High accurate squareness measurement squareness method for ultra-precision machine based on error separation

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High accurate squareness measurement squareness method for ultra-precision machine based on error separation Tao Lai a,b , Xiaoqiang Peng a,b,∗ , Guipeng Tie a,b , Junfeng Liu a,b , Meng Guo a,b a b

College of Mechatronics and Automation, National University of Defense Technology, Deya Road, Changsha, Hunan Province, 410073, China Hunan Key Laboratory of Ultra-Precision Machining Technology, 47 Yanzheng Street, Changsha, Hunan Province, 410073, China

a r t i c l e

i n f o

Article history: Received 9 October 2016 Received in revised form 27 December 2016 Accepted 16 January 2017 Available online xxx Keywords: Squareness Ultra-precision machine Profilometer Measurement accuracy Error separation

a b s t r a c t Traditional measurement methods of squareness for ultra-precision motion stage have many limitations, especially the errors caused by the inaccuracy of standard specimens. On the basis of error separation, this paper presents a novel method to measure squareness with an optical square brick. The angles between the guideways and the four lines of brick section are measured based on the fact that sum of interior angle of a quadrilateral is 2␲, and the squareness is obtained. A squareness measurement experiment was performed on a profilometer with a modified optical square brick. Experimental results show that the squareness accuracy between X and Y axes is not influenced by the accuracy of brick, and the measurement repeatability reaches 0.22 arcsec. Finally, a verification experiment to the proposed method was carried out with a high accurate standard specimen, and the error between the two methods is 1.06 arcsec. According to the error results and simulation analysis of the measurement system, the measurement error based on error separation is 0.06 arcsec. The proposed method is able to achieve a very high accurate squareness measurement with auxiliary components of normal accuracy, and can be applied to measure the accuracy class of sub-arcsec squareness. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Squareness is the very susceptible factor to the profilometer, which accounts for more than 30% of all geometric errors, especially in ultra-precision motion stages [1]. Geometric error of a motion axis of general-precision motion stage and can be measured by laser interferometers. However, interferometers cannot be used to measure squareness of ultra-precision profilometers, such as Renishwa XL-80 interferometer. Therefore, other methods for squareness measurement are demanded. The commonly used methods for squareness measurement can be divided into three categories. The first category includes parametric modeling methods, such as the integrated geometric error modeling method [2] and the product of exponential modeling method [3]. These models can predict the geometric errors on the condition that the basic geometric errors are measured and identi-

∗ Corresponding author at: College of Mechatronics and Automation, National University of Defense Technology, Deya Road, Changsha, Hunan Province, 410073, China. E-mail addresses: laitao [email protected] (T. Lai), [email protected] (X. Peng), [email protected] (G. Tie), [email protected] (J. Liu), gmfi[email protected] (M. Guo).

fied. Basic geometric errors can be acquired by 9-line method [4], 12-line method and step-diagonal measurement method [5] with an interferometer, but the limitations of the laser diagonal measurements are obvious [6]. Generally, the measuring instruments are developed at home and abroad, such as the 3D probe-ball [7,8], double-ball-bar and R-test [9,10]. Given a measurement point, the geometric error measurement methods make use of matrices to transform an array of every degree of freedom (DOF) into array at the point of interest. In this case, the accuracy class of estimating the errors greatly depends on the accuracy class of the kinematic model as well as the class of each DOF error. Because the first category belongs to comprehensive error measurement methods, this measurement is performed with special instruments, and the key factors are calibration of the instruments and error modeling. In addition, squareness measurement accuracy by the first category is lower than the following two categories. The second category is a method uses standard specimen. This category is a very common, accurate and convenient method in markets currently. However, this method becomes expensive and inaccurate when measurement range is large because a specimen that exceeds commercially available dimensions must be custommade and the cost is relevant to the required accuracy class. Principle of this method is given by the GB/T 17421.1-1998 General Test Machine [11]. Jie [12] and Wang [13] measured the squareness

http://dx.doi.org/10.1016/j.precisioneng.2017.01.005 0141-6359/© 2017 Elsevier Inc. All rights reserved.

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Fig. 1. Four positions of the brick.

of guideways with L-square and dial gauges. Additionally, squareness of CMM (Coordinate Measuring Machine) is measured with an interferometer and a pentagonal prism [14]. The third category is based on error separation. The basic idea and benefit of this method is that low-quality specimens can be used in this measurement because errors are separated from the error of interest. Therefore, error-separation method is the very cost-effective and large-range measurement method. In addition, if the procedure is executed well, the achievable accuracy can reach the class of system repeatability. Error-separation method is established for measuring the straightness of guideways, but is rarely used for measuring the squareness. This method is also known as reversal method or error separation technique in the literature. Hocken R J and Borchardt B R [15] used this method to measure the squareness, Lieberman [16] first introduced the concept of rotating a specimen in an earlier e-beam lithography system calibration, Ruijl [17] applied an error separation technique for the straightness calibration of a mirror, and Hume’s angular measurement method [18] is used for measuring squareness, Evans [19] introduced several accurate feature measurements without reference to an externally calibrated artifact. An artifact reversal method usually belongs to a part of self-calibration. In this study, Ruijil and Evans’ methods are improved and applied for measuring the squareness between the X and Y axis of the profilometer. Error analysis and simulation are used to verify the accuracy and validity of the method. In the 2008 conference of “Collège e International pour la Recherché Productique” (CIRP), measurement effective of error was considered as the important study fields within the foreseeable 10 years. Therefore, the methods of improving the measurement accuracy have great practical significance to ultra-precision motion stage. 2. Principle The basic idea of error separation in this paper is that the sum of interior angle of a quadrilateral is 2␲, which is applied to squareness measurement of guidways. No matter what value of every interior angle is, the suqareness of guidways can be obtained accurately. In order to obtain the squareness between X and Y axes, an optical brick with four faces is used. Specific geometric principle is shown in Fig. 1. In position I, angle axy is the squareness between X and Y axes. Interior angles ˇA ,  1 and  2 satisfy the following relationship PositionI :ˇA +  1 +  2 = axy

(1)

where ˇA is bounded by the peripheral face D and face A of the brick,  1 is an angle bounded by axis X and face D,  2 is an angle bounded by axis Y and face A. After continuously rotating the brick at an angle of approximately 90 ◦ along Z axis, three positions II, III and IV are acquired, and the following equations can be obtained

PositionIV :ˇB +  7 +  8 = axy

(4)

where ˇD is bounded by peripheral face A and face B of the brick,  3 is the angle bounded by axis X and face A,  4 is an angle bounded by axis Y and face B. ˇC is bounded by the peripheral face B and face C of the brick,  5 is an angle bounded by axis X and face B,  6 is an angle bounded by axis Y and face C. ˇB is bounded by the peripheral face C and face D,  7 is an angle bounded by axis X and face C,  4 is the angle bounded by axis Y and face D. The angles of the brick meet the following requirement: ˇA + ˇB + ˇC + ˇD = 2

(5)

Summation of (1) to (4) and substitution of (5) yields: 2 + axy =

8 

k

k=1

(6)

4

Eq. (6) shows that angles  i (i = 1, 2, 3, 4, 5, 6, 7, 8) between faces and guideways are measured before getting the squareness axy of X and Y axes. In order to obtain  i (i = 1, 2, 3, 4, 5, 6, 7, 8), displacement transducers and corresponding displacement jig transducers are used. When the brick is located in position I, an actual measurement model for measuring  1 and  2 is built and shown in Fig. 2. Brick angle ˇA is bounded by the peripheral face D and face A. Accordingly,  1 is determined by measuring the brick face D. Specifically, the brick face D at any point satisfies the following equation: eD (ti ) + p1 ti + 1 = m1 (ti ) − eY (ti )

(7)

where, the deviation of the straightness of the brick face D is denoted as eD (ti ); eY (ti ) is the deviation of the flatness; m1 (ti ) measured by the displacement transducer is the distance between face D and the guideway; ti is a step distance value, which is an independent variable in the whole measurement procedure; the wedge between the reference line and the reference plane is defined by parameters p1 and 1 . Angle  1 is determined by p1 which is calculated based on the following equation: 1 = tan−1 (p1 )

(8)

Definition of the reference lines and planes leads to be less susceptible to random measurement errors by a least squares definition [20]. Solution of this least squares definition can be obtained from a linear operation according to a simple matrix equation. A useful least squares definition is as follows: n 

ti ej(j=A,B,C,D) (ti ) =

i=−n

n 

ej(j=A,B,C,D) (ti ) = 0

(9)

i=−n

PositionII :ˇD +  3 +  4 = axy

(2)

Therefore, all the points along the reference line of face D which is measured at t can be expressed using a matrix as:

PositionIII :ˇC +  5 +  6 = axy

(3)

eD +P1 k1 =m1 −eY

(10)

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Fig. 2. Schematic diagram of the measurement principle in position A.

Fig. 3. The experimental setup for the squareness of the X and Y axes.

where, eD = [eD (t-n ) eD (t-n+1 ) . . . eD (tn )]T ; eY = [eY (t-n ) eY (t-n+1 ) . . . eY (tn )]T ; m1 = [m1 (t-n ) m1 (t-n+1 ) . . . m1 (tn )]T . Accordingly, matrix P1 and column matrix k1 are:

 P1T =

−nt

(−n + 1)t

...

nt

1

1

...

1



 ; k1 =

p1

 (11)

1

Multiplying matrix Eq. (10) by P1 T , yields: P1 T eD +P1 T P1 k1 =P1 T m1 −P1 T eY

(12)

In addition, matrix P1 T can be used to constitute a least squares definition for the reference lines of the brick faces, which means Eq. (9) can be written as: P1 T eD = 0

(13)

Then Eq. (12) is written as: −1

k1 =(P 1 T P 1 )

P1 T m 1

(14)

Angle  1 can be calculated by the first element p1 in the column matrix k1 . Specifically, Eq. (8) can be written as: −1

1 = tan−1 (p1 ) = tan−1 (k1(1) ) = tan−1 ((P1 T P1 )

P1 T m1(1) )

(15)

The other  i are obtained using same way as the  1. Additionally, the symbol of each angle  i needs to be considered. According to the definition given in Fig. 1, the sign of the two adjacent  i is different. Finally, the angle axy is obtained using Eq. (6). 3. Experiment 3.1. Experiment instrument The experimental setup is shown in Fig. 3, where two transducers are the STIL and CL2 respectively, and controller of the transducers is a Prima STIL Initial. Measurement range is 400 ␮m and the maximum linearity error is 0.023 ␮m. Straightness of the X and Y axes of the profilometer is 0.1 ␮m on 250 mm length (0.1 ␮m/250 mm). A displacement transducer to Y axis is installed on the slider and can move together with the slider. The other displacement trans-

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4 Table 1 Values of the four angles of the brick. Angle

ˇA

ˇB

ˇC

ˇD

Original accuracy First modification Final modification

89◦ 40 59.73 89◦ 41 13.2 89◦ 59 57.37

90◦ 19 0.18 90◦ 18 58.68 90◦ 00 2.35

89◦ 41 12.05 89◦ 40 59.52 89◦ 59 57.37

90◦ 18 47.20 90◦ 18 47.52 90◦ 00 2.4

ducer to X axis is fixed on the adjustment and cannot move. The configurations of the two transducers are identical. 3.2. Optical brick In order to measure the squareness between the X and Y axes, a brick with four faces was used for error separation, and was modified to be the standard specimen. Ddefinition of the four angles of the brick and six faces are shown in Fig. 4. Original accuracy of the brick was not quite high, as shown in Table 1. The accuracy of brick faces and the values of the four angles are shown in Fig. 5 and Table 1, respectively. The face accuracy was measured by a digital wave-front laser interferometer phase shift Zygo GPI-XP/D. Meanwhile, the angle is obtained by a precision goniometer Prism Master MOT from German TRIOPTICS. Values of the angles of the brick are improved by modifying the surface figure with CCOS(Computer Controlled Optical Surfacing) and MRF(Magneto Rheological Finishing) as shown in Figs. 6 and 7. Therefore, the squareness errors of the angles of the brick are improved to be arcsec accuracy class and this brick can be regarded as a standard specimen. However, improvement of brick accuracy via the modification procedure is costly. As listed in Table 1, summation of the four angles is not 360◦ at each modification because of the measurement error of the MOT. But the measurement results based on error separation model are not affected by the error of the four angles, which can be inferred from the method principle.

Fig. 4. Definition of the angles and faces.

3.3. Experimental data and results 3.3.1. Squareness between X and Y axes based on the error separation In order to reduce random errors in the experimental procedure, the sampling frequency and time at each measurement point are set at 2000 Hz and 10 s respectively. Hence, 20000 sampling data is obtained and the actual value of that point is the average value

Fig. 5. The original accuracy of the four faces of the brick (face A, face B, face C, face D, from left to right).

Fig. 6. Changes of the PV value on face B.

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Fig. 7. Changes of the PV value on face C.

Fig. 10. Results of squareness with the alteration of the four angles of the brick.

Fig. 8. One of the 5 groups of experimental data.

Fig. 9. The results of squareness (␲/2-axy ) based on the error separation.

of the 20000 data. Length of the brick is 145 mm, and the interval distance t is 10 mm. Therefore, there are totally 13 or 15 measurement points on each brick face. From Eq. (11), the number of the point is odd. With these parameters, corresponding experimental data is calculated and shown in Fig. 8. Making use of the 5 group experimental data, angles  i were calculated and listed in Table 2. Squareness between the X and Y axes can be calculated with equation (6), which is shown in Fig. 9. Results in Fig. 9 shows that squareness between the X and Y axes is 19.71 arcsec. Measurement repeatability r is determined by Bessel Formula (16) and the value is 0.22 arcsec.

  n=5  1  r= (xi − x¯ )2 n−1

(16)

i=1

3.3.2. Effect of the alteration of the four angles of the brick on the measurement To determine whether the measurement accuracy is affected by the accuracy of the angles of the brick, an experiment was per-

formed. Alterations of the four brick angles are listed in Table 1. Squareness between the X and Y axes is measured under three types of angle accuracy which are listed in Table 2. The results of the squareness measurement with the alteration of the four angles of the brick are shown in Fig. 10, where the squareness value of the three measurements are 19.72, 19.77 and 19.71 arcsec, respectively. Therefore, the measurement accuracy based on the error separation is not limited by the accuracy of the angles of brick. From Eqs. (9) and (13), a least squares definition leads to the definition of the reference lines and planes to be less susceptible to random measurement errors as explained in Section 2. The flatness eB (ti ) and eC (ti ) shown in Figs. 6 and 7 could be neglected by the results shown in Fig. 10. Because squareness is rarely unchanged when PV value of face C changes from 854 nm to 136 nm, within this extend of the flatness, the least squares approximation do not affect the measurement results. 3.3.3. Squareness between the X and Y axes based on the standard specimen In order to verify the trueness of squareness obtained by the error separation, the experiment of squareness obtained by a standard specimen was carried out. Principle of this method given by the GB/T 17421.1-1998 General Test Machine is shown in Fig. 11. The accuracy of four angles of the brick is improved after modification, as shown in Table 1(Final modification). Accordingly, the brick can be regarded as the standard specimen to measure the squareness between the X and Y axes. Measuring instruments are the same for both methods. The results of the straightness with tilt of X and Y axes of one group are shown in Fig. 12. Experimental data is analyzed by the least squares fit, and straightness error with tilt is obtained. Finally, the squareness is calculated according to the relationship between the guideways and the angle of the brick. Different squareness measurement results obtained by the method of error separation and the method of standard specimen are compared and shown in Fig. 11.

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6 Table 2 5 groups of experimental results of angles  i. Angle (arcsec)

1

2

3

4

5

6

7

8

/2-axy

1 2 3 4 5

−124.92 −44.18 −46.62 −103.78 −103.78

145.50 65.83 68.88 125.40 125.40

−70.13 −31.16 −106.58 −46.64 −46.64

88.97 49.77 125.47 65.65 65.65

35.37 110.89 −120.72 −116.80 −36.13

−11.86 −87.56 143.96 140.34 58.78

−92.46 66.22 −69.02 49.43 −57.59

107.34 −50.29 83.93 −34.00 72.24

19.45 19.88 19.82 19.90 19.48

Fig. 13. Comparison of the different squareness measurement resulted by the two methods.

Fig. 11. Schematic diagram of the measurement principle based on stander specimen.

Results shown in Fig. 13 reveal that the squareness obtained with a standard specimen is 18.65 arcsec, which is 1.06 arcsec less than that obtained by error separation (19.71arcsec). The angle of the brick is obtained by an accurate goniometer Prism Master MOT from German TRIOPTICS whose measurement error is approximately 0.5 arcsec. Indeed, even though the standard specimen method can be used to measure the squareness of the profilometer, the accuracy is not very high and the brick requires costly work to be a standard specimen according to the results presented in section 3.2. In section 4, experimental error of the proposed method will be analyzed to further validate the accuracy of the method.

measurements. In this paper, the measurement is dynamic, and error analysis based on dynamic measurement is given. 4.1. Algorithm error In Eqs. (9) and (13), this approximate part is neglected and the algorithm error is generated. The simulation is implemented based on perfect guidways and flatness of brick. The simulated experiment is performed to establish a relationship between the errors and the actual angle between the X and Y axes. As shown in Fig. 14, the simulation model is simplified from an experimental model shown in Fig. 2; simulation results are shown in Fig. 15 and error u1 is 10−4 arcsec. Therefore, the algorithm error is much smaller compared with the actual measurement. 4.2. Error of the t

4. Error and simulation analysis Under technical conditions, all kinds of errors for design of the measurement system are no larger than the permissible errors. Accordingly, various potential errors of the measurement system need to be analyzed, and a method of compensation for the corresponding errors needs to be applied to improve the system accuracy. The measurement system includes static and dynamic

The interval value t is the distance between two adjacent measurement points on the face of the brick and is controlled by the grid of the profilometer. Therefore, error  of t is produced by the positioning error of the profilometer as shown in Fig. 16(a). Eq. (11) gives a mathematical relation between the error and the effect on the squareness measurement. In the simulation procedure, angle between X and Y axes is set at 90◦ . Then, squarenees error increases

Fig. 12. Straightness with tilt of X and Y axes of one group.

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Fig. 14. Schematic diagram of simulation with different values of axy.

reveals the relationship between the error u3 of the angle axy and the ϕx , as shown in Fig. 18. Angle ϕx ‘given at about 77◦ , which not affect the measurement. But the value of ϕx ‘is different in various measurement situation and cannot be actual angle applied in measurement as shown in Fig. 17(a). The right angle ϕx is carefully controlled, as shown in Fig. 17 (b), and the error of ϕx is 10 ␮m/30 mm. Accordingly, the error u3 of the angle between the X and Y axes is 0.06 arcsec. 4.4. Tilt of the brick

Fig. 15. Comparison between the error and the actual angle between the X and Y axes.

linearly with the positioning error, as shown in Fig. 16(b). The positioning error is 10 ␮m. According to the relationship between the error and the positioning error shown in Fig. 13, the error u2 is 0.02 arcsec. 4.3. Error caused by the installation of the transducer The transducer should be kept perpendicular to the guideway when the transducer is installed on the guideway. In fact, the error of the right angle ϕx effects the distance m1 (ti ). The simulation

Tilt (utiltx and utilty ) of the brick causes experimental error, as shown in Fig. 19. The tilt is controlled at 20 ␮m in the X and Y directions, as shown in Fig. 17 (b). Thus, u is 20 ␮m/55 mm when utitlex is 20 ␮m and the height of the brick is 55 mm. Finally, given a single , the error ux caused by the tilt of the brick is obtained from the following equation: ux = (u /L)utiltx

(17)

For all  i , the error u4 is obtained according to Eq. (6) and can be calculated using the following equation:

 1 var( ux ) = 2ux 2 4 8

u4 2 =

(18)

x=1

Fig. 16. Error of the angle between the X and Y axes against the change of the positioning error.

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Fig. 17. Error u3 caused by the installation of the transducer and the adjustment.

According to Eq. (19), measurement error u based on error separation is 0.06 arcsec. In fact, u is obtained by a simulation model and r is acquired under an actual experiment environment. Therefore, u is much smaller than the experimental repeatability r. The value of u shows a minimum measurement error based on error separation, but the value of r shows the actual measurement error. r should reach the accuracy class of u if all errors were better controlled in the measurement procedure. u shows that error of the method based on error separation is much smaller under normal conditions than the standard specimen. 5. Conclusion In this paper, a measurement squareness method based on the error separation was studied and analyzed. Simulation and experiments verified that:

Fig. 18. Error of the angle axy with the alteration of .

where, L is the length of the brick at 145 mm, so u4 is 0.01 arcsec. The four kinds of errors are determined by: u =

u21 + u22 + u23 + u24

(19)

a The squareness between the X and Y axes of profilometer was measured, and the measurement accuracy was not affected by the accuracy of standard specimen. Even though four interior angles changed at about 17 , the squareness kept nearly unchanged.

Fig. 19. Error caused by tilt of the brick.

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b Measurement repeatability of the squareness between X and Y axes is 0.22 arcsec, which is better than that by a standard specimen. Through the simulation analysis, simulation errors of the squareness measurement based on error separation is 0.06 arcsec and contains the following errors: algorithm error (one in ten thousand arcsec), error of the longitudinal position of every two adjacent measurement points (positioning error at 10 ␮m generated 0.02 arcsec), error caused by installation of the transducer (the error of ϕx at 10 ␮m/30 mm generated 0.06 arcsec) and error caused by the tilt of brick (0.01 arcsec). This method can be applied to measure the squareness at sub-arcsec class. Acknowledgements This research has been supported by the New Century Excellent Talents Scheme (NCET-12-0144) and the National Natural Science Fund (No. 51005259). References [1] Zhang B. Research on Key Thechniques of Form-Free Measurement in Precision Engineering. Beijing University of Technology; 2012. p. 6. [2] Zhu S, Ding G, Qin S. Integrated geometric error modeling, identification and compensation of CNC machine tools. Int J Mach Tools Manuf 2012;52(1):24–9. [3] Fu G, Fu J, Xu Y. Product of exponential model for geometric error integration of multi-axis machinetools. Int J Adv Manuf Technol 2014;71(9/12):1653–67. [4] Hu JZ, Wang BL, Wang M. The study of geometric error identification of DMl007milling machine based on laser interferometer. Manuf Technol Mach Tool 2010;(4):103–6.

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