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

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

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

Cross coupling of compliant mechanisms at

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

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.

Lever amplifier based on flexure hinges possesses the advantages of high
amplification ratio and simple structure

The two-stage amplifier.

According to the mechanical principle, supposing that an outside force

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.

Symmetrical structure with two 2-stage amplifiers.

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.

1-DOF symmetrical mechanism.

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

Simplified principle diagram of mechanism.

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

The 3-D diagram of the 2-DOF compliant mechanism.

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

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 (

The flexure hinges:

Supposing that an input displacement

Due to the symmetry, only half of the amplifier mechanism is analyzed.
According to the assumptions of literatures

Force diagram:

Substituting Eqs. (7) and (8) into Eqs. (5) and (6), the

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

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

For calculating the values of

The stiffness of output mechanism:

Output displacement of the mechanism,

As shown in Fig. 8b, the stiffness

According to the series-parallel relationship of the mechanism, the output
stiffness

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

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

Geometric parameters (mm) and properties of material of the mechanism.

For a 2-DOF motion platform, supposing that the

Assuming that the maximal input displacement is

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

Deformations:

To analyze free vibration of the 2-DOF micro positioning stage, the natural
frequencies are achieved by utilizing Lagrange's method. The coordinate
vector

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

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

For analyzing the amplification ratio of the proposed mechanism, an input
displacement with 20

Considering the workspace within the allowable maximum stress, the maximal
input displacement of the Eq. (39) is simultaneously applied at the

The stress distribution and deformation diagrams:

In order to analyze parasitic motion of the mechanism, the maximal input
displacement is separately applied to the

Output displacements and parasitic motions:

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

The first four modal shapes of the mechanism:

The verification of FEA:

Comparison results between calculated values and simulated values.

For further indicating the rationality of the design

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

Cross coupling in two axes.

The parameters of the architecture by using the Genetic Algorithm (GA) method
are optimized to obtain the best kinematic characteristic of the

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:

For the mechanism with a special thickness (

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

Maximum: Amplification ratio (

Variables to be optimized:

Subjecting to:

Input stiffness value

Natural frequency

Theoretical amplification ratio

Suffering from the restrictions of Eq. (45);

The ranges of parameters: 1.5 mm

The experimental setup. (1) Host computer, (2) signal amplifier and controller, (3) compliant mechanism, (4) laser sensors, (5) laser collectors, (6) dSPACE controlling system, (7) DAQ board, (8) PZT actuators.

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

After optimization, simulation is carried out for demonstrating performances
of the optimized micromanipulation stage. Maximum input displacements are
applied at

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

After fabricating, experimental setup of the micro-positioning stage is shown
in Fig. 14. Two PZTs with stroke of 90

The output displacements and parasitic motions:

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

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

The output displacements of

To further illustrate the kinematic performance of the micro-positioning
stage, the testing results of the

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.

Property comparisons for proposed

The tracking result and error:

From the Figs. 15 and 16, we can see that the output displacements of the

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

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

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