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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Design of a six-degree-of-freedom geometric errors measurement system for a rotary axis of a machine tool Chien-Sheng Liu a,∗, Hung-Chuan Hsu b, Yu-Xiang Lin b a b

Department of Mechanical Engineering, National Cheng Kung University, Address: No. 1, University Road, East District, Tainan City 70101, Taiwan Department of Mechanical Engineering, National Chung Cheng University, Address: No. 168, University Road, Minhsiung Township, Chiayi County 62102, Taiwan

a r t i c l e

i n f o

Keywords: Geometric errors Motion errors Skew-ray tracing Multi-degree-of-freedom measurement Rotary axis Crosstalk

a b s t r a c t This paper proposes a new kind of non-contact optical measurement system, which combines with a polygon mirror and a conical lens to achieve the purpose of simultaneously measuring six-degree-of-freedom (6 DOF) geometric errors for a rotary axis of a machine tool. At the beginning, we used the Zemax software to construct the proposed optical structure and observe the change of light spots on the detectors. Then, we built the proposed optical structure on the Matlab software by using the skew-ray tracing method, which used to present the actual condition of light spot change and compare with the simulation results. Then, we designed the appropriate ﬁxtures to place the system structure on a rotary axis in a laboratory environment. After completing the laboratory-built prototype, we analyzed the eﬀects of diﬀerent geometric errors on the detectors at diﬀerent angular positions of a rotary axis of a machine tool. The proposed measurement system simultaneously measured 6 DOF geometric errors of the rotary axis, and the experimental results were compared with those of two dial gauges by using the measurement method of ISO 230–7. The experimental results show that the maximum deviation ranges of the radial motion error in X axis, the radial motion error in Y axis, the tilt motion error around X, and the tilt motion error around Y are ±4.7 μm, ±6.0 μm, ±3.6 arcsec, and ±8.1 arcsec, respectively.

1. Introduction The processing technology of machine tools has developed for many years. With the requirement of workpiece accuracy, geometric error measurement techniques become more and more important. The geometric error measurement technology of machine tools can traditionally be divided into two type: contact type and non-contact type. The former has low cost and high-precision characteristics, but the probe must be touched point by point to contact the gauge surface on the machine tool, which not only takes a lot of time, but also consumes the probe [1,2]. So a probe radius compensation needs to be done for the probe after a while or it will aﬀect the measurement accuracy. In the non-contact measurement technology, the laser interferometer is the most widely used measuring instrument. However, the interferometer is expensive and it can only measure one-degree-of-freedom (1 DOF) geometric error of machine tools at a time. If we want to measure the diﬀerent DOF geometric error, we must use diﬀerent mirror groups and experimental setup. As a result, measurement eﬃciency is obviously low, and the measurement environment may change to aﬀect measurement accuracy during long measuring periods [3,4]. There are diﬀerent kinds of geometric error sources in machine tools due to the cutting tools, the rotary shafts, the linear stages, and so on. In ∗

Corresponding author. E-mail address: [email protected] (C.-S. Liu).

this paper, we just discuss the geometric error sources from the rotary axis of a machine tool. The motion freedom of an object can be deﬁned as 6 DOF, including three linear motions moving along X, Y, and Z axes, respectively, and three rotational motions around X, Y, and Z axes, respectively. The rotary axis of a machine tool also has 6 DOF geometric errors, namely radial motion error in X direction (𝛿 X ), radial motion error in Y direction (𝛿 Y ), tilt motion error around X axis (ɛX ), tilt motion error around Y axis (ɛY ), axial motion error (𝛿 Z ), and angular positioning error (ɛZ ) [5]. In recent years, many researchers have developed a wide variety of methods to measure multi-DOF geometric errors for a linear axis and rotary axis of machine tools [6]. The former has developed very well [7–10], and divided 6 DOF geometric errors of a linear axis into horizontal straightness error, vertical straight line error, positioning error, roll error, pitch error, and yaw error [11,12]. Although the latter has less research literature than that of the linear axis, but it has also begun to develop recently [13]. For example, Murakami et al. proposed a new kind of spindle error measuring system, which was combined with a cylindrical lens and a spherical lens as their moving module and they can measure 5 DOF geometric errors [14]. He et al. used two laser displacement meters and self-designed ﬁxture to measure 6 DOF geometric errors of rotary shaft of the machine tools [15]. Li et al. utilized a single-mode ﬁber-coupled laser and two retro-reﬂector as their novel method to measure 5 DOF geometric errors of rotary shaft [16]. The application of the single-mode ﬁber not only improves the spatial stability but also the energy distribution of the measurement laser beam

https://doi.org/10.1016/j.optlaseng.2019.105949 Received 2 October 2019; Received in revised form 3 November 2019; Accepted 18 November 2019 0143-8166/© 2019 Elsevier Ltd. All rights reserved.

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Optics and Lasers in Engineering 127 (2020) 105949

Fig. 1. Structure of proposed measurement system.

signiﬁcantly. Other kinds of measuring equipment like laser tracers and laser scanners also can measure multi-DOF geometric errors of rotary shaft [17–19]. Researchers can also construct the whole machine tool using multi-body system with a homogeneous transfer matrix (HTM) methodology for the error modelling and error compensation [20–22]. On the other hand, the application of interferometry method can be used to measure the correlation errors [23–26]. Some researchers have conducted research on sensors and uncertainty of machine tools [27–30]. In this paper, we have proposed a novel optical measurement system to simultaneously measuring 6 DOF geometric errors for a rotary axis of a machine tool, which uses a conical lens and a polygon mirror as its moving module. The mathematical model of the proposed measurement system was built using a HTM and the skew-ray tracing method developed by our research groups [31–35]. The proposed measurement system can simultaneously measure 6 DOF geometric errors of a rotary axis. 2. Structure layout and measuring principle of proposed measurement system 2.1. Structure layout As shown in Fig. 1, the architecture of the proposed measurement system consists of two parts: one ﬁxed part and one moving part. The ﬁxed part is composed of three laser sources, a mirror, a beam splitter (BS), three position sensitive detectors (PSDs), and some connecting ﬁxtures. The moving part includes a conical lens, a polygon mirror, a precise cylinder gauge, and some connecting ﬁxtures. The precise cylinder gauge is designed to connect the conical lens and polygon mirror. The function of the precise cylinder gauge is to verify the feasibility of the proposed measurement system. Detailed explanation is addressed in Section 4. The idea of using the polygon mirror comes from the measurement method of the optical auto-collimator for angular positioning measurement of a rotary axis. Due to its simple structure, the proposed measurement system has the advantages of low cost. The ﬁxed part was placed outside of the rotary axis to avoid the vibration errors generated by the rotary axis and receive the light spots reﬂected from the moving part. On the other hand, the moving part was placed on the measured rotary axis of the machine tool. The laser light beams of these laser sources therefore hit the moving part and return to incident on the PSDs,

Fig. 2. Light path (a) with radial motion error in X axis and Y axis, (b) with tilt motion error around X axis, and (c) with axial motion error.

respectively. When the measured rotary axis starts to rotate, the signals of light spots on the detectors will change due to its geometrical errors, and the amounts of change can be inputted into the proposed mathematical model to calculate the 6 DOF geometric errors of the measured rotary axis.

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Optics and Lasers in Engineering 127 (2020) 105949

Fig. 3. 3D optical model of proposed measurement system.

Fig. 4. Variation of light spot locations on three PSDs with diﬀerent geometric errors.

Fig. 5. Skew-ray tracing and each optical component (i) of Path2 .

2.2. Measuring principle The proposed measurement system can be divided into three light paths with three laser sources and three PSDs, as shown in Fig. 1. In the ﬁrst light path (Path1 ), the laser light passes through the BS and hits the polygon mirror of the moving part. For PSD1 , the angular positioning

error (ɛZ ) and tilt motion error around Y axis (ɛY ) have higher sensitivity than other four errors to PSD1 , and they aﬀect the change of image centroid coordinates of the light spot on PSD1 in the X-direction and Y-direction, respectively. This design can reduce the error crosstalk of other four errors to PSD1 . In the second and third light paths (Path2 and Path3 ), two laser beams pass through the conical lens of the moving part

C.-S. Liu, H.-C. Hsu and Y.-X. Lin

Optics and Lasers in Engineering 127 (2020) 105949

Fig. 6. Calculated geometric errors from the mathematical model: (a) radial motion error in X axis, (b) radial motion error in Y axis, (c) axial motion error in Z axis, (d) tilt motion error around X axis, (e) tilt motion error around Y axis, (f) angular positioning error.

and are orthogonal to each other. Expect for angular positioning error, other 5 DOF geometric errors will cause variation of Path2 and Path3 . Fig. 2a–c show the variation in the light paths with the radial motion error in X axis and Y axis, with the tilt motion error around X axis, and with the axial motion error, respectively. When the moving part moves

with the rotary axis, the errors in all 6 DOF will simultaneously induce changes in the positions of the light spots on the PSDs. In other words, the light spot positions on the PSDs contain the 6 DOF coupled geometric errors. To eﬃciently and accurately analyze the relation between the individual 6 DOF geometric errors and light spot position information,

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Optics and Lasers in Engineering 127 (2020) 105949

Fig. 7. (a) Moving part and (b) ﬁxed part of laboratory-built prototype. Table 1 Analyzed results of leading coeﬃcients. Radial motion error (𝛿 X ) XPSD1A1 1.16E-05 YPSD1B1 1.16E-05 XPSD2C1 0.0037 YPSD2D1 −2.2E-05 XPSD3E1 −8.6E-06 YPSD3F1 1.14E-05

Radial motion error (𝛿 Y )

Axial motion error (𝛿 Z )

Tilt motion error (ɛX )

Tilt motion error (ɛY )

Angular positioning error (ɛZ )

A2 1.13E-09 B2 1.13E-09 C2 1E-04 D2 −1.1E-05 E2 −0.2223 F2 5.28E-05

A3 2.81E-05 B3 1.23E-05 C3 −6.5E-06 D3 0.11915 E3 −1.1E-09 F3 0.11915

A4 −4.1E-05 B4 4.13E-05 C4 −8.5E-05 D4 0.0406 E4 0.1891 F4 5.91E-05

A5 −3.6E-05 B5 −0.4262 C5 −0.1915 D5 −1.2E-05 E5 −2.6E-06 F5 0.04

A6 −0.4262 B6 4.13E-05 C6 −8.1E-13 D6 2.32E-13 E6 −6.3E-13 F6 −1.5E-13

Fig. 8. Proposed prototype and measured ﬁve-axis machine tool.

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Optics and Lasers in Engineering 127 (2020) 105949

tilt motion error around X axis (ɛX ), the tilt motion error around Y axis (ɛY ), the axial motion error (𝛿 Z ), and the angular positioning error (ɛZ ), respectively. These simulation results express the feasibility of adopting this system. The position change of the light spots can also be calculated by entering our constructed mathematical model and thus reduce the operation time due to the speciﬁc sensitivity of the proposed measurement system. 3.2. Mathematical model On the other hand, the mathematical model of the proposed measurement system was built using a HTM and the skew-ray tracing method [31–35]. With the algorithm of the ﬂat boundary surface and spherical boundary surface of skew-ray tracing method, the transformation matrix R Ai represents the transfer matrix of each optical component (i) coordinate system from the reference (R) coordinate system, and is as follows: 𝑅

Fig. 9. Schematic of veriﬁcation system.

⎡𝐼𝑖𝑥 ⎢ 𝐼 𝐴𝑖 = ⎢ 𝑖𝑦 ⎢𝐼𝑖𝑧 ⎢0 ⎣

𝐽𝑖𝑥 𝐽𝑖𝑦 𝐽𝑖𝑧 0

𝐾𝑖𝑥 𝐾𝑖𝑦 𝐾𝑖𝑧 0

𝑡𝑖𝑥 ⎤ ⎥ 𝑡𝑖𝑦 ⎥ 𝑡𝑖𝑧 ⎥ 1 ⎥⎦

(1)

Fig. 5 illustrates the detailed skew-ray tracing and each optical component (i) of Path2 . As shown in Fig. 5, the i th surface of the optical components can be represented as 𝑃𝑖−1 = [ 𝑃𝑖−1𝑥 𝑃𝑖−1𝑦 𝑃𝑖−1𝑧 1 ]𝑇 and the laser beam has the unit directional vector 𝓁𝑖−1 = [ 𝓁𝑖−1𝑥 𝓁𝑖−1𝑦 𝓁𝑖−1𝑧 0 ]𝑇 . If 𝜆i is the distance between 𝑃𝑖−1 and Pi , then Pi and 𝜆i can be expressed as follows: [ ]𝑇 𝑃𝑖 = 𝑃𝑖𝑥 𝑃𝑖𝑦 𝑃𝑖𝑧 1 [ ]𝑇 = 𝑃𝑖−1𝑥 + 𝓁𝑖−1𝑥 λ𝑖 (2) 𝑃𝑖−1𝑦 + 𝓁𝑖−1𝑦 λ𝑖 𝑃𝑖−1𝑧 + 𝓁𝑖−1𝑧 λ𝑖 1

λ𝑖 =

( ) − 𝐼𝑖𝑧 𝑃𝑖−1𝑥 + 𝐽𝑖𝑧 𝑃𝑖−1𝑦 + 𝐾𝑖𝑧 𝑃𝑖−1𝑧 + 𝑡𝑖𝑧 𝐼𝑖𝑧 𝓁𝑖−1𝑥 + 𝐽𝑖𝑧 𝓁𝑖−1𝑦 + 𝐾𝑖𝑧 𝓁𝑖−1𝑧

=

−𝐵𝑖 𝐺𝑖

(3)

According to Snell’s Law, the unit directional vector 𝓁 i of the reﬂected ray is as follows: [ ]𝑇 𝓁𝑖 = 𝓁𝑖𝑥 𝓁𝑖𝑦 𝓁𝑖𝑧 0 [ = 𝓁𝑖−1𝑥 − 2𝐼𝑖𝑧 𝐺𝑖 𝓁𝑖−1𝑦 − 2𝐽𝑖𝑧 𝐺𝑖

Fig. 10. Photograph of veriﬁcation system.

𝓁𝑖−1𝑧 − 2𝐾𝑖𝑧 𝐺𝑖

]𝑇 0

(4)

Through the method mentioned above, we can calculate the unit directional vector of the surface of the optical components, and then the location of the light spots on the three PSDs can be represented as follows: ( ) 𝑋𝑃 𝑆𝐷1 = 𝐹𝑋1 𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧 (5) ( ) 𝑌𝑃 𝑆𝐷1 = 𝐹𝑌 1 𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧

(6)

( ) 𝑋𝑃 𝑆𝐷2 = 𝐹𝑋2 𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧

(7)

3. Numerical simulation and mathematical model of proposed measurement system

( ) 𝑌𝑃 𝑆𝐷2 = 𝐹𝑌 2 𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧

(8)

3.1. Numerical simulation

( ) 𝑋𝑃 𝑆𝐷3 = 𝐹𝑋3 𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧

(9)

( ) 𝑌𝑃 𝑆𝐷3 = 𝐹𝑌 3 𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧

(10)

the mathematical models of the proposed measurement system were built using a HTM and the skew-ray tracing method developed by our research groups [31–35]. Details of the mathematical models are presented in Section 3.1.

The proposed measurement system were simulated by the optical simulation software, Zemax, to observe the situation when 6 DOF geometric errors were coupled into the proposed measurement system, respectively. Fig. 3 shows the 3D optical model of the proposed measurement system. Fig. 4 depicts the variation of light spot locations on three PSDs with the diﬀerent geometric errors of the radial motion error in X direction (𝛿 X ), the radial motion error in Y direction (𝛿 Y ), the

where XPSDi (i = 1, 2, and 3) and YPSDi (i = 1, 2, and 3) are the image centroid coordinates of the light spot on PSDi in the X-direction and Ydirection, respectively. Eqs. (5)–(10) describe the relationship between the 6 DOF geometric errors of the rotary axis (including the radial motion error in X direction (𝛿 X ), the radial motion error in Y direction (𝛿 Y ),

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Optics and Lasers in Engineering 127 (2020) 105949

Fig. 11. Measurement results for variation of geometric error with position: (a) radial motion error in X axis, (b) radial motion error in Y axis, (c) tilt motion error around X axis, (d) tilt motion error around Y axis, (e) Angular positioning error, and (f) Axial motion error.

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Optics and Lasers in Engineering 127 (2020) 105949

the tilt motion error around X axis (ɛX ), the tilt motion error around Y axis (ɛY ), the axial motion error (𝛿 Z ), and the angular positioning error (ɛZ )) and the locations of the light spots on the PSDs. Eqs. (5)–(10) are a series of nonlinear equations and the 6 DOF geometric errors of the rotary axis for machine tools are very small. So, the Taylor series expansion can be used to expand Eqs. (5)–(10) in linear form. Here, we use sensitivity analysis to observe which errors will aﬀect the value of sensors signiﬁcantly. Table 1 lists all the analyzed results of the leading coeﬃcient (Ai , Bi , Ci , Di , Ei , and Fi , i = 1, 2, 3, 4, 5, and 6). The leading coeﬃcient, which represents the weight by which each linear and rotational motion error contributes to the location of the light spots on the PSDs, is the sensitivity of the corresponding linear and rotational motion errors. To improve the calculation speed and without reducing the calculation accuracy, we remove the low sensitivity term and simplify the entire mathematical model, and the simpliﬁed mathematical equations are as follows: 𝑋𝑃 𝑆𝐷1 = 𝐴6 𝜀𝑍 + 0.0005

(11)

𝑌𝑃 𝑆𝐷1 = 𝐵5 𝜀𝑌 − 0.0005

(12)

𝑋𝑃 𝑆𝐷2 = 𝐶1 𝛿𝑋 + 𝐶5 𝜀𝑌 + 0.003

(13)

𝑌𝑃 𝑆𝐷2 = 𝐷3 𝛿𝑍 + 𝐷4 𝜀𝑋 − 0.0006

(14)

𝑋𝑃 𝑆𝐷3 = 𝐸2 𝛿𝑌 + 𝐸4 𝜀𝑋 + 0.003

(15)

𝑌𝑃 𝑆𝐷3 = 𝐹3 𝛿𝑍 + 𝐹5 𝜀𝑌 − 0.0019

(16)

From Eqs. (11)–(16), the radial motion error in X direction (𝛿 X ), radial motion error in Y direction (𝛿 Y ), tilt motion error around X axis (ɛX ), tilt motion error around Y axis (ɛY ), axial motion error (𝛿 Z ), and angular positioning error (ɛZ ) can be obtained. Through the optical simulation software Zemax, the image centroid coordinates of the light spots on the PSDs can be obtained with the different geometric errors of 𝛿 X , 𝛿 Y , ɛX , ɛY , 𝛿 Z , and ɛZ , respectively, and then be inputted into the proposed mathematical model to obtain the calculated geometric error values. Fig. 6 shows the comparison between the ideal geometric errors inputted to Zemax and the calculated geometric errors from the proposed mathematical model. It is observed that the maximal calculated errors of 𝛿 X , 𝛿 Y , ɛZ , ɛX , ɛY , and 𝛿 Z are 0.0006 μm, 0.015 μm, 0.0008 μm, 0.008 arcsec, 0.0005 aresec, and 0.0005 aresec, respectively. These calculated errors are very small. In other words, the numerical results prove that the proposed mathematical model is correct enough for machine tool applications. In this section, the mathematical model derivation is brieﬂy introduced along with its characteristics. Further information about the mathematical model can be found in our previous papers [31–35]. 4. Experimental characterization of prototype model As shown in Figs. 7 and 8, the validity of the proposed measurement system was demonstrated by means of a laboratory-built prototype. The prototype of the proposed measurement system was used to simultaneously measure the 6 DOF geometric errors of a rotary axis (C-axis) of a ﬁve-axis machine tool (NXV560A, YCM, Taichung, Taiwan) in the factory. The experiments are conducted in a temperature-controlled laboratory to reduce the environmental errors and angular position command of 30° interval was inputted to the rotary axis of the machine tool for each measurement. In order to verify the feasibility of the proposed measurement system, two dial gauges (Mitutoyo: 513–471–10E, resolution of 1 μm) by using the measurement method of ISO 230–7 and a precision cylinder gauge were used to measure the geometric errors of the rotary axis with

the simple triangular methodology. Fig. 9 shows the veriﬁcation system including two dial gauges. As shown, L0 represents the distance between the bottom of the precision cylinder gauge and the lower dial gauge; L1 is the distance between two dial gauges. When the rotary axis produces an angular error (ɛ) and a translation error (𝛿), the reading values of these two dial gauges will change. Fig. 10 shows the setup of the veriﬁcation system. Fig. 11 shows a series of measurement results with 5 repeated measurements at each position in comparison to those of the veriﬁcation system. However, it is important to note that the veriﬁcation system can only measure 4 DOF geometric errors (𝛿 X , 𝛿 Y , ɛX , and ɛY ) of the rotary axis. As shown, the measurement curves of the proposed measurement system and the veriﬁcation system have a very similar shape. It proves the feasibility of the proposed measurement system. The measurement results show that the deviation ranges for the radial motion error in X axis (𝛿 X ), the radial motion error in Y axis (𝛿 Y ), the tilt motion error around X axis (ɛX ), and the tilt motion error around Y axis (ɛY ) are ±4.7 μm, ±6.0 μm, ±3.6 arcsec, and ±8.1 arcsec, respectively. The above results demonstrate that the proposed measurement system can simultaneously measure 6 DOF geometric errors for a rotary axis of a machine tool. It is noted that the laser beam instability and misalignments could inﬂuence the measured results when conducting the experiments [32,33]. In the experiments, a sampling time of 10 s for the PSDs was set to average and reduce the laser non-stability eﬀect. In order to reduce the systematic errors due to the misalignments, an adjustment attaching on each component is necessary to ensure the installation errors are as small as possible for the proposed measurement system. As shown in Fig. 7(a), a three-axis manual stage was used to align the precise cylinder gauge. However, there are many other error sources (PSD sensitivity, components position error, aberrations, quantization errors, and vibrations, etc.) could cause imperfections in the measured results. Therefore, these issues must be considered to improve the measurement accuracy of the proposed measurement system in the future. 5. Conclusion In this paper, a new kind of non-contact optical measurement system has been proposed to simultaneously measure 6 DOF geometric errors for a rotary axis of a machine tool. In the proposed approach, a polygon mirror and a conical lens are integrated into to simply its structure and reduce its cost. The performance of the proposed measurement system has been demonstrated using a laboratory-built prototype. The experimental results have shown that the proposed measurement system can simultaneously measure the rotary axis’s 6 DOF geometric errors. As a result, the proposed measurement system provides another solution for a wide range of optical inspection and the geometric error measurement technology of machine tools. Declaration of Competing Interest The authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper. Acknowledgements The authors gratefully acknowledge the ﬁnancial support provided to this study by the Ministry of Science and Technology of Taiwan under Grant nos. MOST 106–2628-E-006–010-MY3, MOST 105–2221E-006–265-MY5, MOST 107–2218-E-002–071, and MOST 107–2218-E194–001. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.optlaseng.2019.105949.

C.-S. Liu, H.-C. Hsu and Y.-X. Lin

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