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Advances in Mechanical Engineering
Volume 2012 (2012), Article ID 962439, 11 pages
http://dx.doi.org/10.1155/2012/962439
Research Article

Pure Nano-Rotation Scanner

1Department of Mechanical Engineering, Dong-A University, Busan 604-714, Republic of Korea
2School of Mechanical Engineering, Yeungnam University, Gyeongsan 712-749, Republic of Korea
3Department of Mechanical System Engineering, Chonbuk National University, Jeonju-si 561-756, Republic of Korea

Received 10 July 2012; Accepted 7 August 2012

Academic Editor: Mehdi Ahmadian

Copyright © 2012 Moo-Yeon Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We developed and tested a novel rotation scanner for nano resolution and accurate rotary motion about the rotation center. The scanner consists of circular hinges and leaf springs so that the parasitic error at the center of the scanner in the X and Y directions is minimized, and rotation performance is optimized. Each sector of the scanner’s system was devised to have nano resolution by minimizing the parasitic errors of the rotation center that arise due to displacements other than rotation. The analytic optimal design results of the proposed scanner were verified using finite element analyses. The piezoelectric actuators were used to attain nano-resolution performances, and a capacitive sensor was used to measure displacement. A feedback controller was used to minimize the rotation errors in the rotation scanner system under practical conditions. Finally, the performance evaluation test results showed that the resonance frequency was 542 Hz, the resolution was 0.09 μrad, and the rotation displacement was 497.2 μrad. Our test results revealed that the rotation scanner exhibited accurate rotation about the center of the scanner and had good nano precision.

1. Introduction

Advanced technologies such as semiconductors and liquid crystal displays are at the forefront of the information technology industry. The fundamental requirement of this industry is ultra precision. Ultra precision technology that relates to precise nano motion is known as nano-positioning technology and has been used in various devices, such as ultra-precision machine tools, semiconductor steppers and scanners, atomic force microscope (AFM) actuators, and three-dimensional (3D) measuring instruments, with the goal of ensuring precision at the nanometer level. Various precise positioning stages have been developed, such as piezoelectric-driven and flexure-based Scott-Russell linkages [13], a compact precision XY scanner with a voice coil motor and double compound linear spring flexure guide mechanism [4], and a two-dimensional (2D) metrological AFM system with minimal parasitic errors [5]. To obtain a curved-face structure and a free curved-face diffraction grid when molding an aspheric lens, scanners with 6 degrees of freedom (DOF) are required. A nano-rotational scanner should rotate at the rotation center with no parasitic errors. Even though there have been extensive attempts to develop scanners capable of linear displacement, as well as studies verifying their performance, there have been few studies of rotational scanners. Those that have been performed have focused on rotation with coupled planar stages. For example, Yao et al. [6] reported on the parallel kinematics of a multiaxis nano-positioning stage, and Wu and Zhou [7] used an inchworm type actuator for the XYθ mechanism. Voice coil motors and air bearing guides, a dual-stage actuation system including a voice coil motor, and a compliant mechanism consisting of quad-symmetric simple parallel linear springs and quad-symmetric double compound linear springs have also been reported [810]. Moreover, Chang and Sun [11] presented an analysis and control procedure for a monolithic piezoelectric nanoactuator of a six DOF manipulator and Chang et al. [12] reported the development of a three DOF piezo-driven micropositioner along with parametric analyses and FEM for low crosstalk interference. Hsiao and Lin [13] created a monolithic flexure hinge with three DOF using a model for the analysis and design of the flexure and finite elements since a planar scanner, where , , and direction motions are coupled on a flat surface, is a simple structure. Because coupled planar stages usually have parasitic motion errors, each motion axis should be independent and decoupled from the others. A pure rotational scanner should have an exact rotation axis with no parasitic motion errors.

In this paper, we describe the design of a pure rotational scanner that has an accurately defined rotation center with minimal parasitic errors. We first derive the theoretical motion equations for the scanner, then describe the design optimization process, showing how parasitic motion errors other than those in a rotary direction were minimized based on the derived equations. Also, we describe our verification of the design optimization using finite element analysis (FEA). The final version of the scanner is designed using iterated simulations. The scanner is implemented and verified via the various experimental techniques. Finally, the experimental parasitic motion error about the rotation center is shown.

2. Modeling

2.1. Presimulation

Figure 1 is a schematic diagram of the proposed rotational scanner. Two multilayered piezo-electric (PZT) actuators generate opposing forces on both sides of the rim to generate a rotation moment around the rotation center. Four L-shaped flexure leaf springs are located outside the rim to improve -axis stiffness. Circular hinges located inside the rim provide the rotation center with enhanced parasitic direction stiffness and enable smooth rotation. A fixed base is located inside the rim to provide a single DOF for the scanner. Accordingly, if the moment were applied through the PZT actuators, the rotation center would rotate with minimal parasitic motions.

962439.fig.001
Figure 1: Schematic diagram of the pure rotational scanner. 1: Rim, 2: rotation center, 3: circular hinge, 4: L-shaped flexure leaf spring and 5: fixed base. is the actuated force by pair of the PZT actuators.

Figure 2 shows the FEA results for the rotation center in detail. The center of the scanner is not fixed at the base, but it does not move away from the rotation center and rotates accurately when the rotary moment is applied to the rim. Because the rotation center is not physically fixed, this is termed a virtual pivot point.

962439.fig.002
Figure 2: Virtual pivot point for the center of rotation.
2.2. Theoretical Modeling of the Rotational Scanner

The vertical and horizontal stiffness values for the L-shaped flexure leaf spring guide, shown in Figure 3, were derived as follows [14]: Here, is the elastic modulus, is the moment of inertia, and is the length of the leaf spring. We used the same lengths of element 1 and element 2. The horizontal stiffness is the same as the vertical stiffness due to the symmetry of the structure.

962439.fig.003
Figure 3: Schematic diagram of the flexure element. is the horizontal force, is the force acting on each axis, is the fictitious vertical force, and is the fictitious vertical moment.

The schematic diagram used for modeling rotation stiffness due to the four L-shaped flexure springs is shown in Figure 4. If the radius of the external diameter of the rim of the scanner is defined as and is displacement of , then the moment, , is expressed by (2). Further, the central angle of the rim is obtained by (3):

962439.fig.004
Figure 4: Schematic representation of rotation stiffness. F is Force exerted by the PZT actuator, is Rotation displacement of the main body.

The total rotation stiffness of the scanner is given by

To determine the stiffness of the circular hinge shown in Figure 5, we used a formula based on Schotborgh’s experimental test [15]. This formula was derived using several factors, including thickness , diameter , and width .

962439.fig.005
Figure 5: Schematic representation of the circular hinge. is circular hinge diameter, is flexure hinge thickness.

Dimensionless stiffness along the -axis, , is given by

Dimensionless stiffness along the -axis, , is given by

Dimensionless rotation stiffness, , is given by

Dimensionless rotation stress, , is given by

The total stiffness of the rotational scanner was calculated from the stiffness equations deduced for each direction. The total rotation and translational ( and direction) stiffness values are given by the sum of the stiffness values for the L-shaped flexure guide and for each of the 8 circular hinges inside the rim. Here, the direction stiffness is the same as that of the direction due to the symmetrical structure of the rotational scanner:

The first three resonance frequencies of the scanner were determined by (11) and (12):

To verify the theoretical modeling results, the FEA was conducted using commercial software (Pro-Mechanica, PTC Corp.), as shown in Table 1. The overall stiffness of the rotational scanner had an error of less than 10%, and hence (9)–(12) are acceptable. Additionally, the L-shaped flexure leaf springs were predominantly responsible for the rotary stiffness of the scanner, whereas the circular hinges were predominantly responsible for stiffness in the and directions.

tab1
Table 1: Comparison of theoretical and FEA results (diameter of circular hinge = 10 mm, neck thickness of circular hinge = 0.48 mm, length of L-shaped flexure leaf spring = 10 mm, thickness of L-shaped flexure leaf spring = 0.4 mm).

3. Design Optimization

To optimize the performance of the rotational scanner using the aforementioned equations, we ran a design optimization process in MATLAB (Mathworks Corp.). The objective was to minimize translational parasitic errors at the rotation center. Hence, the cost function aimed to maximize the ratio between the translational and rotational stiffness of the rotational scanner as follows:

Table 2 lists the constraints used in the optimal design. To ensure high-speed performance, resonance was fixed at >200 Hz, and the maximum allowable stress was limited to <200 MPa in consideration of the material used (Al-6061; maximum allowable stress 240 MPa). To limit the overall size of the scanner, the diameter was fixed at 100 mm and the diameter of the circular hinge at 20 mm. These constraints were also used to maximize the relative rotational displacement.

tab2
Table 2: Optimal design constraints.

Table 3 shows the nine optimal design parameters: length and thickness of the flexure guide and L-shaped flexure leaf springs; thickness and diameter of the circular hinges; the total thickness of the scanner used.

tab3
Table 3: Design variables of the rotational scanner.

The sequential quadratic programming (SQP) routine of MATLAB was employed in the optimization process. Random values were used as the initial upper and lower boundary values. The convergence plot resulting from the optimization process is shown in Figure 6. To ensure a global minimum value, the optimization process was performed 10 times with the random initial values shown in Figure 7.

962439.fig.006
Figure 6: Convergence plot of the optimization process.
962439.fig.007
Figure 7: Convergence of cost function at 10 different initial values. This plot shows that optimized results are global minimums.

Table 4 lists the optimal and implemented design values for the optimized design. The values used in production are shown in the column on the right. The theoretically obtained resonance frequency was 517 Hz, which is very high. The final rotation angle was 0.03°.

tab4
Table 4: Optimization results.

Table 5 lists the results of FEA simulations obtained using the implemented design values. Stiffness in the and directions was considerably higher than that in the rotary direction, indicating that parasitic errors were minimized during rotation. Errors between the theoretical and FEA results are <10%. Based on the above design values, the design used for production was as shown in Figure 8. A capacitive sensor was used to measure the rotation of the scanner. Internal strain gauge sensors were attached to the PZT actuators for sensing PZT expansion.

tab5
Table 5: Comparison between theoretical and FEA results.
962439.fig.008
Figure 8: Final design of the pure rotational scanner. 1: PZT actuator, 2: main body, and 3: capacitive sensor.

4. Implementation

Figure 9 shows an image of the scanner built for the experiment. Each part for nano resolution was manufactured by using wire electrical discharge machining. To control the scanner, two PZT actuators with built-in strain gauges for monitoring piezo displacement (AE0505D16F, NEC-Tokin Corp.) were used; these were controlled with LABVIEW software (National Instruments). A capacitive sensor (Probe 2812, ADETech) was also used to measure the final rotational displacement of the scanner for feedback purposes. This sensor was located in the inner rim of the sensor (Figure 10). As indicated in Figure 10, the capacitive sensor is located in the vicinity of the virtual pivot center and measures the linear displacement of the located point. Because the distance ( mm) between the pivot and the sensor center is known, the rotation angle () is calculated as , for the small rotation angle, approximately .

962439.fig.009
Figure 9: Photo of the pure rotational scanner.
962439.fig.0010
Figure 10: Position of the capacitive sensor in the scanner.

The equipment used in this experiment is described in Table 6.

tab6
Table 6: Specifications for the experimental components.

To evaluate the performance of the scanner, we used a closed-loop control system that decreased errors between the reference input and output displacement of the scanner, hysteresis, and creep of the PZT actuator. A block diagram of the control system is shown in Figure 11. The capacitive feedback sensor location is shown in Figures 9 and 10. PZT control resolution was about 3 nm (peak to peak) before low-pass filtering. The maximum noise band was approximately 2 nm (peak to peak, 90 nrad) after low-pass filtering (Figure 12).

962439.fig.0011
Figure 11: Block diagram of closed-loop control system.
fig12
Figure 12: Capacitive sensor noise resolution. Sensor signal before (a) and after (b) 80 Hz low-pass filtering.

Multistep response, indication of the final feedback resolution, was as shown in Figure 13. The performance of the capacitive sensor was expected to be more than sufficient for this experiment. As shown in Figure 14, the resonance frequency of the scanner was measured by the sine-swept test (0.1–1 kHz) and found to be 542 Hz, which is close to the 517 Hz expected based on the simulation. In addition, a high scanning speed is ensured because the resonance frequency is high.

962439.fig.0013
Figure 13: Capacitive sensor noise resolution in multiple steps.
fig14
Figure 14: Resonance frequency of the pure rotational scanner.

Figure 15 shows the nonlinearity and hysteresis properties of the PZT actuator. The nonlinear movement and the hysteresis (0.83 m) of the PZT actuator during the open loop test are clearly shown in Figure 16. These unwanted errors were eliminated using a PI closed-loop control. As shown in Figure 15, the maximum displacement was 497.2 rad (7.8% error), which differs from the expected simulation value of 6 m obtained when the maximum input voltage of the PZT was set to 150 V. Therefore, we concluded that the result of the simulation was reasonably satisfactory.

962439.fig.0015
Figure 15: Nonlinearity and hysteresis of the PZT actuator after PI controller design.
fig16
Figure 16: Saw-tooth response in the open loop (a) and closed loop (b) tests.

Triangular responses in the open and closed loop states were compared as shown in Figure 16. The sharp apices were clearly tracked and nonlinearities were absent in the case of the closed loop control, providing further proof of the feedback performance.

Figure 17 indicates the step responses in the open loop test. Creep, a phenomenon that varies with time, was observed in transient responses. Accordingly, a steady-state error was created. To improve the step response, a closed loop control was used during another observation. As shown in Figure 18, the transient response improved considerably and creep was decreased.

962439.fig.0017
Figure 17: Step response in the open loop test.
962439.fig.0018
Figure 18: Step response in the closed loop test.

Open and closed loop multistep responses are shown in Figures 19(a) and 19(b), respectively. The closed multistep responses indicate that the overshoot in the open loop test was decreased, ensuring stable movement. Agreement between the starting and ending locations was also improved, ensuring good performance in consecutive movements, consistent results over repeated tests, and increased reliability of overall performance.

fig19
Figure 19: Multistep response: multistep response in the open (a) and closed loop (b) tests.

Figures 20(a) and 20(b) show the tracking responses for the open and closed loop tests in the case of a 3 Hz sine wave signal. In the open loop test, the phase lagged behind the reference signal. In contrast, the output tracked the reference signal correctly in the closed loop test.

fig20
Figure 20: Sine wave tracking at 3 Hz in the open loop (a) and closed loop (b) tests.

Observation of the tracking responses of a 10 Hz sine wave signal shown in Figures 21(a) and 21(b), as a measure of fast response, showed that the error increased in the open loop test but was decreased in the closed loop test.

fig21
Figure 21: Sine wave tracking at 10 Hz in the open (a) and closed loop (b) tests.

Finally, the parasitic motion error of the virtual rotational pivot center was measured as shown in Figure 22. Two capacitive sensors were located symmetrically near the pivot center inside the rim as shown in Figure 9. Here, each sensor measures simultaneously but inversely along the -axis movement of the same rotation center. If the two measured sensor displacements during operation are same, it can be concluded that there are no parasitic motions of the pivot point. From Figure 22, which shows the measured output of both of the sensor, the measured discrepancy between the two sensors was only 10 nm at full stroke. The above result shows that the manufactured rotation scanner can rotate the virtual pivot center accurately with minimal parasitic motion error.

962439.fig.0022
Figure 22: Parasitic motion error of the rotation center.

5. Conclusion

We reported on a novel rotational scanner with minimized center shift errors. The scanner consists of circular hinges and L-shaped leaf springs to minimize parasitic errors at the center in the and directions, and to optimize rotation performance. Rotational parasitic errors are minimized as stiffness in the x and y directions is maximized by means of circular hinges inside the rim of the scanner. The center of the scanner was not mechanically fixed on a base; however, we ensured that rotation occurred at a virtual pivot point by thorough theoretical and experimental investigations. The derived theoretical modeling results were compared and refined with those obtained using FEA. Design optimization also minimized parasitic errors at the rotation center. Simulation results showed an error rate of below 10%, which implies good theoretical agreement. The 3D production design was created using the optimized results. The rotational scanner was then fabricated and assembled to assess its performance. We conducted a fast Fourier transform (FFT) of the whole system by applying a sine-swept signal to the actuator, proving that a resonance frequency of 542 Hz corresponded to an error rate of 4.5%, the value expected from numerical estimations and simulations. A PI controller was also designed to minimize steady-state error and the overshoot. Step, multistep, triangular, and sine responses were measured to evaluate the performance of the scanner. In summary, the rotational scanner had a resonance frequency of 542 Hz, rotational displacement of 497.2 rad, resolution of 90 nrad, and the parasitic error of 10 nm. These test results reveal that the rotational scanner rotated accurately about the center and had good nano precision. It is expected that rotational scanners based on the proposed approach will be applicable for use in various nano positioning systems.

Authors’ Contribution

The two authors, M. Lee and E. Park, contributed equally to this work.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0002258, 2012-004056, and 2011-002856).

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