An investigation into the aspheric ultraprecision machining using the response surface methodology

Precision Engineering 44 (2016) 203–210

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An investigation into the aspheric ultraprecision machining using the response surface methodology Yung-Tien Liu a,∗,1 , Liangchi Zhang b a Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, No. 1, University Rd., Yanchao Dist., Kaohsiung 824, Taiwan, ROC b School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney 2052, NSW, Australia

a r t i c l e

i n f o

Article history: Received 17 December 2015 Accepted 18 December 2015 Available online 29 December 2015 Keywords: Ultraprecision machining Aspheric surface Response surface methodology Compensation cutting Form error

a b s t r a c t Aspheric ultraprecision machining is increasingly important to the manufacturing industry. The performance of aspheric optical components manufactured by mass-production is largely dependent on the form error of molds and dies. It is believed that productivity of a machining process could be improved if the form error is predictable. In this study, the response surface methodology (RSM) was employed to derive predictive models of rough and compensation cuttings for an aspheric convex mold, with an outer aperture of 12 mm and curve height of 0.6 mm. Two control factors, the depth of cut and spindle speed, were selected for study. The 2K factorial design with four center points was adopted. Two linear models for both rough and compensation cuttings were derived experimentally based on the form errors obtained. The models adequacy was examined through ANOVA (analysis of variance) results for the surface responses. It was found that the linear model of rough cutting is adequate, reflected by the significant regress coefficients and the high R2 value. However, the model of compensation cutting was found to be inadequacy. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Ultraprecision machining is to use the state-of-the-art machine tool to perform material removal process through extremely precise motion control. Due to remarkable machining performance, various applications of ultraprecision machining are well found in engineering industry. One of the most important applications is to manufacture aspheric molds and dies [1] for imaging and illumination optical lenses. That is mainly because the optical system using aspheric lenses might feature small number of lenses required and be capable of avoiding image distortion. Therefore, the requirements for aspheric lens are increasing in engineering industry [2]. Although there are already many commercially available systems capable of machining aspheric molds and dies precisely, the process is sensitively affected by complex factors, such as tool de-centering, ground profile, on-machine measurement, and operational parameters [3–5]. In practice, for obtaining a satisfactory form accuracy of an aspheric surface, it is very likely to perform several compensation cuttings following rough cutting. To examine

∗ Corresponding author. Tel.: +886 7 6011000x2220; fax: +886 7 6011066. E-mail address: [email protected] (Y.-T. Liu). 1 JSPE Member (No. 2970378). http://dx.doi.org/10.1016/j.precisioneng.2015.12.006 0141-6359/© 2015 Elsevier Inc. All rights reserved.

how process factors affect the form error in the stage of compensation, several studies were performed [6–8]. To reduce the number of compensation cuttings required, thus to increase productivity, it is expected that the form error in rough cutting is as small as possible. In the authors’ previous study [9], the response surface methodology (RSM) was employed to establish an empirical mathematical model of form error in terms of spindle speed and depth of cut in rough cutting. In this paper, the approach based on the RSM is extended to the compensation cutting to provide a complete RSM study for ultraprecision machining. In the next section, the RSM will be briefly introduced, followed by experimental and modeling details. 2. Response surface methodology 2.1. Response model The RSM is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes. The objective and typical applications of RSM are: (1) mapping a response over a particular region of interest, (2) optimization of response, and (3) selection of operating conditions to achieve specifications [10]. In metal cutting processes such as turning, milling, grinding, and lapping, RSM is well used to optimize tool wear,

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surface finish, tool force, grinding force, geometry error, and tool geometry [11–14]. Using the technique of design of experiment (DOE), the predictive model between output and input parameters, or response and control factors can be derived. The optimization approach can be therefore performed based on the derived model. In this paper, RSM is used to quantify the relationship between the form error of an aspheric surface and process parameters. In general, a response variable y can be represented by a function of independent variables xi (i = 1, . . ., k) as follows,

Table 2 Aspheric coefficients.

y = f (x1 , x2 , x3 , . . ., xk ) + ε,

x1 =

ln S − ln (S)center , ln (S)high − ln (S)center

x2 =

ln D − ln (D)center , ln (D)high − ln (D)center

(1)

where ε is a random error. If the expected response is denoted by E(y) =, then the surface being represented by  =  f x1 , x2, x3 , . . ., xk is called a response surface. Usually, a lower order polynomial (first- and second-order models) is employed to present the response surface. The first-order model, or a linear model, can be written as follows,



k

c = −1/r0

c2

c4

c6

−6.44

−1/31

−2.5E − 3

1.05E − 8

9.83E − 7

The levels of (−1, 0, 1) factors mean the lowest, central, and highest levels of the investigated variables with the transforming equations as follows [15],

(5)

where (·)high and (·)center indicate the highest and central levels of the corresponding control factors, respectively. 3. Experimental works

k

y = ˇ0 x0 +

ˇi xi + ε,

(2)

i=1

where x0 is a dummy variable with an usual value of 1, and ˇi parameters of the polynomial are to be derived by the method of least-squares. The matrix approach of solving ˇi parameters is to describe the relation between the response and independent variables in the matrix form as, Ym = X␤,

(3)

where Ym is a (n × 1) vector of the measured response, i.e., the measured form errors, X is the design matrix (n × p) of levels of independent variables xi , and ␤ to be (p × 1) vector of parameters to be estimated. Using the least-squares method to minimize the error term, ␤ can be derived based on the following equation, ␤ = (XT X)

−1

XT Ym ,

(4)

where XT represents the transpose of matrix X, and (XT X)−1 is the inverse matrix of (XT X).

2.2. Control factors and levels Although there are complex factors that would affect the form accuracy in aspheric ultraprecision machining, for a given expected roughness (usually under 0.05 ␮m) of workpiece that will be machined by a fixed tool, the controllable process parameters are limited to spindle speed (x1 ) and depth of cut (x2 ) only, i.e., k = 2 in Eq. (2). To investigate how these two control factors affect the response of the form error, the 2K factorial design with center points, or central composite design (CCD), can be adopted. The control factors and their levels are given in Table 1. The ranges of spindle speed (S) and depth of cut (D) are assigned as 500 to 2000 rpm and 2.4 to 15 ␮m, respectively.

Table 1 Control factors and levels. Factor

Symbol (unit)

Coding

1. Spindle speed 2. Depth of cut

S (rpm) D (␮m)

x1 x2

Level of factors (coding) −1

0

1

500 2.4

1000 6

2000 15

3.1. Aspheric curve If the optical axis and its perpendicular axis are denoted as z and x, respectively, the curve used to generate the aspheric surface is expressed as follows, z=

1+



cx2 1 − c 2 (k + 1)x2

+

12 

ci xi ,

(6)

i=2

where c = 1/r0 , k = 1 − e2 , r0 is radius curvature at the vertex of the aspheric curve, e eccentricity, and ci are aspheric coefficients. The coefficients used in this study are listed in Table 2. The radius of curvature r0 is 31 mm. The minus sign of c means a convex aspheric curve. Based on the designed aspheric coefficients, an aspheric surface can be produced by a machine tool through G-code commands. The generation of the G-code commands needs to consider the operational parameters such as the nose radius of the diamond tool, outer aperture of the workpiece and surface increment for each command step, as listed in Table 3. 3.2. Experimental setup The machine tool used was an ultraprecision machine (Nano 350 FG, Moore Nanotechnology Systems, USA) at the Laboratory for Precision and Nano Processing Technologies in the University of New South Wales, Australia. The travel ranges of the three linear axes (xyz) are 350 × 300 × 150 mm3 , with straightness of 0.3 ␮m along x- and y-axis, and 0.5 ␮m along z-axis, over their corresponding full travel ranges. The maximum feed rate of the linear axes is 2000 mm/min, and the speed range of the rotational c-axis is 50 to 10,000 rpm. The program resolutions for the linear and rotational axes are 0.01 nm and 0.0000001◦ , respectively. A stud workpiece made of brass was with the dimension of 12.7 × 20 mm. The machined aspheric surface was measured with an on-machine LVDT (linear variable differential transformer) ruby probe having a tip radius of 0.5 mm. Before the measurement, the on-machine device was calibrated by a standard sphere. Since the LVDT probe tip was guided by air bearing, the measurement process might feature a small measuring force. Table 3 Parameters used for G-code generation. Nose radius (mm)

Outer aperture (mm)

Surface increment (mm)

0.523

12

0.01

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205

Table 4 Design of experiments and results for rough cutting.

1 2 3 4 5 6 7 8

Spindle speed (S, rpm)

500 2000 500 2000 1000 1000 1000 1000

Depth of cut (D, ␮m)

Level

2.4 2.4 15 15 6 6 6 6

3.3. Experimental configuration and results

Profile zm (mm)

(12 mm)

−1 1 −1 1 0 0 0 0

−1 −1 1 1 0 0 0 0

0.341 0.244 0.223 0.099 0.160 0.143 0.194 0.168

Form error ze

0.6

(ePV = 0.397 μm)

-0.2

0.4

-0.4

0.2

-0.6

0.0 -8

-4

0

4

8

Rad ial position x (mm) (

(a) Measured profile and form error

4 0.4

Form error ze (μm)

3.3.2. Data processing of measured form error The machined aspheric profile was measured with the onmachine LVDT probe. Fig. 2(a) shows the measured profile zm for Trial No. 1 in Table 4. The curve height of the profile is 0.6 mm which was measured at the radial position of x = ±6 mm, i.e., the outer aperture of 12 mm. The form error ze could therefore be obtained by comparing the measured profile with the designed. The peakto-valley (PV) form error ePV was calculated as 0.397 ␮m. However, since the LVDT probe was guided by an air bearing thus with a floating measuring condition, the form error was usually accompanied with a tilt angle with respect to the x-axis. To evaluate the form error under a constant condition, the tilt angle was removed by the least-squares method. As a result, the PV form error was reduced from 0.397 to 0.341 ␮m but yet with an unsymmetrical profile as shown in Fig. 2(b). All the PV form errors after tilt removal for eight experimental trials are noted in Table 4. The minimum and maximum values are 0.099 and 0.341 ␮m, respectively. The PV form errors obtained by rough cutting are employed for developing a predictive model described in the next section. To investigate the relation between the form error and the outer aperture A, several sections in the measured raw data were drawn out for profile processing. Fig. 3 shows an example of three different apertures of 12, 8, and 4 mm with their corresponding PV form errors of 0.341, 0.187, 0.073 ␮m after tilt removals. Fig. 4 presents all the PV form errors obtained by eight experimental trials with

x2

Measured profille zm

0.0 3.3.1. Factorial design for rough cutting and results According to the control factors and levels listed in Table 1, the factorial DOE with two factors for rough cutting is configured as presented in Table 4. The center level repeating four times is for estimating the pure error. Following the experimental configuration, a total of eight experimental trials were performed. The machined workpiece with a mirror surface is shown in Fig. 1.

ePV (␮m)

x1

Measured form error (ePV = 0.397 μm m)

2 0.2 Form error (tilt removal, ePV = 0.341 3 μm)

0.0 0 -8

-4

0

4

8

Radial position x(mm)

(b)) Tilt removal Fig. 2. Measured profile and tilt removal. (a) Measured profile and form error and (b) tilt removal.

ePV = 0.341 μm (φ12 mm)

Form error ze (μm)

0.4

ePV = 0.187 μm (φ8 mm)

ePV = 0.073 μm (φ4 mm)

0.2

0.0 -8

-4

0

4

Radial position x (mm) Fig. 1. Machined workpiece with a mirror surface.

Error ze (μm)

Trail no.

Fig. 3. Form error evaluated by reducing the aperture.

8

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PV form error ePV (μm)

0.5 φ12 mm

0.4 0.3

Average

0.2 0.1 0.0 0

1

2

3

Compensation number

Fig. 4. PV form error versus aperture.

Fig. 6. Comparison of PV form errors. Table 5 Design of experiments and results for compensation cutting* . Trail no.

1 2 3 4 5 6 7 8 *

Rough cutting

Level

ePV (␮m)

ePV (␮m)

x1

x2

1st

2nd

3rd

0.225 0.204 0.133 0.273 0.127 0.147 0.106 0.162

−1 1 −1 1 0 0 0 0

−1 −1 1 1 0 0 0 0

0.408 0.098 0.183 0.087 0.241 0.177 0.168 0.382

0.072 0.095 0.082 0.069 0.103 0.068 0.073 0.132

0.087 0.084 0.089 0.122 0.071 0.098 0.081 0.077

The value with underline indicates an effective compensation.

the relationships to the apertures. Although the data are scattering, there is a trend that a larger aperture would result in a larger error. 3.3.3. Factorial design for compensation cutting and results The approach of compensation cutting based on the RSM is to follow the same control factors and levels as listed in Table 1. The 2K factorial design with CCD was adopted as detailed in Table 5. In every experimental trial, it was configured with one rough cutting and three sequent compensation cuttings. To maintain the same initial condition, the rough cutting before compensation was given by the same spindle speed (S = 2000 rpm) and depth of cut (D = 15 ␮m). The PV form errors obtained by rough cuttings vary from 0.106 to 0.273 ␮m with a mean value of 0.172 ␮m and a standard deviation of 0.053 ␮m. The compensation algorithm is to eliminate the form error by a corrected curve, which is obtained by mirror mapping and averaging the measured unsymmetrical form error. Fig. 5 shows one of the first compensation results by reducing the PV form error from the rough cutting of 0.204 to 0.098 ␮m, with

4. Model derivation for rough cutting 4.1. Predictive model

0.3

Form error ze (μm)

a compensation percentage of 51.96% (=(0.204 − 0.098)/0.204). Then, the second (2nd) and third (3rd) compensations were performed sequentially with the results of PV form error being 0.095 ␮m and 0.084 ␮m (not shown), respectively, as indicated by Trail No. 2 in Table 5. All the results of the rough and compensation cuttings are summarized in Table 5 and schematically shown in Fig. 6. Referring to the results of the 1st compensation, however, only Trials Nos. 2 and 4 were capable of reducing the form errors from 0.204 to 0.098 ␮m and 0.273 to 0.087 ␮m, respectively. Other trials failed to achieve the compensation function. This can be considered to be due to the change of the operational conditions from rough cutting to compensation cutting. In the rough cutting, the machining system was required to start up a liquid cooling system since the spindle speed was as high as 2000 rpm. However, the compensation cuttings other than Trials Nos. 2 and 4 were all performed below 2000 rpm without the additional liquid cooling function, which gave rise to different operational conditions. Therefore, it is known that to achieve an effective compensation, the operational conditions for both the rough and compensation cutting stages should be the same. This can be further verified by the results of the 2nd compensation, which shows that all of the form errors at the 1st compensation could be successfully reduced subject to the same operational condition. However, at the 3rd compensation, only three experimental trials with their initial PV form errors being larger than 0.095 ␮m could achieve effective compensations. This could be due to the performance limitation of the machining system including the sensing device and environmental condition. According to the above, the RSM approach will be performed only based on the 2nd compensation in Section 5.

Referring to Table 4 configured with eight experimental trials and results, the regress parameters ˇi (i = 1, 2) for two control factors could be derived based on Eq. (4). The predicted response yˆ (␮m) obtained is as follows,

ePV = 0.204 μm (φ12 mm)

0.2

0.1

yˆ = 0.197 − 0.055 x1 − 0.066 x2 .

0.0

By using the transforming equations Eq. (5), the predictive model can be expressed in terms of the spindle speed S and depth of cut D as follows,

ePV = 0.098 μm (After compensation)

-0.1 -8

-4

0

(7)

4

Radial position x (mm) Fig. 5. Measured form error after compensation.

8

Yˆ = 2.397S −0.079 D−0.072 ,

(8)

where Yˆ = eyˆ . This model shows that increasing the spindle speed and depth of cut would reduce the form error, and that the effects

Y.-T. Liu, L. Zhang / Precision Engineering 44 (2016) 203–210

Depth of cut D (μm)

A = φ12 mm

ePV = 0.075 μm

15

ePV = 0.10 μm 10

ePV = 0.15 μm ePV = 0.20 μm

5 500

1000

1500

2000

Spindle speed S (rpm) Fig. 7. Response contours for form error in the S–D parameter plane.

Table 6 ANOVA for response surface function of the form error (12 mm). Source

SS

DF

MS

F

P

R2

Regression

29,502

2

14,751

8.33

0.026

0.769

8855 7503 1353

5 2 3

1771 3751 451

8.32

0.060

38,358

7

5480

Residual error Lack-of-fit Pure error Total

207

Referring to the result of F-test for significance of regression, because F = 8.33 > F0.05, 2, 5 = 5.79, we would reject the null hypothesis that the regress coefficients are zero, i.e., H0 : ˇ1 = ˇ2 = 0. Also, note that the P-value (0.026) for F is smaller than ˛ = 0.05. The result of ANOVA incorporating the lack-of-fit test is shown in the same Table. The lack-of-fit test statistics is F = 8.32, which means that the lack-of-fit sum of squares is a larger component than that of the pure error of the residual. However, since F = 8.32 < F0.05, 2, 3 = 9.55 and the P-value (0.060) > 0.05, we fail to reject the null hypothesis (H0 : There is no lack of linear fit). That is, there is insufficient evidence at the level of ˛ = 0.05 which could lead to the conclusion that there is lack of linear fit. In other words, the derived linear model could suitably describe the data obtained. The final step of the approach is to check the adequacy of the predictive model. The coefficient of multiple determination (R2 ) is defined as the ratio of the SSR (sum of square residual) to the SST (sum of square total). R2 is an index to evaluate the closeness of the derived model to the actual. A higher value of R2 implies a more precise model. The coefficient R2 = 0.769 implies that the linear model could explain 76.9% of the variability observed in the PV form error. In ultraprecision machining, since there are uncountable factors that would affect the machining accuracy, the model having R2 = 0.769 could be considered to be good for predicting the PV form error in some particular ranges of process parameters. 4.3. Comparison of measured results and model predictions

of these parameters are almost equivalent. This predictive equation can be plotted as contours for each of the responses shown in Fig. 7. Using the contour plot, a suitable combination of parameters (spindle speed and depth of cut) can be determined for a given form error. In addition, the 3D response surfaces of the dependent variable (PV form error) can be plotted with respect to the independent coding factors and dimensional factors, respectively, as shown in Fig. 8. The models demonstrate how the spindle speed and depth of cut affect the form error. It can be seen that the smallest PV form error obtainable is 76 nm when the spindle speed is 2000 rpm and the depth of cut is 15 ␮m. Therefore, without the need of further compensation cutting, the result that the PV form error being under 0.1 ␮m was confirmed through this study. In some applications of optical lenses, this level of form accuracy is acceptable for dies and molds [16,17]. 4.2. Adequacy of model By converting the unit of PV form error from ␮m to nm, the ANOVA result for the surface response is shown in Table 6.

The difference between the measured results and the model predictions for the aperture of 12 mm is shown in Fig. 9(a), and the variation percentages of the difference are presented in Fig. 9(b). It can be seen that the trial numbers of 4 and 6 are with large variation percentages of 31.1% and 27.2%, respectively. 4.4. Predictive model vs. aperture To establish the predictive models for different outer apertures, the regress parameters bi (i = 0, 1, 2) are derived by referring to all the results shown in Fig. 4. The predictive equation in unit of nm can be expressed in terms of the dimensional factors of S and D as yˆ = b0 + b1 S + b2 D.

(9)

Fig. 10 shows the parameters bi with the relations to the outer aperture (A, 2 to 12 mm). It can be seen that the model coefficients are highly related to the aperture. The smaller the aperture is, the smaller the intercept (b0 ) and the larger the negative slopes (b1 , b2 ). However, the large values of the negative slopes for the spindle speed and depth of cut imply that they are less dominant

Fig. 8. 3D-plot of surface response of the form error for rough cutting. (a) 3D plot based on coding factors and (b) 3D plot based on dimensional factors.

Y.-T. Liu, L. Zhang / Precision Engineering 44 (2016) 203–210

0.5

PV form error ePV (μm)

A = φ12 mm

0.4 Experiment

0.3 0.2 0.1

Model

0.0 1

2

3

4

5

6

7

8

Variation percentage of PV error (%)

208

100 A = φ12 mm

80 60 40

31.1%

27.2%

20 0

Trail number

1

2

3

4

5

6

7

8

Trail number

(b) Variation percentage

(a) PV form error

Fig. 9. Comparisons of measured results and model predictions for rough cutting. (a) PV form error and (b) variation percentage.

400

Coefficient b0 (nm)

b2 300

0

200

-50

b1

b0 100

-100

Coefficient b1(nm/rpm), b2 (nm/μm)

50

-150

0 2

4

6

8

10

although two control factors of the spindle speed and depth of cut are capable of being used to predict the form error of aspheric machining, the developed model will become inadequate if it is derived based on the PV form errors smaller than 0.1 ␮m. This can be due to the limitation of the current operational conditions. 5. Model derivation for compensation cutting Similar to the previous approach for rough cutting, the predictive model for the form error of the 2nd compensation cutting is derived in this section. The predictive model and model adequacy are briefly described in the followings.

12

Outer aperture A (mm)

5.1. Predictive model

Fig. 10. Derived model coefficients.

in affecting the form error when the aperture is small. That is, the regress parameters are insignificant.

According to the PV form errors obtained by the 2nd compensations, the linear model can be expressed in terms of the independent coding factors as yˆ = 0.0868 + 0.0025x1 − 0.0040x2 ,

4.5. Coefficients of determination

and in terms of the S and D as

The coefficient of determination R2 is examined for different apertures with the results shown in Fig. 11. The results show that R2 can remain at a large value of over 0.6 when the aperture is larger than 8 mm. However, it significantly drops from 0.572 to 0.146 when the aperture is reduced from 6 to 2 mm. Referring to Fig. 4, it can be seen that the form errors range from 0.047 to 0.106 ␮m and from 0.027 to 0.050 ␮m for 6 and 2 mm, respectively. From these examinations, it could be concluded that

1.0

Yˆ = 1.1270S −0.0036 D0.0044 ,

(11)

This model shows that increasing the spindle speed and reducing the depth of cut would decrease the form error, and that the effect of depth of cut is larger. In addition, the 3D response surfaces of the PV form error with respect to the coding factors and dimensional factors are plotted, respectively, as shown in Fig. 12. It can be seen that the smallest PV form error obtainable is 80 nm when S = 2000 rpm and D = 2.5 ␮m. 5.2. Adequacy of model

2

Coefficient of determination (R )

(10)

The ANOVA result for the surface response is shown in Table 7. Referring to the result of F-test for significance of regression, because F = 0.067 < F0.05, 2, 5 = 5.79, we fail to reject the null hypothesis that the regression coefficients are zero, i.e., H0 : ˇ1 = ˇ2 = 0.

0.8 0.6 0.4

Table 7 ANOVA for response surface function of compensation cutting. F

P

R2

89

2

44.5

0.067

0.936

0.0256

Residual error Lack-of-fit Pure error

3387 745 2642

5 2 3

677.3 372.5 881

0.423

0.689

Total

3476

7

496.5

Source

0.2

Regression

0.0 2

4

6

8

10

Outer aperture A (mm) Fig. 11. The coefficient of determination (R2 ).

12

SS

DF

MS

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209

Fig. 12. 3D-plot of surface response of the form error for compensation cutting. (a) 3D plot based on coding factors and (b) 3D plot based on dimensional factors.

Fig. 13. Comparisons of measured results and model predictions for compensation cutting. (a) PV form error and (b) variation percentage.

Also, note that the P-value (0.936) for F-test is much larger than ˛ = 0.05. The R-squared statistic also indicates that the first-order linear model could explain only 2.56% of the variability observed in the PV form error. From these examinations, it could be concluded that the model is inadequate, implying that when the PV form error is near 0.1 ␮m, the effects to the compensation performance due to the spindle speed and depth of cut become trivial. This result is consistent to that obtained in the rough cutting. Therefore, to obtain a more precise model for predicting the PV form error below 0.1 ␮m, further improvements in measurement device, workpiece, tool, and operational environment, etc. are required. 5.3. Comparison of measured results with model predictions The difference between the measured results and model predictions is shown in Fig. 13(a), and the variation percentages of the difference are presented in Fig. 13(b). It is found that the trial number 8 has the a largest variation percentage of 52.2% while the others are under 21.6%. 6. Conclusions In this paper, mathematical models for predicting the form errors of aspheric ultraprecision machining including rough and compensation cuttings based on the RSM have been developed, using two control factors, i.e., depth of cut and spindle speed. The main results are summarized as follows: (1) The model prediction for rough cutting indicates that the PV form error will reduce while increasing the spindle speed and depth of cut. However, for compensation cutting, it shows that

the PV form error will reduce while increasing the spindle speed and reducing the depth of cut. (2) When the aperture is 12 mm, a precision model for rough cutting with R2 = 0.769 could be obtained. However, when the PV form errors are under 0.1 ␮m, the regress parameters for both rough and compensation cuttings become insignificant. (3) To achieve an effective compensation, the operational conditions for both the rough and compensation cutting stages should be the same. This study has demonstrated that the derived model for rough cutting based on the RSM could be used to predict the PV form error of aspheric ultraprecision machining. However, further improvements in experimental conditions are required if a model resolution is to be less than 0.1 ␮m. Acknowledgements The first author wishes to acknowledge the Ministry of Science and Technology, Taiwan, ROC, for financial supports with grant Nos. NSC 101-2911-I-327-501 and MOST 103-2221-E-327-010. The authors appreciate the experimental assistance offered by Mr. Evan Yang at the University of New South Wales, Australia. The second author wishes to thank the Australian Research Council for financial support to this research through its Grants LE077564, LE110100016, DP130100101 and DP140103476. References [1] Suzuki H, Moriwaki T, Yamamoto Y, Goto Y. Precision cutting of aspherical ceramic molds with micro PCD milling tool. Ann CIRP 2007;56(1):131–4.

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