Closed-Form Modeling and Analysis of an XY Flexure-Based Nano-Manipulator
Qin et al. Chin. J. Mech. Eng.
Closed-Form Modeling and Analysis of an XY Flexure-Based Nano-Manipulator
Yan‑Ding Qin 0
Xin Zhao 0
Bijan Shirinzadeh 2
Yan‑Ling Tian 1
DaW‑ei Zhang 1
0 Institute of Robotics and Automatic Information System (Tianjin Key Laboratory of Intelligent Robotics), Nankai University , Tianjin 300350 , China
1 School of Mechanical Engineering, Tianjin University , Tianjin 300072 , China
2 Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University , Clayton, VIC 3800 , Australia
Flexure‑ based mechanisms are widely utilized in nano manipulations. The closed‑ form statics and dynamics modeling is difficult due to the complex topologies, the inevitable compliance of levers, the Hertzian contact interface, etc. This paper presents the closed‑ form modeling of an XY nano‑ manipulator consisting of statically indeterminate symmetric (SIS) structures using leaf and circular flexure hinges. Theoretical analysis reveals that the lever's compliance, the contact stiffness, and the load mass have significant influence on the static and dynamic performances of the system. Experiments are conducted to verify the effectiveness of the established models. If no piezoelectric actuator (PEA) is installed, the influence of the contact stiffness can be eliminated. Experimental results show that the estimation error on the output stiffness and first natural frequency can reach 2% and 1.7%, respectively. If PEAs are installed, the contact stiffness shows up in the models. As no effective method is currently available to measure or estimate the contact stiffness, it is impossible to precisely estimate the performance of the overall system. In this case, the established closed‑ form models can be utilized to calculate the bounds of the performance. The established closed‑ form models are widely applicable in the design and optimization of planar flexure‑ based mechanisms.
Flexure‑ based mechanism; Statically indeterminate structure; Dynamics; Lever mechanism; Piezoelectric actuator
The integrations of piezoelectric actuators (PEAs) and
flexure-based mechanisms have been widely utilized in
nano-positioning and manipulations [1–5]. On the one
hand, the shape of a PEA changes if charge or voltage
is exerted, and thus generating sub-nanometer
resolution actuation. However, PEAs suffer from the inherent
hysteresis and creep nonlinearities [6–8]. Many
feedforward and feedback methodologies have been proposed to
compensate for the hysteresis and creep nonlinearities of
PEAs [9, 10]. On the other hand, flexure-based
mechanisms are capable of transmitting high-precision motions
via the elastic deformations of the flexure hinges, making
it ideal in building the transmission chains for PEAs [11,
12]. Widely utilized flexure hinge profiles include circular
[13–16] and leaf [17, 18].
A single flexure hinge can be treated as a revolute
joint during micro- and nano-scale motions. In
literature, many analytical and empirical models have been
established for the compliance/stiffness of a single
flexure hinge [19–21]. In order to improve the performance,
multiple flexure hinges are generally combined in various
configurations, such as the parallelograms [22–24] and
the statically indeterminate symmetric (SIS) structures
. In these structures, it is common to treat the flexure
hinges as flexible, and all the other components as rigid.
Considering the widely-utilized lever mechanism as an
example, the lever is frequently assumed to be rigid [26,
27] so as to facilitate the design and modeling processes.
However, this assumption may increase the estimation
error of the analytical model, especially when the lever is
long or the compliance of the lever is not negligible.
A PEA is brittle and very weak when subjected to large
lateral forces or torques. As a result, a PEA is not allowed
to be firmly fixed to the mechanism during the
installation. Many commercial PEAs use ball tips to eliminate
the bending torques. In this case, a Hertzian contact
interface forms between the tip and the mechanism. One
significant drawback of Hertzian contact is its low
contact stiffness that consumes large portion of the PEA’s
displacement. The contact stiffness is highly dependent
on the material properties and the contact status.
Currently, there is no effective and reliable model to estimate
the contact stiffness. Thus, the contact stiffness is
frequently identified from the measured data .
As a flexure-based mechanism is generally light and
compact, its performance is likely to be affected by the
load mass, including the sensors, end-effectors, fixtures,
and other accessories installed on the mechanism. The
load mass increases the effective mass and moment of
inertia of the system, leading to a slow response. Thus,
the influence of the load mass should be taken into
consideration in the design and modeling of flexure-based
This paper presents the closed-form modeling of an
XY flexure-based nano-manipulator developed in our
previous work . In this nano-manipulator, the
flexure hinges are arranged in SIS configurations to transmit
linear or angular motions. Analytical modeling reveals
that the lever’s compliance significantly increases the
estimation error. Thus, a threshold is proposed to
determine whether the lever’s compliance can be neglected
or not. Subsequently, a systematic modeling
methodology is established to investigate the behavior of the
nano-manipulator during linear and angular motions.
Experimental results show that the modeling accuracy is
significantly improved if the influence of the lever’s
compliance, the contact stiffness, and the load mass is taken
2 Design of the XY Nano‑manipulator
An XY flexure-based nano-manipulator was
developed for nano manipulation tasks in our previous work
. The schematic diagram and the geometric
parameters of the nano-manipulator are presented in Figure 1,
where a central platform is connected to four rigid
linkages (consecutively labeled as linkage 1‒4) and then to
the fixed frame through leaf springs. The
nano-manipulator is symmetric in the x- and y-axes, thus
attenuating thermally induced errors and guaranteeing uniform
characteristics. In each axis, the displacement of a PEA
(P-843.30 from Physik Instrumente) is magnified by a
lever mechanism. Ball tip is selected to form a Hertzian
contact interface so as to protect the PEA against the
lateral forces and torques. Leaf and right circular
flexure hinges are adopted in the manipulator. These flexure
hinges are arranged into four different groups, namely
Structure I-IV (labeled as I-IV in Figure 1). Except for
Structure IV, the other structures are SIS structures with
“clamped-clamped” boundary conditions. Experimental
results showed that the cross-axis coupling ratio of the
nano-manipulator is below 1% .
3 Characteristics of the SIS Structures
3.1 In‑plane Compliance of a Single Flexure Hinge
Leaf and right circular flexure hinges are utilized in the
nano-manipulator. As illustrated in Figure 2, the
geometric parameters are the hinge length 2L0 and the
minimum thickness t. The shape functions of these hinges are
defined in Eqs. (
) and (
y(x) = t/2,
y(x) = r + t/2 − r 1 − x2/r2,
x ∈ [−L0, L0], (
where r is the radius of the circular profile, and for the
right-circular flexure hinge, r = L0.
As shown in Figure 2, the in-plane loads of the
flexure hinge are the moment about the z-axis (Mz) and two
forces in the x- and y-axes (Fx and Fy). The angular
deflection about the z-axis and the linear deflections in the
xand y-axes are denoted as θB, uB, and vB, respectively. The
bending moment Mz(x) and shear force Q(x) generated at
position x may be written as
Mz(x) = Mz + Fy(L0 − x),
Qy(x) = −Fy.
In the x-axis, the linear deflection at point B is defined
by the following equation:
2Ed −L0 y(x)
dx = FxP1, (
where E is the Young’s modulus, A(x) = 2dy(x) is the
cross sectional area of the hinge, and
For hinges with a rectangular cross section, κ = 5/6.
) can be rewritten into a matrix form:
uB P1 0 0
vB = 0 κEG P1 + L20P2 + P3 L0P2
θB 0 L0P2 P2
= C Fy ,
where matrix C is defined as the in-plane compliance
matrix of the flexure hinge.
In this paper, P1‒P3 are only dependent on the
geometric parameter of the hinge, and thus they are named as
3.2 Stiffness Modeling of the SIS Structures
Structure I-III can be schematically illustrated in
Figure 3. It is obvious that static indeterminacy causes axial
tension in lateral deformations, resulting in nonlinear
load-deflection relationship. However, the deflection of
a flexure-based mechanism is very small when compared
to the dimension of the mechanism. Thus, the above
structural nonlinearity can be treated as negligible .
This is also adopted in this paper.
The reaction forces and moments of the SIS structure
are defined in Figure 3. The static equilibrium conditions
lead to the following equations:
Fx − FAx + FBx = 0,
Fy + FAy + FBy = 0,
Mz − MAz + MBz − (FAy − FBy)(2L0 + L1) = 0.
There are six unknown variables in the above
equations. For this statically indeterminate problem, the
reactions of the structure can be solved using the flexibility
method. If we remove the constraints from point B and
treat the reactions FBx, FBy, and MBz as additional loads,
the original statically indeterminate structure can be
transformed into a statically determinate structure. The
transformed structure is equivalent to the original
structure only when the deflections of the transformed
structure at point B are the same as the original structure. As a
result, another three equations are derived:
2Ed −L0 y(x)
2Ed −L0 y3(x)
Subsequently, the angular deflection of point B about
the z-axis can be modeled as
L0 Mz(x) dx = 3(Mz + FyL0)
= (Mz + FyL0)P2,
where Iz(x) = 2dy3(x)/3 is the second moment of area
with respect to z-axis, and
Timoshenko beam theory is utilized to calculate the
linear deflection in y-axis:
= L0θB +
θ (x)dx −
2Ed −L0 y3(x)
dx + Fy κEG P1
= L0(Mz + FyL0)P2 + FyP3 + Fy κEG P1,
where θ(x) denotes the angular deflection at position x,
G is the shear modulus, κ is the Timoshenko shear
2Ed −L0 y3(x)
contribution of hinge 1
where L01 = L0+L1.
Linear superposition is used to facilitate the
calculation, and contribution of hinge 1 (or 2) refers to the
deflections of point B when only hinge 1 (or 2) is treated
as flexible. Utilizing Eqs. (
) and (
), we obtain the
reaction forces and moments as below:
FAx = 12 Fx, FBx = − 12 Fx,
FAy = − 12 Fy + 12 κEG P1L+0L120P12P2+P3 Mz,
FBy = − 12 Fy − 21 κEG P1L+0L120P12P2+P3 Mz,
MAz = L20 Fy + 21 κEκGEGPP1−1+LL0L2010P1P2+2+PP33 Mz,
MBz = L20 Fy − 12 κEκGEGPP1−1+LL0L2010P1P2+2+PP33 Mz.
Similarly, we can obtain the deflections at point O using
the linear superposition method, as shown below:
1 0 0
vO = 00 01 l11 C
Fx + FBx
Fy + FBy
= 0 κEG P1+P3
( κEG P1+P3)P2
2[ κEG P1+L201P2+P3]
Subsequently, the in-plane stiffness of the SIS structure
can be derived from Eq. (
kL = Fx uO = 2 P1,
kT = Fy vO = 2
kR = Mz θO = 2 P2 + kTL021,
κEG P1 + P3 ,
stiff in the longitudinal and transverse directions. Thus,
Structure III can be treated as an ideal revolute joint.
3.3 Stress Concentration of the SIS Structure
In Section 3.2, an SIS structure can be treated as rigid
in the longitudinal direction. Thus, the normal stress
caused by the axial load is not investigated herein.
During the lateral deformations, the normal stress caused by
the bending effect is the dominant stress. Thus, the
maximum stress locates on the outer surface of the hinge. The
maximum stress on the outer surfaces can be expressed
using the following equation:
σmax(x) = −kbσn(x) = − kby(x)Mz(x) = − 3kbMz(x)
where kb is the stress concentration factor for
bending, σmax and σn are the actual and nominal maximum
stresses, respectively. For the leaf hinges in Structure I
and II, the stress concentration has little influence on the
bending compliance calculation according to DU’s work
. As a result, kb can be set to 1 for Structure I and
II. For circular hinges in Structure III, according to the
generalized model established in CHEN’s work , the
stress concentration factor is calculated to be 1.030.
Due to the symmetry, hinge 1 is selected to
calculate the maximum stress. Based on Figure 3, the inner
moment at position x1 within hinge 1 is
Mz(x1) = MAz + FAy(L0 + x1),
where kL, kT and kR are the longitudinal, transverse, and
angular stiffness, respectively.
Substituting P1‒P3 into Eq. (
), the in-plane stiffness
of Structure I-III are obtained and provided in Table 1.
It is found that the difference between the
longitudinal and transverse stiffness is over 350 times. Therefore,
these structures can be treated as rigid in the
longitudinal direction. Table 1 also shows that Structure III is very
On the other hand, Structure III functions as a revolute
joint, and thus vO = 0. Substituting Eqs. (
) and (
), the relationship between the σx and vO is established:
−3kb[kT(L0 + L1)P2x1 + 2]
4dP2 r + t 2 − r
1 − x12 r2
) and (
) are utilized to obtain the maximum
allowable deflections of the SIS structure. For Structure
I and II, Eq. (
) shows that the maximum stress locates
at both ends of the hinge. For Structure III, the location
of the maximum stress can be obtained by differentiating
) to x1. Taking the yield strength of the material
into consideration, the maximum allowable deflections
of Structure I-III are calculated to be 1.46 mm, 1.30 mm,
and 5.349 mrad, respectively.
4 Statics and Dynamics Modeling
Monolithic flexure-based mechanisms exhibit
frictionless motions, resulting in an extremely low damping
ratio. Hence, the nano-manipulator can be approximated
as an undamped system. Based on Lagrange’s equation,
the dynamics of a system can be expressed as follows:
= Qi, i = 1, 2, . . . , N , (
where T and V denote the total kinetic and
potential energy of the system, respectively; qi and Qi are
the ith generalized coordinate and non-conservative
force, respectively; and N is the number of generalized
The first three modes of the nano-manipulator are
the linear motions in the x- and y-axes and the angular
motion about the z-axis. The linear motions in each axis
are the primary motions, whereas the angular motion
about the z-axis is an unexpected motion degrading the
motion accuracy. In this section, the dynamics models in
both linear and angular motions are established.
4.1 Influence of the lever’s compliance
The lever in flexure-based mechanisms is typically treated
as a rigid element [26, 27, 31] to facilitate the modeling
process. This approximation may affect the estimated
parameters of the overall system, e.g., the displacement
amplification ratio. Figure 4(a) examines the lateral
deformations of the lever in the nano-manipulator when
a lateral force is applied at the free end. The
contribution of Structure III is equivalent to a revolute joint with
a torsional stiffness of kR3. If the lever is assumed to be
rigid, the free end moves to point C′. However, the lever
is flexible, and the actual position of the free end is point
C, with a distance of δ to point C′. The distance δ is
negligible in very short levers while it becomes noticeable in
long levers. In this paper, the equivalent structure shown
in Figure 4(b) is proposed to account for the lever’s
compliance, where klever is the equivalent lateral stiffness of
the lever with “clamped-free” boundary conditions. The
modeling of klever is straightforward using the mechanics
of materials, and thus it is omitted for the conciseness of
this paper. In the lever mechanism, klever is calculated to
be 1.111 N/μm.
Figure 5(a) shows the deformation of the manipulator
in the x-axis, where linkages 2 and 4 remain stationary,
and the central platform, linkages 1 and 3 generate the
same displacement. This corresponds to the first/second
mode. The masses of the central platform, linkages 1 and
3 are denoted as m0, m1, and m3, respectively. In the
contact interface, point A is the end of the PEA and point
B represents the contact point on the lever. The lumped
mass model in the x-axis can be depicted in Figure 5(b),
where mL is the load mass, and Ilever denotes the moment
of inertia of the lever. In Figure 5(b), the equivalent
stiffness and the effective mass of the central platform are
defined in the following equation:
The output stiffness is the linear stiffness of the central
platform in the x or y axis that can be modeled as
kout = keq +
1+η η h2kPEA + kR3 klever
1+η η h2kPEA + kR3 + h23klever
The displacement amplification ratio of the lever
mechanism is the ratio between the displacements of points C
and B, which can be expressed as
= h3 · klever
h2 klever + keq
The influence of the lever’s compliance is significant. If
the lever is assumed to be rigid, klever converges to
infinity. In this case, kin, kout, and kamp will be overestimated.
On the contrary, if the lever’s compliance is considered,
the modeling complexity will increase significantly.
Therefore, a criterion is necessary to to decide whether
the lever’s compliance can be neglected or not. Based on
), klever can only be neglected if the following
two conditions are satisfied:
klever > 100keq,
h3klever > 100 h2kPEA + kR3 .
keq = 2(kT1 + kT2),
meq = m0 + m1 + m3.
From the static and dynamic point of view, the PEA
is equivalent to an active force generator. In Figure 5(b),
the equivalent stiffness, the driving force, and the
effective mass of the PEA are denoted as kPEA, FPEA, and mPEA,
respectively; and kcon represents the equivalent contact
stiffness of the contact interface. Further, a
dimensionless parameter, η = kcon/kPEA, is proposed to characterize
the contact stiffness. Three generalized coordinates are
defined in Figure 5(b), namely, the displacement of the
PEA (xPEA), the rotation angle of the lever (θlever), and the
linear displacement of the central platform (xeq).
In this nano-manipulator, the connection between the
PEA and the lever is not firm, and preload force is utilized
to keep the PEA and the lever in contact. The input stiffness
kin is defined as the linear stiffness at point B when no PEA
is installed. If kin is too low or the preload force is not large
enough, the detachment phenomenon may occur in large
step motions. However, if the input stiffness is too high,
the displacement of the PEA will be significantly reduced.
Based on Figure 5(b), kin can be calculated as follows:
keqklever · h232 + kR3 .
keq + klever h2 h22
) is the criterion to decide whether the lever
can be treated as rigid or not. In this nano-manipulator,
klever = 11.7keq and h3klever = 0.618(h22kPEA+kR3). As a
result, the lever’s compliance must be considered.
4.2 Dynamics of the Nano‑manipulator in the x‑axis
Based on Figure 5(b), the total kinematic and potential
energy of the nano-manipulator are given below:
T = 12 mPEAx˙PEA + 12 Ileverθlever + 12 meq + mL x˙e2q.
V = 21 kPEAxP2EA + 12 ηkPEA(h2θlever − xPEA)2
+ 12 kR3θl2ever + 12 klever h3θlever − xeq 2 + 21 keqxe2q.
Substituting Eqs. (
) and (
) into Eq. (
nanomanipulator’s equations of motion in the x-axis are
established as follows:
Mx¨ + Kx = u,
M = 0
meq + mL
As the nano-manipulator is not designed as a
highspeed scanner, only the first natural frequency is
investigated, and all the higher order dynamics is neglected.
The influence of the contact stiffness and the load mass
on the first natural frequency is analyzed and shown in
Figure 6. The variation range of η is 10−3 to 103,
corresponding to the cases of low and high contact stiffness,
respectively. When the contact stiffness is low, the first
natural frequency converges to its lower bound,
corresponding to the case when no PEA is installed. When
the contact stiffness increases, the first natural frequency
gradually converges to its upper bound. When η > 100,
the first natural frequency starts to converge. This
indicates that the contact interface can be treated as rigid if
η > 100. As the load mass increases the effective mass of
the nano manipulator, its influence is also obvious in
Figure 6: the first natural frequency decreases when the load
4.3 Static Analysis of the Nano‑manipulator
In static modeling, the velocities and accelerations are
zero. Substituting these into Eq. (
), we can solve for the
static relationships between the outputs and the input of
the nano-manipulator, as shown below:
xPEA = kin+η(kin+kPEA) · xPEA0,
θlever = kin+ηη(kkPinE+AkPEA) · xPhE2A0 ,
xeq = kin+η(kin+kPEA) · xPEA0.
where xPEA0 = FPEA/kPEA is defined as the nominal
displacement of the PEA (free extension without loads).
In order to investigate the influence of the contact
stiffness on the nano-manipulator’s static characteristics, the
following three dimensionless ratios are introduced to
characterize the actual displacement of the PEA, the
displacement applied to the lever, and the displacement of
the central platform, respectively:
xPEA , g2(η) = h2θlever , g3(η) =
As Figure 7 illustrates, if the contact stiffness is low,
g1 converges to its upper bound of 1, and both g2 and g3
converge to zero. As a result, the majority of the PEA’s
displacement is not transmitted to the lever, but
consumed in the contact interface. In contrast, if the
contact stiffness is high, both g1 and g2 converge to kPEA/
(kin+kPEA), and g3 converges to its upper bound of
kPEAkamp/(kin+kPEA). Therefore, in practice, it is desirable
to improve the contact stiffness so as to achieve larger
4.4 Angular Motion of the Nano‑manipulator
As illustrated in Figure 8(a), when a moment Mz is
applied on the central platform, the central platform and
all the linkages experience almost the same rotations,
denoted as θeq. Figure 8(a) actually shows the third mode
of the nano-manipulator. Based on the computational
analysis, the lever mechanisms are almost stationary.
Thus, this angular motion of the central platform has no
effect on the PEAs, and the installations of the PEAs will
not affect the angular behavior of the central platform.
The effective moment of inertia of the central platform,
linkages 1 and 3 are denoted as I0, I1, and I3, respectively.
The corresponding rotation centers of linkages 1‒4 are
labeled as Oi (i = 1‒4). During the angular motion, the
load mass will increase the central platform’s moment
of inertia. If the load mass is assumed to be distributed
uniformly across the central platform, the total kinematic
energy of the nano-manipulator can be written as
where IL≈mLI0/m0 is the moment of inertia of the load
For Structure I, the deformation is a rotation about the
z-axis. The deformation of Structure II is separated and
illustrated with dashed lines in Figure 8(b1). The
respective rotation centers are labeled as O′, O2′, and O′4. Since
the leaf springs and linkages are connected in
parallelogram configurations, during the angular motion, the
distance between points O′ and O′4 is constant and equal to
the distance between points O and O4:
′ 4′ = |OO4| = (t3 + 2l2 + l3) 2.
In Figure 8(b1), solid lines show a transformed
structure obtained by counter rotating Structure II by an angle
of −θeq. A further transformation is illustrated in
Figure 8(b2): flipping the deformation of the lower flexure
hinge. The above operations do not change the potential
energy of Structure II. Thus, the angular deformation of
Structure II is transformed to a linear deformation vO.
The relationship between vO and θeq can be established
based on the geometric constraints as below:
′ 4′ · sin θeq ≈ |OO4|θeq.
vO = O O
As illustrated in Figure 8(a), Structure IV rotates about
point O3 during the angular motion. If an identical copy
of Structure IV is connected to the opposite side, a
new structure is obtained, as shown in Figure 8(c). The
topology of the new structure is the same as SIS I. The
deformation is an angular displacement of θeq about the
Based on the above analyses, the total potential energy
of the nano-manipulator is derived as follows:
5 Experimental Verification
5.1 Experimental Setups
The static and dynamic characteristics of the
nanomanipulator are experimentally investigated to verify the
established models. Figure 9(a) shows the
experimental setup for the static test, where the applied force and
the resultant displacement of the central platform are
measured. This corresponds to the output stiffness. The
applied force is measured by a force gauge (HF-10 from
ALGOL), and the displacement is measured using a
displacement probe (GT21 from TESA Technology).
Figure 9(b) shows the experimental setup for the dynamic
test, where a modal hammer (ENDEVCO 2301-10 from
MEGGITT) is used to excite the nano-manipulator, and
the response is measured by two accelerometers (4507B
from Brüel & Kjaer) installed on the central platform.
During the experimental tests, the parameters of the
nano-manipulator with and without the PEAs installed
are measured individually. In the installations of the
PEAs, each PEA is bolt-fixed on the nano-manipulator,
and the preload force is manually adjusted. Based on the
previous analyses, higher contact stiffness is preferred
during the installation. The load mass is not measured
in the static test because it has no influence on the static
parameters of the nano-manipulator. In the dynamic test,
the load mass is measured to be 53.4 g, including the
fixtures and two accelerometers.
5.2 Statics of the Nano‑manipulator
The measured and estimated output stiffness of the
nano-manipulator are listed in Table 2. If no PEA is not
installed, the analytical results are obtained by
substituting kPEA = 0 and η = 0 into Eq. (
). In this case, the
estimation error of the analytical model (Analytical 1) is only
−2%. If the PEAs are installed, the output stiffness of the
nano-manipulator increases. As η is an unknown
variable, the lower and upper bounds of the analytical results
are provided. The measured output stiffness in each axis
is close to the upper bound of the analytical results,
indicating that high contact stiffness is achieved.
In order to investigate the influence of the lever’s
compliance, the analytical results with rigid lever assumption
(Analytical 2) are also presented in Table 2 for the
comparison. These analytical results are obtained by
assigning klever a large value according to the criterion defined
in Eq. (
). When no PEA is installed, the estimation
error with rigid lever assumption is 42%. Such a high
overestimation is not acceptable.
5.3 Dynamics of the Nano‑manipulator
The frequency responses of the nano-manipulator are
presented in Figure 10. Only the experimental results in
the x-axis are presented due to the symmetry of the
nanomanipulator. There are three peaks in the magnitude plot.
The first peak corresponds to the first (or second) mode
and the other two peaks are the unmodeled higher order
dynamics. It is clearly illustrated that the installation of
the PEAs only increases the first natural frequency.
The first natural frequencies in the x- and y-axes are
given in Table 3. Without the PEAs, the first natural
frequency in the x-axis is measured to be 320 Hz. If the
lever’s compliance is taken into consideration
(Analytical 1), the estimation errors in the x-axis is 3.3%. When
the PEAs are installed, the first natural frequency in the
x-axis increase to 429 Hz. In this case, the lower and
upper bounds of the analytical result are listed in Table 3.
The measured first natural frequency is very close to the
upper bound of the analytical result. This also
demonstrates that high contact stiffness is achieved. Similarly, if
the lever is assumed to be rigid (Analytical 2), the
modeling accuracy is significantly affected. When no PEA is
installed, the estimation error is 20.9%.
In current experimental setup, the third mode
(rotations about the z-axis) doesn’t show up in the measured
data. Therefore, the computational analysis is employed
to evaluate the nano-manipulator’s behavior during the
angular motions. Based on the computational results, the
third natural frequency with the 53.4 g load mass is found
to be 769.2 Hz. The analytical result is 754.1 Hz,
corresponding to an estimation error of 2%, and thus
validating the analytical model.
1. An XY flexure-based nano-manipulator is presented
in this paper. Two PEAs are employed to generate
actuations and the cross-axis couplings are
attenuated in the kinematic chains. The flexure hinges,
arranged in SIS configurations, function as prismatic
and revolute joints. Lever mechanism is utilized to
magnify the displacement of the PEA. It is found
that the lever’s compliance may significantly affect
the estimated parameters of the nano-manipulator,
such as the input/output stiffness and the first
natural frequency. In this paper, a criterion is proposed
to decide whether the lever’s compliance can be
neglected or not. The lever’s compliance can be
modeled by cascading a linear spring at the end of the
lever. Although simple in formulation, this
methodology is effective in improving the modeling accuracy,
as verified through experimental results.
2. The dynamics of the nano-manipulator in linear
and angular motions is analyzed. The influence of
the contact stiffness and the load mass is
analytically investigated. Higher contact stiffness results in
improved performances, such as larger workspace
and higher first natural frequency. The influence of
the load mass is also significant as it adds extra
inertia to the nano-manipulator.
3. The nano-manipulator is monolithically fabricated
using wire electrical discharge machining technique.
During the installation of the PEAs, the preload
forces of the PEAs are manually tuned for a high
contact stiffness. The analytical results show good
modeling accuracy in comparison with the experimental
results, and thus verifying the established models.
The methodologies proposed in this paper are
applicable in the design and optimization of flexure-based
YDQ designed the prototype, carried out the experiments and wrote the
paper. XZ participated in the revision of the paper. BS participated in the
design of experiments and revision of the paper. YLT and DWZ participated in
the mechical design and manufacture of the prototype. All authors read and
approved the final manuscript.
Yan‑Ding Qin is currently an associate professor at Institute of Robotics and
Automatic Information System, Nankai University, China. He received his PhD
degree from Tianjin University, China, in 2012. His research interests include
micro/nano manipulation and 3D bio‑printing.
Xin Zhao is currently a professor at Institute of Robotics and Automatic
Information System, Nankai University, China. He received PhD degree from
Nankai University, China, in 1997. His research interests include micro operation
robotics, MEMS, and biological pattern and tissue formation.
Bijan Shirinzadeh is currently a professor at Department of Mechanical and
Aerospace Engineering, Monash University, Australia. He received his PhD degree
from The University of Western Australia, Australia, in 1990. His research interests
include micro/nano manipulation, systems kinematics and dynamics, haptics
and robotic‑assisted surgery and microsurgery, and advanced manufacturing.
Yan‑Ling Tian is currently a professor at School of Mechanical Engineering,
Tianjin University, China. He received his PhD degree from Tianjin University,
China, in 2005. His research interests include micro/nano manipulation,
mechanical dynamics, surface metrology and characterization
Da‑ Wei Zhang is currently a professor at School of Mechanical Engineering,
Tianjin University, China. He received his PhD degree from Tianjin
University, China, in 1995. His research interests include micro/nano positioning
techniques, high speed machining methodologies, and dynamic design of
Supported by National Natural Science Foundation of China (Grant Nos.
61403214, 61327802, U1613220), and Tianjin Provincial Natural Science Foun‑
dation of China (Grant Nos. 14JCZDJC31800, 14JCQNJC04700).
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Springer Nature remains neutral with regard to jurisdictional claims in pub‑
lished maps and institutional affiliations.
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