 (a) Stacked PZT actuators from PI (Physik Instrumente 2014). (b) A 1-DOF PZT-driven compliant stage with
an amplification level and preloaded springs and its schematic representation (Ho et al. 2004)
The major disadvantages of PZT actuators include the small and nonlinear displacement characteristics. The nonlinearity of PZT actuator is contributed by the hysteresis and creep behavior of the
ceramic materials during the dimensional change. As a result, the forward and backward paths of
a cyclic PZT actuator are different and require mechanical means or advanced control schemes to
minimize such nonlinearities. One approach is to preload the moving stage with springs in the
direction that opposes the PZT actuation as shown in Fig. 46b. However, preloading with springs can
only linearize a portion of the stroke. Hence, the maximum displacement of the stage is limited to
250 mm (Ho et al. 2004). Another approach is to model the hysteresis and creep of the PZT actuation
and linearize the displacement through advanced control schemes (Liu et al. 2013). The biggest
limitation is the limited displacement of a PZT actuator. By stacking the PZT ceramic disks, the
maximum achievable stroke is a few hundred micrometers. In addition, this stacking approach
generates accumulative errors at the end of the stack and increases the stress within each PZT
ceramic disks. Most importantly, this approach increases the internal resistance which in turn
increases the applied voltage requirements, e.g., >200 VDC.
Shape-Memory Alloy
SMA material has a unique memory (known as the “shape memory effect”) of its pre-deformed
shape. At low temperature, the SMA returns back to its original shape when the temperature
increases. This behavior makes SMA as a form of solid-state actuation. At high temperature, an
applied force can cause a large deformation, which can also easily recover by releasing the applied
force. This effect is termed as the “superelasticity.” The use of SMA on the compliant manipulators
is unique too. Instead of using conventional materials to develop the flexure joints, SMA is used as
the flexure material creating interesting flexible members that double up as both the limbs and
actuators of the compliant manipulators (Reynaerts et al. 1995; Hesselbach et al. 1997; Bellouard
and Clavel 2004). This is because the SMA flexure joint exhibits higher flexibility as compared to
the 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, SMA
can also be considered as a high damping metal, which is a favorable characteristic for springdominated systems, i.e., compliant mechanisms.
The main limitation of SMA is its slow rate of cooling, which is usually limited to a few hertz. As
a result, the difference between heating and cooling transition creates a hysteresis effect during the
forward and reverse motions as shown in Fig. 47. The slow cooling rate also limits the bandwidth of
any SMA-based compliant manipulator. Hence, these manipulators are only suitable for tasks or
applications that require slow yet precise motion delivery.
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a
b
Force (mN)
300
Steel X220CrVMO13–4
Max. range 0.34 mm
200
Binary Ni-Ti
Annealed @550°C
15 min / Water Quenched
100
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Range motion (mm)
Fig. 47 A monolithic SMA compliant linear spring and its force-displacement characteristic (Bellouard and Clavel
2004)
Fig. 48 Illustrations of the (a) bimorph and (b) chevron thermal actuator designs
Thermal Actuators
Thermal actuation is one of the popular schemes in driving the MEMS-based compliant manipulators (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 microscaled
functional systems. The basic working principle of the thermal actuation is to generate a small
amount of thermal expansion in a material through joule heating. Simultaneously, this small thermal
expansion will be amplified to produce a deflection motion. Such amplifications are realized through
the 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 been
joined together. During operation, one segment will be heated while the other remains cool. The
temperature difference between two segments causes both segments to expand differently. As
a result, a bending deflection occurs since both segments are fixed at one end. In addition, two or
more thermal actuators can be connected together in parallel to enhance the force output or generate
a linear motion (Comtois and Bright 1997). The chevron design can be articulated by one or more
pairs of “V”-shaped cantilever beams. For each pair of beams, the applied current passes through
from one beam to the other, thus causing them to expand, buckle, and create a linear motion. In
general, the bimorph thermal actuator provides a bending motion, while the chevron thermal
actuator provides a translation motion.
Figure 49a shows a 6-DOF compliant manipulator, mHexFlex, that is driven by three pairs of 2axes 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 is
driven 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 (Zhu
et al. 2006). The main advantage of the thermal actuation is having higher force generation as
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Fig. 49 (a) A 6-DOF microscale nanopositioner, mHexFlex, driven by six 2-axes bimorph thermal actuators (Chen and
Culpepper 2006). (b) A MEMS-based compliant stage driven by a chevron thermal actuator (Zhu et al. 2006)
compared to the electrostatic actuation. However, it has lower bandwidth and requires good heat
dissipation for continuous operations.
Electrostatic Actuators
Electrostatic 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; Chiou
and Lin 2005; Borovic et al. 2006). Unlike the thermal actuation, it consumes very small amount of
power and can operate in high bandwidth. Electrostatic actuator can also be easily fabricated and
integrated as part of the MEMS devices. Governed by Coulomb’s law, the electrostatic actuation
uses 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
1 2
A
(87)
F e ¼ V eair 2
2
d
where V represents the voltage and eair represents the permeability of air, while E ¼ A/d2 where A is
the 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 of
two sets of capacitor banks where each bank comprises of a parallel array of capacitors. Each
capacitor has a pair of parallel surface. However, there can be many variations in the geometry of
these comb-drives such as the typical parallel configuration shown in Fig. 50b, the spiral (Tang
et al. 1989) and the T-bar configurations (Brennen et al. 1990). Figure 50a shows an example of an
electrostatic-driven MEMS-based X-Y-yz micro-/nanopositioning stage articulated by a parallel-
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Fig. 50 (a) Schematic of an electrostatic force actuation. (b) An X-Y-yz MEMS-based 3PRR parallel compliant
manipulator 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 motion
compliant manipulator and its schematic plot of all three motion (Chen et al. 2002)
kinematic architecture. Each electrostatic achieved a travel of 27 mm at 85 V, while the overall
workspace of the end effector is 18 mm2 1.72 when all three actuators are energized.
Electromagnetic Actuators
Electromagnetic (EM) actuation is a driving scheme that offers noncontact, frictionless, and long
travel characteristics. Over the past two decades, compliant manipulators driven by EM actuation
have achieved millimeters of displacement and subnanometer positioning resolutions with high
accelerations and speed responses. To realize an EM-driving actuation, two types of techniques can
be employed, i.e., the solenoid actuation and the Lorentz-force actuation.
Solenoid Actuation Solenoid actuation is an EM technique based on the attraction of
a ferromagnetic moving part. Based on Fig. 51a, the fixed stator generates an attraction force to
propel the ferromagnetic moving part towards it. This force is expressed as
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Fig. 52 (a) Conventional VC actuators and other variations of VC actuators from (b) H2W Technologies (H2W
Technologies 2014) and (c) BEI Technologies (BEI Kimco Magnetics 2014)
F¼
mN 2 I 2 A
2g 2
(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 stator
and the moving part. The main advantage of this technique is that it generates higher force as
compared to the Lorentz-force actuation. However, it exhibits nonlinear characteristic due to the
square 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 solenoid
actuation. Three pairs of solenoid actuators are used to produce 3-DOF decoupled planar motion
with a minimum positioning resolution of 50 nm over a workspace of 80 mm2 3.52 mrad. Each
solenoid actuator generates a driving force of 50 N with an input current of 0.5 Amp at an air gap of
250 mm. This example also demonstrated that the solenoid actuation is not suitable for large
traveling range and has a nonlinear force-displacement and current-force relationship.
Lorentz-Force Actuation Lorentz-force actuation has a contrast characteristic as compared to the
solenoid actuation. This EM technique offers a linear current-force relationship and can be configured to deliver large traveling range with linear force-displacement relationship. Governed by
Lorentz’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 the
applied current, N represents the number of coil turns within the effective air, and L represents the
coil 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 of
magnetic flux density emanates from a permanent magnetic source, i.e., PM. In this air gap, the
magnetic flux density should be well distributed. Due to the low permeability of air, the magnitude of
the magnetic flux density is usually lower than the remanence magnetic flux density of a PM. The
best approach to overcome this limitation is to increase the size of a PM while maintaining a small
effective air gap. Examples can be found in the VC actuators from H2W Technologies (Stroman
2006) show in Fig. 52b and BEI Technologies (Speich and Goldfarb 2000) shown in Fig. 52c. These
magnetic circuits increase the magnetic flux density through larger PMs. Nevertheless, they still
require very small effective air gaps because the magnitude and uniformity of the magnetic flux
density decreases with respect to the increment in distance from the magnet-polarized surface.
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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
Due to its linearity and large displacement characteristics, there is a growing trend of using
Lorentz-force actuation to drive the compliant manipulators (Teo et al. 2010a; Yang et al. 2008; Teo
et al. 2013; Fukada and Nishimura 2007; Bacher 2003; Helmer 2006; Culpepper and Anderson
2004). These multi-DOF compliant manipulators have demonstrated nanometric positioning resolution capability over a few millimeters and degree workspace. Figure 53a shows an example of a 3DOF parallel compliant manipulator driven by three VC actuators from BEI Technologies. Using the
VC 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 3DOF out-of-plane motion compliant manipulator shown in Fig. 53b. Using customized Lorentzforce actuators, it achieved 20 nm positioning resolution and 0.05 arcsec angular resolution over
a workspace of 5 mm 5 5 . With the customized actuators, the manipulator produces 150 N/
Amp of output force.
Performance Trade-Offs
Electrostatic and thermal actuators are commonly used in the MEMS-based compliant manipulators.
The trade-off of using the electrostatic actuators is poor force generation. Switching to the thermal
actuator may provide larger force generation but at the expense of lower-frequency responses. For
micro- to macroscale compliant manipulators, the PZT actuators and SMA materials are the potential
solutions for driving them. Both offer limited travel range, but PZT actuators can deliver very high
driving force and bandwidth. On the other hand, SMA materials can be made into a monolithic
compliant manipulator. It is an attractive solution for minimizing moving masses and stiffness in
actuating direction. From macroscale onwards, EM actuation can be an alternate solution to the PZT
actuators. Solenoid actuators may have the advantage of large force-to-size ratio, but the Lorentzforce actuator has linear characteristic and large traveling range. Although PZT actuators have
limited stroke, high force generation and bandwidth are desirable characteristic for driving the
compliant manipulators. Being a solid-state actuator, PZT can still provide additional non-actuation
stiffness (even when power-off) to the compliant manipulators. This benefit can never be provided
by the EM actuation. To summarize the review of high-resolution positioning actuators, the
performance and trade-offs of each actuator are listed in Table 8.
High-Resolution Sensors
A sensor is a transducer that converts one form of energy to another form, for example, a device that
responds to or detects a physical quantity and transmits the resulting signal to a controller (Slocum
1992). Hence, sensors are also considered as the “eyes” or “ears” of a control system that controls
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Table 8 Performances of various actuators that are suitable for driving the compliant manipulator
Actuator
Piezoelectric
Electrostatic
Thermal
SMA
Solenoid
Lorentz-forcea
a
Max. range (mm)
~500
~100
~20
~10
~300
~50
Stiffness
High
0
High
Medium
0
0
Linearity
Poor
Square law
Poor
Poor
Poor
Good
Accuracy (nm)
<10
<1
<10
<100
<1
<1
Bandwidth
High
High
Low
Low
Low
Medium
Cost
Medium
Low
Low
Medium
Low
Medium
Single-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
a manipulator. Hence, it defines the achievable positioning resolution of a compliant manipulator
even 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 quantity
change by optical means, while the nonoptical sensors use other means of measurement. Such kind
of sensors include capacitive sensor, hall effect sensor, inductive displacement sensor, variable
differential sensor, inclinometers, magnetic scales, magnetostrictive sensor, and PZT-based sensor.
Among these nonoptical sensors, the capacitive sensor is commonly used in the compliant manipulators due to its high-resolution nature. For the same reason, the optical sensors, which include the
optical encoders and interferometric sensors, are the popular choices for the compliant manipulators.
Capacitive Sensors
A capacitive sensor determines the gap between a probe and a target by measuring the amount of
capacitance formed between the two parallel surfaces, i.e., the face of the probe and the target, as
shown in Fig. 54a. By applying voltage to one surface, an electric field will exist between the two
surfaces. Electric field is the result of the difference between the electric charges that are stored on
the surfaces. Hence, the capacitance is the “capacity,” which is formed between the two surfaces, to
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Fig. 55 (a) Illustration of an optical measurement principle. (b) A commercial optical linear encoder from
HEIDENHAIN Corporation (Heidenhain 2014)
hold the charges. The effect of the gap, g, and the surface area, A, on the capacitance, C, between two
parallel surfaces is expressed as
C¼
eA
g
(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 only
provide a measuring range of up to 0.13 mm or subnanometer resolution and the frequency
response is up to 20–40 Hz. The measuring range can be increased but at the expense of losing the
nanometric resolution. A unique characteristic of the capacitive sensor is the ability to detect a wide
range of materials, e.g., metals, dielectric, and semiconductors. The sensor output is only affected by
different types of material surfaces but will not be affected with different contents with the same
material. For example, a capacitive sensor calibrated over a stainless steel target can also be used to
measure brass or aluminum target. Apart from the optical means, the capacitive field sensing is the
only nonoptical means of providing subnanometer resolution measurement capability. Hence, it is
widely used in high-precision compliant manipulators.
There are a couple of stringent requirements when using the capacitive sensors. First, there will be
stray capacitance that affects the accuracy of the measurement. However, the stray capacitance can
be easily minimize by adding a guard around the sensing electrode and collimate the electric field
lines between the sensor and the target as shown in Fig. 54c. The trade-off is that the size of the probe
will increase due to the presence of the guard. Second, it is important to keep the surfaces of the
sensor and the target parallel. Misalignment in parallelism of two surfaces will affect the accuracy of
the measurement since the capacitance is proportional to the sensing area and gap between the probe
and 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 and
resolution. Other benefits include minimizing the effect of electromagnetic waves on the accuracy
and providing an averaging effect to the output. The forth requirement is to minimize environmental
disturbances 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 to
measure a fixed object to record a reference signal that captures the noise or drift cause by the change
in barometric pressure, humidity, and temperature. Subtracting the measurement signal with this
reference signal concurrently will produce the actual displacement signal.
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Fig. 56 Illustration of (a) a 1-DOF measurement setup and (b) a 3-DOF planar motion measurement setup using the
laser interferometer sensor
Optical Encoders
Optical encoder operates based on the principle of counting the slits or windows within a scale
through a light source and a photodiode as shown in Fig. 55a. Simply imagine a LED that shines
light on a scale, which has an array of equally spaced windows. Behind the scale and directly
opposite the LED is a photodiode. If the LED light transmits through these windows, the photodiode
will receive the transmitted light and generate signals or pulses. On the other hand, no signal will be
generated by the photodiode if the light is block by the material in-between two windows. This
measuring scheme can be configured to linear or angular optical encoder that is immune to electric
noise. 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 scale
that can pack more equally spaced slits (gratings) will have higher resolution. Based on the
commercial available optical encoders from HEIDENHAIN Corporation, a scale with the absolute
grating can deliver approximately 10- to 12-bits positioning resolution, a scale with the incremental
grating can deliver 10- to 16-bits positioning resolution, and a diffraction grating can deliver up to
21-bits positioning resolution. Like a ruler with equally spaced numbered markings, the absolute
grating has a fixed and low resolution. For the incremental grating, the quadrature and interpolation
methods are used to enhance the resolution of the encoder. For example, an incremental grating with
20 mm pitch can deliver 5 nm per encoder count resolution based on 4,000 steps interpolation. With
such interpolation approach, the diffraction grating provides even higher positioning resolution due
to its smaller pitch size. The scale also comes in different materials, i.e., stainless steel tape, glass
scale, and zerodur scale. The glass scale has a thermal expansion coefficient of 5.9 106/K, while
the zerodur material has a thermal expansion coefficient of 0 0.007 106/K from 0 C to
50 C. The selection of the material depends on the applications and desired specifications. For highprecision positioning tasks, the zerodur scale may be the best option, but it is the most expensive
among the rest.
Laser Interferometer Sensors
Laser interferometer sensor is the most accurate measurement system that can be applied to today’s
mechatronics and robotic systems. Hence, it can be used as working standards for machine
calibrations, measurement, and feedback control. Based on the typical size of a photon, the
resolution of a laser interferometer sensor is up to 0.15 nm. It also has a measuring range that is
beyond 10 m and a measuring speed of up to 4.2 m/s. Most laser interferometer sensors use HeliumNeon as the laser source (wavelength, l, ¼ 633 nm). The accuracy of laser interferometer sensor is
dependent on very stringent metrology conditions:
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Fig. 57 Measurement setups using (a) two single-axis interferometric encoders from Renishaw and (b) a three-axes
interferometric sensor from SIOS (SIOS 2014)
1.
2.
3.
4.
Temperature stability of 1 C
Humidity variation of less than 10 %
Pressure variation of less than 0.25 mmHg
Target mirror flatness of l/10 PV
Temperature and humidity can be actively controlled in an enclosed environment, e.g., a clean
room, while the mirror flatness can be achieved through high-precision polishing technology. As
barometric (environmental) pressure is more difficult to control, the usual approach is to setup
a 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 not
give an absolute displacement value but rather a value relative to the previous value. In addition, it
can only measure relative displacement change beyond a dead path, which represents the minimum
length requirement between laser interferometer sensor and the target. Setting up the laser interferometer sensor requires optics components such as the beam splitters (BS) and retroreflectors to
orientate the laser beam from the laser source to the moving stage. Figure 56a illustrates a setup to
measure a 1-DOF moving stage via the laser interferometer sensor. A BS is used to split the beam
into 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
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position of the moving stage relative to the fixed reference position. Figure 56b illustrates another
setup that measures a 3-DOF moving stage. After the first BS from the laser source, 50 % of the beam
goes to another BS that further splits it into two beams (25 % each) where one goes to a wavelength
tracker and the other measures the position along the x-axis. The other 50 % of the beam is also split
into two beams (25 % each) that measure the position along the y-axis. All three 25 % beams also
measure the angular change about the z-axis. The wavelength tracker is used to monitor the accuracy
of the beam, which could easily be affected by changing metrology conditions such as the
temperature, humidity, and pressure.
Fiber optics can be used to deliver the laser beam directly to the remote units with built-in optics
and detector. This technology minimizes the needs of using optic components and the setup time to
orientate the beam from the laser source and can channel the beam from one laser source to multiple
remote units. These units, termed the interferometric encoders, can be placed near the end effector
for direct measurements. Figure 57a shows a pair of single-axis interferometric encoders from
Renishaw 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 two
separated interferometric encoders without any loss in the resolution, i.e., 10 nm per count. The
transmitter, optics, and detector are all built inside each encoder to simplify the setup, while the end
effector carried the metrology mirrors to reflect the transmitted laser beams back to individual
encoder. 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 three
individual measuring beams. These beams are equally spaced in a predefined arrangement that is
capable of measuring 3-axes out-of-plane motion and provides a resolution of 1 nm per count from
each beam. This encoder system also provides a metrology station that has a similar function as
a wavelength tracker.
Performance Trade-Offs
Laser interferometer sensor offers the most accurate measurement in modern world and has the
ability to provide a positioning resolution of up to 0.15 nm. It is a noncontact and relative
displacement sensor that is commonly used to feedback to position of the end effector rather than
the joints of a manipulator. The main limitation of laser interferometer sensor is that its accuracy can
be easily affected by changing metrology conditions. On the other hand, these changing metrology
conditions have less effects on an optical encoder since the scale and encoder head are placed in
close proximity. However, this arrangement limits the use of the optical encoders to single-axis or
two-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 the
length of the scale, which typically range from 10 cm to 3 m. Although interpolation offers high
resolution, the accuracy is always limited to 1 % of the grating pitch size. One method to overcome
this limitation is to directly relate the change in analog signal with the change in displacement. Such
calibrations are also commonly done in motion systems that use the analog capacitance sensors. Just
like 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 capacitance
sensor can be calibrated for larger measuring range but sacrifices on the resolution due to the effects
of 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 environment
and methodology conditions.
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Future Advancement
The first generation of compliant manipulators offered single-axis translational motion, while the
second generation delivered multi-DOF planar or spatial motion through the aid of parallelkinematic architectures. In both generations, the workspace of the compliant manipulators is limited
by the notch-hinge flexure joints and PZT actuators. By exploring the EM actuation and more
flexible beam-based flexure joints, the third generation of compliant manipulators has successfully
broken through the millimeter-range barrier encountered by the previous generations but limited to
low payload positioning applications. More recent research work has introduced the latest generation 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 next
generation of compliant manipulators is projected to be an integrated system fabricated through 3D
printing technology. Termed as the Flextronics, these single monolithic compliant manipulators will
have 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 optimization
of compliant manipulators. Using the 3D printing technology, the structure arrangement of the
compliant limbs or flexure joints will be isotopic rather than homogenous. Hence, new theoretical
models and modeling approach will be needed to predict the stiffness characteristic of the
Flextronics. The advancement in material science will be the key to the realization of Flextronics
since this advanced technology requires new material that can be actuated and sense while providing
necessary strength and stiffness to the overall structural integrity. Most importantly, the new
engineered material must have high endurance limit and predictable fatigue life. Potential applications of Flextronics can be advanced surgical tools and miniature robotic systems or even providing
macro- to nanoscale positioning tasks in vacuum and harsh environment.
Summary
Development of a compliant manipulator, which is articulated by the flexure joints, involves good
understandings on several fundamental topics associated with this advanced robotic system. Material science plays an important role in developing a reliable compliant manipulator. Accurate
identification of the fatigue life of those flexure joints is crucial as it varies according to the types
of materials, surface finishing, geometries, and loadings. Material properties also affect the stress
level that each flexure joint can sustain under loading, bending, and torsional bending. Although
lower Young’s module properties will lessen bending stiffness and thus the stresses, the compliant
manipulators may lose desirable stiffness in the non-actuating directions. Hence, the major challenge in designing a compliant manipulator is to fulfill its desired functions and deflection
workspace while maintaining the stresses well below the yield strength of the material. Solid
mechanics provide important analytical tools for modeling the deflection stiffness of the flexure
joints. Classical bending-moment-curvature equation brought several key formulations in predicting
the stiffness and stresses of functional compliant mechanisms, while the large deflection theorem
covers the nonlinear characteristics of the flexure joints. Advancement in the theoretical studies on
large deflection characteristic gave birth to the pseudo-rigid-body approximation model, which also
became the bridge between the classical rigid-body linkage mechanism and the compliant mechanism. This model was further enhanced by the semi-analytic approximation model, which provides
a simple, quick, and generic solution for any form of flexure joint configuration. With these
approximation models, the knowledge of rigid-body linkage mechanism, particularly the parallelPage 57 of 63
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kinematic architectures, can be applied in synthesizing the compliant mechanism. Synthesizing of
the compliant mechanism can also be conducted through the classical exact-constraint design
approach and the state-of-the-art topological optimization techniques. The synthesized compliant
mechanism has to be driven by actuators to form a compliant manipulator. The actuators will
ultimately decide the traveling range, output force, stiffness, and even size or footprint of the
manipulator. 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 as
a 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 35
3PPS parallel compliant mechanism 39
Capacitive sensor 52
Circular-shaped notch hinges 17
École Polytechnique Fédérale de Lausanne (EPFL) 7
Compliant manipulator 1–3, 9–11, 14, 18, 21, 25, 32, 34, 40, 44, 52, 57
advantages 1
benefits 3
classification 3
compliant mechanism 1
DOF compliant manipulator 2
exact constraint design approach 34
fatigue failure 14
flexure-based compliant bearing 3
future advancement 57
high positioning resolution actuators 52
high resolution sensors 52
history 9
non-contact bearings 3
nonlinear large deflection theorems 18
parallel-kinematic architecture 40
pseudo-rigid-body model 21
semi-analytical modeling 25
small deflection theorems 18
stiffness modeling 32
strength and stiffness 10
stress failure 11
topology optimization approach 44
Compound linear spring mechanism 16
Corner-filleted notch hinge 18
Degrees-of-Freedom (DOF) compliant manipulator 2
Delta3 compliant manipulator 8
Double compound linear spring mechanism 16
Electromagnetic (EM) actuation 49
Electrostatic actuator 48
Elliptical notch hinge 18
Exact constraint design approach 32
Fiber optics 56
Finite Element Modeling (FEM) platform 36
Flexure joints 3
Flexure-based Electromagnetic Linear Actuator (FELA) 39
Forward kinematic modeling 37
Fully compliant manipulators 2
High positioning resolution actuators 45–49, 51
electromagnetic actuators 49
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Handbook of Manufacturing Engineering and Technology
DOI 10.1007/978-1-4471-4976-7_102-1
# Springer-Verlag London 2014
electrostatic actuators 48
performance trade-offs 51
piezoelectric (PZT) actuator 45
shape memory alloy 46
thermal actuators 47
High resolution sensors 52–54, 56
capacitive sensor 53
laser interferometer sensor 54
optical encoder 54
performance specifications 52
performance trade-offs 56
Laser interferometer sensor 54
Leaf-spring compound linear spring mechanism 5
Lorentz-force actuation 50
MEMS-based micro-actuator 9
Michelson interferometer mirror 5
Monolithic compound linear spring mechanism 5
Nonlinear large deflection theorems 18
Notch flexure joints 17
Optical encoder 54
Parallel linear spring mechanism 15
Partially compliant manipulator 2
Piezoelectric (PZT) actuators 45
Planar motion compliant manipulators 7
Pseudo-Rigid-Body (PRB) model 21
Semi-analytical modeling 25
Shape memory effect 46
Slender strips 4
Solenoid actuation 49
Spatial joint compliant module 2
Spherical-Prismatic-Spherical (SPS) serially-connected compliant limbs 7
Stiffness modeling 32
Superelasticity 46
Thermal actuators 47
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