the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Design and optimization of full decoupled micro/nano-positioning stage based on mathematical calculation

### Yangmin Li

### Min Hu

Nano-positioning is widely used in Micro-electromechanical Systems (MEMS),
micromanipulator and biomedicine, coupling errors and tiny output
displacements are the main disadvantages of the one. A totally uncoupled
micro/nano-positioning stage with lever amplifiers is designed and tested in
this paper. It is fully symmetrical along with the *x*- and
*y*-directions.
For obtaining large output displacements, two fully symmetric two-stage lever
displacement amplifiers are utilized to amplify output displacements of
piezoelectric actuators (PZTs). The established models for performances
evaluation of the stage, in terms of kinetostatics, amplification ratio,
reachable workspace, the input and output stiffness, are verified by finite
element analysis (FEA). After that, the dimensional optimization is also
carried out through the genetic optimization algorithm.The prototype of the
mechanism is fabricated by using Wire-Electrical-Discharge-Machining (WEDM)
process. Testing results indicate that the proposed micromanipulator
demonstrates good performance.

Flexible hinge, which possesses these advantages of no backlash, no friction, simple structure, and easy manufacture, is widely applied in the micromanipulation system including micro/nano-positioning stages, micromanipulators, and high-accuracy alignment instruments (Tian et al., 2009). Micromanipulation system has been paid more and more attention in recent years, especially the parallel compliant mechanisms own some inherent advantages as big load capacity, high velocity, and high precision compared with serial ones (Dong et al., 2016). Piezoelectric actuators (PZTs) are usually selected as the actuators of the compliant mechanisms due to these advantages of fast response and high precision (Yong et al., 2009). Unfortunately, small output displacements of the PZTs and cross coupling motions of the end-effector have badly restricted the further application (Yu et al., 2011) and development of the compliant mechanisms (Yu et al., 2015).

At present, some amplifiers between the actuator and motion stage have also been proposed to overcome the disadvantage of small output displacement. Such as the lever displacement amplifiers (Tang and Li, 2015) and bridge-type amplifiers (Wu and Li, 2014) are commonly used as the bridge for amplifying the output displacements.

Cross coupling of compliant mechanisms at *x*- and *y*-directions is an
inevitable property in parallel mechanism (Dong et al., 2007). It can seriously
influence the precision of motion (Hao et al., 2016). The 2-degree-of-freedom
(2-DOF) compliant stages include two types of structure: One is in series and
another is in parallel (Fan et al., 2018). For example, Lin and Lin (2012) have designed a
series of *x**y* compliant mechanisms with the maximal cross coupling error of
0.12 µm and the frequency of 50 Hz. Additionally, Polit and Dong (2011) have
also proposed a high-bandwidth *x**y* positioning stage with the cross coupling
motion of 0.2 % at the total workspace of
15 µm × 15 µm in two directions. A kind of novel
*x**y* micromanipulation stage for micro/nano positioning and manipulation is
also developed by Qin et al. (2013), which has the workspace of
8 µm × 8 µm with the first natural frequency of
665.4 Hz, while the cross-axis coupling error is 2 % more than previous
results. In addition, Liu et al. (2015) have also designed a novel *x**y* parallel
compliant mechanism with the first natural frequency of 763.23 Hz more than
other papers.

Based on the aforementioned analyses, the micromanipulation stage is proposed in this paper with low cross coupling, large reachable workspace, high stiffness and high bandwidth. It aims to improve the positioning precision of micromanipulator. The theoretical calculation, simulation analysis, and prototype tests demonstrate that the presented 2-DOF micromanipulator with mechanical amplifiers owns the performance for cross coupling error of under 0.1 % at the full range of 169.6 µm × 165.3 µm, with the frequency of 348.31 Hz, which can be used in high frequency positioning micromanipulation system. Additionally, the positioning system will be controlled easily by reducing the coupling errors, which have the important applying value in the field of micro-nano manufacturing platform or micro 3-D printing system combining with the micro gripper. The main contribution of this paper is described as follows: (a) the structural design, kinematics and statics modeling analysis, and prototype test of this novel mechanism; (b) the lower cross coupling errors with larger workspace compared with other papers.

The remainder of this paper is organized as follows. The concept of mechanical amplifier and the process of mechanism design is described in Sect. 2. Then in Sect. 3, the kinematics and dynamics analyses of the mechanism with amplification ratio, stiffness, reachable workspace and natural frequency are presented in details. Besides, the performance evaluation and model verification implemented by the finite element analysis (FEA) are conducted in Sect. 4. Afterwards, structure optimization is carried out through genetic optimization algorithm in Sect. 5. Furthermore, prototype fabrication and performance tests are presented in Sect. 6. Finally, the conclusions and future work are presented in Sect. 7.

## 2.1 The concept design of lever amplifier

Lever amplifier based on flexure hinges possesses the advantages of high amplification ratio and simple structure (Choi et al., 2007). A two-stage amplifier is shown in Fig. 1.

According to the mechanical principle, supposing that an outside force
*F*_{in} is applied at free-end, the corresponding deformation
displacement is *δ*. Therefore, output displacements *d*_{out}
and *δ*_{out} are obtained. Considering Fig. 1, it can be
observed that the two-stage amplifier is composed of two one-stage
amplifiers. Assuming that the input displacement of two-stage magnifier is
also *δ*, the total theoretical magnification ratio can be written by

For enhancing the input stiffness, the parallel structure is used as an amplification mechanism for amplifying the input displacement as shown in Fig. 2. The mechanism is fully symmetric along with the central line and is connected by two amplifiers in parallel. The symmetrical structure of input-end of the mechanism can protect the PZT from damage.

## 2.2 2-DOF positioning stage

To eliminate cross coupling of the end-effector, symmetrical 1-DOF structure is shown in Fig. 3, where the double four-bar parallelogram mechanism is designed to connect the motion stage. The 4P-joint is composed of the four prismatic joints.

Simplified principle diagram of the 2-DOF positioning stage is presented as
shown in Fig. 4, where the four mechanical displacement amplifiers and two
double four-bar parallelogram mechanisms are utilized to design a fully
symmetrical mechanism along with *x*- and *y*-directions. Particularly, the
vertical distribution structure of two double four-bar mechanisms can
effectively decrease cross coupling.

After a series of elaborated designs, three-dimension (3-D) model of the
mechanism driven by PZT is proposed as shown in Fig. 5, input displacements
of the motion platform, which are conducted by the PZTs installed in the *x*-
and *y*-directions, can be magnified by the displacement amplifiers. Due to
the symmetry of the structure and the four-bar parallelogram, parasitic
motions of the positioning stage can be reduced or even eliminated. Circular
hinges are adopted in the mechanical amplifier to avoid the parasitic
motions.

There are many modeling methods to analyze the kinematic performance of compliant mechanism, such as the numerical model method, the pseudo-rigid-body (PRB) model method, and the compliance matrix model method (Tang and Li, 2013). In this section, kinetostatics and dynamics modeling of the compliant mechanism, including the amplification ratio (AR), input stiffness, reachable workspace and natural frequency, are analyzed by the numerical modeling method and Lagrange's method.

## 3.1 Amplification ratio and input stiffness analysis

Firstly, the amplification ratio and input stiffness of the mechanism are
analyzed. The flexure circular notched hinge can be regarded as the general
spring with bending stiffness (*k*_{b}) created by lateral force,
torsion stiffness (*k*_{t}) caused by the torque and linear stiffness
(*k*_{l}) produced by the axial force. Additionally,
${k}_{\mathrm{t}}^{\prime}$ and ${k}_{\mathrm{l}}^{\prime}$ are the torsion stiffness
and linear stiffness of the prismatic beam (Koseki et al., 2002). Therefore, the
rotational and linear deformations may simultaneously occur when a force is
applied at the free-end of the beam. As shown in Fig. 6, the simplified
diagrams of the circular notched hinge and prismatic beam are presented.
Additionally, the torque *M*_{t} is produced after a small
deformation angle Δ*θ* occurs along with the center of the hinge
from inside of the flexure hinge. Therefore, stiffness of different
directions for flexure hinge can be represented in the following equations:

where *E* is the elastic modulus, *b* is the thickness of the flexure hinge,
*r* and *t* are radius and smallest thickness of the circular notched hinge,
*a* and *l* are the width and length of the prismatic beam, respectively.

Supposing that an input displacement *δ*_{in} is applied at
the input-end of the mechanism, the corresponding input force and output
displacement can be expressed by

where *λ*_{AR} and *K*_{in} are the amplification
ratio and input stiffness of the micromanipulator, respectively.

Due to the symmetry, only half of the amplifier mechanism is analyzed. According to the assumptions of literatures (Su and Yang, 2001a), force analysis simplified diagrams of the mechanical amplifiers are drawn up as shown in Fig. 7, where the Fig. 7b and c are two kinds of different one-stage amplifiers (Su and Yang, 2001b). Therefore, in consideration of the force and moment balance at the equilibrium state of the beam 2, the following equations can be derived as

where *M*_{t} and ${{M}_{\mathrm{t}}}^{\prime}$ are bending moments of
the points *O*_{2} and *B*. Additionally, the forces ${F}_{{O}_{\mathrm{2}}y}$ and
${F}_{{B}_{y}}$ can be calculated through multiplying stiffness by displacement.
They are expressed by the following formulas

where ${k}_{{O}_{\mathrm{2}}y}$ and ${k}_{{B}_{\mathrm{l}}}$ are the lateral bending stiffness
caused by force ${F}_{{O}_{\mathrm{2}}y}$ and output stiffness of the point *B*,
respectively. ${k}_{{O}_{\mathrm{2}}\mathrm{t}}$ and ${k}_{{B}_{\mathrm{t}}}$ are the
rotational stiffness of the points *O*_{2} and *B*.

Substituting Eqs. (7) and (8) into Eqs. (5) and (6), the *θ*_{2} and
*δ*_{2} can be derived as

Therefore, the amplification ratio and input stiffness of the beam 2 can be calculated by the following equations

Then, substituting Eqs. (9) and (10) into Eqs. (11) and (12), following formulas can be obtained as

Similarly, considering the relationship of force and moment balance at the equilibrium state for the beam 1 as shown in Fig. 7b, the following formulas can be obtained by

where ${k}_{{O}_{\mathrm{1}}\mathrm{l}}$ and ${k}_{{O}_{\mathrm{1}}\mathrm{t}}$ are axial stiffness and
rotational stiffness of the point *O*_{1}; ${k}_{{O}_{\mathrm{3}}\mathrm{t}}$ is
rotational stiffness; *k*_{2} is proposed as the stiffness of the series
connection between the flexure hinge *O*_{3} and the beam 2, which can be
expressed by

where ${k}_{{O}_{\mathrm{3}}\mathrm{l}}$ is axial stiffness of the flexure hinge *O*_{3}.
Combining the Eq. (15) into Eq. (18), the *θ*_{1} and *δ*_{1} can
be calculated by

Therefore, according to the Fig. 7b, amplification ratio and input stiffness of the beam 1 can be represented by

Substituting Eqs. (19) and (20) into Eqs. (21) and (22), they can be rewritten by

Consequently, the total amplification ratio of the two-stage lever displacement mechanism is calculated by

Additionally, total input force can be obtained by ${F}_{\mathrm{in}}=\mathrm{2}{F}_{{A}_{y}}$, since the amplifier is symmetric and is connected in parallel. Therefore, input stiffness can also be calculated by

For calculating the values of *k*_{in} and *λ*_{AR},
the stiffness (${k}_{{B}_{\mathrm{l}}}$) of the output point *B* must be firstly
computed. According to the Fig. 4, it can be observed that the output
stiffness of the micromanipulator is the stiffness of components except from
the driven amplifier. Simplified diagrams of output components are shown in
Fig. 8 when the platform is driven in *y*-directions. The marked signs,
${{k}_{\mathrm{l}}}^{\prime}$ and ${{k}_{\mathrm{t}}}^{\prime}$, are axial
stiffness and rotational stiffness of the prismatic beam, respectively. And
*k*_{1} is the input stiffness of the two-stage amplifier when the input-end
is at the point *B* and output-end is at the point *A* as shown in Fig. 7a.

Output displacement of the mechanism, *d*_{out}, can be obtained
when an input displacement *d*_{in} is applied at *y*-direction,
whereas the amplifier of the *x*-direction will still hold since they have
large transverse stiffness (Tang and Li, 2014). In addition, all rotational angles of
the prismatic beams, (*θ*), are the same due to the identical
structure. The potential energy stored in the flexible hinges can be derived
as follows:

where *α*_{1}, *β*_{1} and *γ*_{1} are rotational angles of
flexible hinges of the amplifier and can be written by

As shown in Fig. 8b, the stiffness *k*_{N} can be achieved by

According to the series-parallel relationship of the mechanism, the output
stiffness *k*_{out} can be expressed by the following formula

where $M=\frac{\mathrm{1}}{{l}_{\mathrm{4}}^{\mathrm{2}}}+\frac{\mathrm{2}}{\mathrm{9}{l}_{\mathrm{2}}^{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{25}{l}_{\mathrm{1}}^{\mathrm{2}}}$, ${{k}_{\mathrm{t}}}^{\prime}$ and ${{k}_{\mathrm{l}}}^{\prime}$ are rotational stiffness and axial stiffness of the prismatic beam.

Because circular notched hinges and prismatic beams used in this study are identical, thus their stiffness can be expressed by

Due to the symmetry of the mechanism, the stiffness ${k}_{{B}_{\mathrm{l}}}$ can be obtained by

Combining the Eqs. (25) and (32), at the same time considering Eq. (2), total
amplification ratio and input stiffness of the mechanism, which can be
represented by the designed parameters including *r*, *t*, *a*, *b* and
${l}_{i}(i=\mathrm{1},\mathrm{\dots},\mathrm{5})$, are not provided here due to page limitations. To
conveniently calculate the corresponding calculations, geometric parameters
and properties of the material (AL7075-T6) of mechanism are listed in
Table 1.

## 3.2 Reachable workspace analysis

For a 2-DOF motion platform, supposing that the AR_{m} is
denoted as the amplification ratio of the mechanism and *S* is looked as the
stroke of the PZT actuator, the workspace of the micromanipulation stage is
represented by AR_{m}*S*×AR_{m}*S*. In
this study, the bending stress only occurs on the cross section of flexible
hinge when axial tension and shearing effects are not taken into
consideration. Thus stress (*σ*_{r}) can be calculated as the following
equation:

where *σ*_{y} and *s*>1 are the yield strength and the safety factor
of the material. In addition, maximal stress ${\mathit{\sigma}}_{r}^{max}$ occurs when
the angular displacement *θ*_{max} is maximum. It can be expressed
using the following formula

where *k*_{c} and *f*(*β*) are the dimensionless concentration
factor and compliance factor, respectively. They are expressed by
(Smith, 2000).

where $\mathit{\beta}=t/\mathrm{2}r$ is the dimensionless geometry factor.

Assuming that the maximal input displacement is ${\mathit{\delta}}_{\mathrm{in}}^{max}$, corresponding to the output displacement of the motion stage is ${\mathit{\lambda}}_{\mathrm{AR}}{\mathit{\delta}}_{\mathrm{in}}^{max}$. According to the mechanical
principle, the maximal stress may occur on the hinge *O*_{3} connecting the two
lever beams, since it not only suffers from the angular deformation created
by output displacement, but may be also subjected to the angular deformation
produced by the input displacement. Therefore, the maximal deformation angle
can be derived by

Substituting Eqs. (37) into (34) and considering the Eqs. (36) and (33) at the same time, the maximal input displacement can be calculated as follows

Substituting values of the Table 1 into the Eq. (38) and letting the safety factor be 1.78. Maximal input displacement can be calculated as follows

Considering the amplification ratio of the mechanism, the output displacement of the stage can be 236.2 µm. Due to the symmetry of the mechanism, so the reachable workspace of the platform can be expressed as 236.2 µm × 236.2 µm.

## 3.3 Natural frequency analysis

To analyze free vibration of the 2-DOF micro positioning stage, the natural
frequencies are achieved by utilizing Lagrange's method. The coordinate
vector $\mathit{\eta}={\left[{\mathit{\eta}}_{\mathrm{1}},{\mathit{\eta}}_{\mathrm{2}}\right]}^{T}$ is used to describe
the motions of *x*- and *y*-axes. The kinetic energy (*T*) and potential
energy (*V*) stored in the mechanism can be expressed by the selected
generalized coordinate and their derivations (Li et al., 2013). And then,
substituting the kinetic and potential energies into the following Lagrange's
equation

where $i=\mathrm{1},\mathrm{2}$, corresponds to the free vibration of *x*- and *y*-directions
of the stage.

where the equivalent mass **M**=diag{M} and
stiffness **K**=diag{k} are the 2×2
diagonal matrices, along with

where ${m}_{i}(i=\mathrm{0},\mathrm{1},\mathrm{\dots},\mathrm{5})$ are shown in the Figs. 2 and 5.

Solving the Eq. (41), the natural frequency of the stage can be obtained as

which has the unit of Hertz.

In this section, the established models to evaluate the properties of the 2-DOF micromanipulation stage on aspects of amplification ratio, input stiffness, reachable workspace, and natural frequencies are verified by using FEA software (Li et al., 2012).

## 4.1 Model verification

For analyzing the amplification ratio of the proposed mechanism, an input
displacement with 20 µm is separately provided in the *x*- and
*y*-directions and the deformation of the mechanism is shown in Fig. 9a and
b, respectively. Therefore, the corresponding output displacements can be
measured by selecting the points of the surface of the central platform and
the average values of them are calculated with 87.58 and 87.57 µm.
Additionally, the force can be also measured with 43.20 *N*. Therefore, the
amplification ratio and input stiffness of the mechanism can be calculated as
4.38 and 2.16 *N* µm^{−1}, respectively.

Considering the workspace within the allowable maximum stress, the maximal
input displacement of the Eq. (39) is simultaneously applied at the *x*- and
*y*-directions. Thus, the maximal stress distribution and the corresponding
displacement deformations are shown in Fig. 10. It can be seen that the
maximal stress is 278.1 MPa at the point *O*_{3}, which is less than the
allowable stress of the material. The Fig. 10b, c and d indicate the maximal
output displacements of the whole mechanism. Based on the aforementioned
analysis, the reachable workspace of the mechanism is measured with
215.50 µm × 215.50 µm.

In order to analyze parasitic motion of the mechanism, the maximal input
displacement is separately applied to the *x*- and *y*-axes. As shown in
Fig. 11a and b, which is the output displacement of *x*-direction with
215.50 µm and parasitic motion at *y*-direction with
0.034 µm, respectively. Similarly, the Fig. 11c and d are the
output displacement at *y*-direction with 215.50 µm and parasitic
motion of *x*-direction with 0.036 µm. The parasitic motions of the
*x*- and *y*-directions may be caused by the model errors and the
deformations of the prismatic joint.

For verifying the dynamic model with the natural frequencies, the first four
modal shapes of the structure are expressed in Fig. 12. The first modal shape
is the rotational motion, which has the frequency with 179.99 Hz. The second
and third modal shapes in the *x*- and *y*-directions are almost the same
with values of 354.21 and 355.32 Hz, respectively. The fourth modal shape of
the *Z*-direction is about 867.23 Hz.

## 4.2 Discussions

For further indicating the rationality of the design (Li and Wu, 2016), a
comparison between the calculated values and the simulated values of FEA, in
terms of amplification ratio, input stiffness, natural frequency, and maximum
output displacement, is listed in Table 2. It can be observed that the
maximal error is the output displacement and amplification ratio by using the
benchmark of FEA. The reasons may be that the transverse deformation caused
by the transverse force as shown in Fig. 7 is not taken into account in the
calculated models and flexure hinge itself. In addition, the maximal stress
is under the truly calculated value. The natural frequencies of the *x*- and
*y*-directions are almost equivalent, which demonstrate that their
performances are identical.

Cross coupling is the key performance of the 2-DOF micro-positioning stage.
As shown in Fig. 11, the results of the maximum output displacements in two
axes are listed in the Table 3. The output displacement of *x*-axis is
215.50 µm when input displacement is separately applied at
*x*-direction, while the parasitic displacement of *y*-direction caused by
*x*-direction is only 0.034 µm. Taking input displacement as the
benchmark, coupling error of the *y*-axis is only for 0.1 %. Similarly,
the parasitic motion of *x*-direction caused by *y*-direction is
0.036 µm with the output displacement is 215.50 µm at
*y*-direction. Thus, coupling error on the *x*-axis is only for 0.1 %
more than that of *y*-axis. Additionally, output displacements of *x*- and
*y*-axes are up to 214.99 and 215.01 µm when the input
displacements are simultaneously applied in the *x*- and *y*-directions.
Based on the aforementioned analysis, the 2-DOF micromanipulation stage has
low cross coupling and owns the best decoupling property.

The parameters of the architecture by using the Genetic Algorithm (GA) method
are optimized to obtain the best kinematic characteristic of the *x**y* stage
(Wang et al., 2014). In practical applications, the actual amplification ratio of
mechanical amplifier is less than the theoretical value, which is arisen from
the combination of the deformation of hinge and lever arm bending.
Additionally, considering analytical models overestimate the performance of
the stage with deviations around 5 %–30 %, a compensation factor,
*ε*=0.75, is used in the optimization process to rebuild the
models.

## 5.1 Optimization problem description

Based on aforementioned equations, which reveal that the performances of the
2-DOF micro-positioning stage rely on the relative parameters. All
established models mainly include these parametric variables: *r*, *t*, *a*,
*b*, *l*_{1}, *l*_{2}, *l*_{3}, *l*_{4}, *l*_{5}, which influence the
synthetical property of the stage.

For the mechanism with a special thickness (*b*=10 mm in this study), only
eight parameters (*r*, *t*, *a*, *l*_{1}, *l*_{2}, *l*_{3}, *l*_{4}, *l*_{5})
need to be optimized. While other parameters can be determined by taking into
account the length and width restrictions of the PZT with the addition of an
appropriate assembling space. Otherwise, input stiffness of the mechanism
should not surpass the minimum stiffness of the PZT. According to the
geometrical relation of the structure, these variables should satisfy the
following equation

With the selection of the amplification ratio of the stage as an objective function, the optimization process can be stated as follows:

- a.
Maximum: Amplification ratio (

*λ*_{AR}≥7). - b.
Variables to be optimized:

*r*,*t*,*a*,*l*_{1},*l*_{2},*l*_{3},*l*_{4}, and*l*_{5}. - c.
Subjecting to:

- 1.
Input stiffness value

*ε**k*_{in}≤*K*_{PZT}; - 2.
Natural frequency

*ε**f*≥150 Hz; - 3.
Theoretical amplification ratio AR≥7;

- 4.
Suffering from the restrictions of Eq. (45);

- 5.
The ranges of parameters:

1.5 mm ≤*r*≤2.0 mm, 0.3 mm ≤*t*≤0.8 mm,

0.3 mm ≤*a*≤1 mm, 5 mm ≤*l*_{1}≤15 mm,

7 mm ≤*l*_{2}≤20 mm, 5 mm ≤*l*_{3}≤15 mm,

20 mm ≤*l*_{4}≤40 mm, 28 mm ≤*l*_{5}≤40 mm.

- 1.

## 5.2 Optimal results

The Genetic Algorithm (GA) is adopted in the current issue due to its
superiority of fast convergence, fewer calculating time and higher robustness
over other method such as simulated annealing algorithm (Mccall, 2005). The
optimization process is implemented by the GA toolbox of the MATLAB software
and the results of optimized parameters are: *r*=1.5 mm, *t*=0.78 mm,
*a*=0.5 mm, *l*_{1}=5.47 mm, *l*_{2}=14.57 mm, *l*_{3}=7.90 mm,
*l*_{4}=27.94 mm, *l*_{5}=28 mm, which will lead to an *x**y* stage with
*λ*_{AR}=7.87, *k*_{in}=13.39 *N* µm^{−1} and the natural frequency *f*=367.07 Hz.

After optimization, simulation is carried out for demonstrating performances
of the optimized micromanipulation stage. Maximum input displacements are
applied at *x*- and *y*-axes and the total deflection and stress distribution
of the mechanism are expressed in Fig. 13. It is observed that the output
displacement of *x*-axis is 94.90 µm, and the corresponding input
force is 214.63 *N*. Therefore, the optimized mechanism has an amplification
ratio of 4.75 with an input stiffness of 10.63 *N* µm^{−1}. In
addition, maximal stress is 263.68 MPa, which is far less than the yield
strength of the material.

Furthermore, the first four natural frequencies are also analyzed and their
values are 174.26, 348.31, 350.14 and 850.65 Hz, respectively. They are all
less than the frequencies before optimizing and the frequencies of *x*- and
*y*-directions are very close to the calculated values, which demonstrate
that the structure optimization is effective.

## 6.1 Experimental setup

After fabricating, experimental setup of the micro-positioning stage is shown in Fig. 14. Two PZTs with stroke of 90 µm (model P-840.60 produced by Physik Instrumente, Inc.) are adopted to drive the micromanipulation stage, and the PZTs are actuated with a voltage of 0–100 V through a three-axis piezo-amplifier and driver (E-509 from the PI). Two laser displacement sensors and collectors (Microtrak II, head model: LTC-025-02, from MTI Instrument, Inc.) are used to measure the end-effector displacements of the two axes. The analog outputs of two sensors are connected to a PCI-based data acquisition (DAQ) board (PCI-6143 with 16-bit A/D convertors, from NI Corp.) through a shielded I/O connector block (SCB-68 from NI) with noise rejections. The digital outputs of the DAQ board are read by a host computer simultaneously.

Since the sensitivity of the laser sensor is 2.0 mm/10 V and the maximum value of 16-bit digital signal corresponds to 10 V, the resolution of the displacement detecting system can be calculated as

However, due to a considerable level of the noise, the resolution of the sensor is claimed ±0.2 µm by the manufacturer. In the process of practical precision motion control experiments, it is found that the limitation of the fabricated prototype mainly arises from the laser displacement sensors which have a not-high resolution of ±0.2 µm.

## 6.2 Open test

To describe the dynamic properties of the proposed mechanism, the
corresponding open-loop tests are carried out by using the dSPACE real-time
simulation control system. As shown in Fig. 15a and b, which demonstrate
respectively the output displacements and parasitic motions of *x*- and
*y*-directions when the input voltage is $u=\mathrm{30}\mathrm{sin}(\frac{\mathit{\pi}}{\mathrm{4}}t-\frac{\mathit{\pi}}{\mathrm{2}})+\mathrm{30}$. The results indicate that the actual amplification
ratio of the mechanical amplifier in *x*-direction by comparing the input and
output displacements is about 3.51 and the parasitic motion for the
*y*-direction is about 0.32 µm. While one of *y*-direction is
calculated as 3.43 and the parasitic motion for the *x*-direction is only
0.31 µm. Based on the aforementioned tests, the experimental
results are less than the analytical results. The main reasons may come from
the assembly errors of the system and the preloaded force of the PZT.
Additionally, parasitic motions of the *x*- and *y*-axes are very small,
which can be ignored. Therefore, testing results demonstrate that the
designed mechanism has excellent decoupled performance.

To further illustrate the kinematic performance of the micro-positioning
stage, the testing results of the *x*- and *y*-axes are shown in Fig. 16 when
the two PZTs at *x*- and *y*-directions are driven simultaneously. The
results indicate that the output displacements of *x*- and *y*-directions are
not identical. The errors may be created by the preloaded force of PZTs and
manufacturing defects between two axes.

## 6.3 Tracking experiments

For further validating the performances of micro/nano-positioning platform, a robust tracking controller is used to obtain the well tracking effect. The reference input displacement is the signal with different frequencies since the hysteresis phenomenon of piezoelectric actuator is a rate-dependent hysteresis. Therefore, the robust controller can effectively eliminate the drawback caused by rate-dependent of the PZT-actuated micromanipulator. The tracking results and corresponding errors is shown in Fig. 17, where output displacement is well tracking with the input displacement and the error is low than 0.01 %. Thus the optimal design is suitable for this compliant mechanism.

## 6.4 Discussions

From the Figs. 15 and 16, we can see that the output displacements of the
*x*- and *y*-axes are not exactly same, the main reasons maybe come from the
errors coming from installation and manufacturing. Thus, the rotation of
end-effector can exist, but it may be minor affections for the other
orientation.

Based on aforementioned modeling, analyzing, and testing, the results demonstrate that the proposed 2-DOF micro-positioning stage with mechanical amplifier owns some advantages, in terms of large motion, and low cross coupling. A comparison with other proposed 2-DOF stages are completed in Table 4. The natural frequencies of the presented stages in Ref. 3 and Ref. 11 are higher than other mechanisms, but their working ranges are very small, which seriously limit their further applications. Additionally, the performances in terms of cross coupling and workspace of the other two stages presented in Ref. 5, Ref. 9 and Ref. 12 are obviously lower than the proposed in this study.

In this paper, a novel fully decoupled *x**y* micro-positioning stage with
lever amplifier has been proposed. The designing process of the stage is
provided in consideration of the decoupled property of the output motions. In
addition, the analytical method and Lagrange's method are adopted for the
kinematics and dynamics modeling of the mechanism with amplification ratio,
stiffness, reachable workspace and natural frequency. The modeling
verification and performance evaluation are carried out by FEA. Considering
the performance requirement, a series of structural optimizations by using GA
method have been implemented to improve the amplification ratio. Finally,
prototype fabrication and experimental tests are implemented in detail. All
results indicate that the maximal cross coupling of the 2-DOF
micro-positioning stage is less than 0.1 % under the workspace for
169.6 µm × 165.3 µm with the natural frequency
of 348.31.

For further study, intelligent controller is going to be considered to control the micro-positioning stage and precise position tracking will be carried out in our future work.

This experimental data can be downloaded at https://pan.baidu.com/disk/home?errno=0&errmsg=Auth Login Sucess&&bduss=&ssnerror=0&traceid=#/all?path=/&vmode=list (last access: 20 November 2018).

The main contribution for ZW includes the structural design, modeling analysis and the control. The contribution for co-author MH includes the structural optimization. And the contribution for co-author YL includes the structural design and the English writing error and grammar modification.

The authors declare that they have no conflict of interest.

This work was supported in part by Science and technology research project of
department of education, Jiangxi, China (GJJ170568), National Natural Science
Foundation of China (51575544, 51275353), Research Committee of The Hong Kong
Polytechnic University (1-ZE97, G-YZ1G).

Edited by: Xichun Luo

Reviewed by: Calin-Octavian Miclosina
and one anonymous referee

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