Int J Adv Manuf Technol (2014) 75:1431–1438 DOI 10.1007/s00170-014-6223-7
ORIGINAL ARTICLE
Restraint of tool path ripple based on the optimization of tool step size for sub-aperture deterministic polishing C. Wang & W. Yang & S. Ye & Zhenzhong Wang & P. Yang & Y. Peng & Y. Guo & Qiao Xu
Received: 12 March 2014 / Accepted: 1 August 2014 / Published online: 22 August 2014 # Springer-Verlag London 2014
Abstract Tool path ripple error (TPR_error) is one of the main reasons due to the medium-high spatial frequency error on the surface of aspheric optics. The purpose of this paper is to analyze the effect of the tool step size to the TPR_error in sub-aperture deterministic polishing (SDP) and study a method which can optimize the tool step size to restrain this error. Three groups of simulation experiments were conducted using three different tool influence functions to simulate the uniform removal of the material. As the TPR_error is influenced by three factors, which are full width at half maximum (FWHM) of tool influence function (TIF), tool step size, and depth of material removed, each group of the experiments was conducted under the fixed TIF and depth of material removed. It C. Wang : W. Yang (*) : S. Ye : Z. Wang : P. Yang : Y. Peng : Y. Guo Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China e-mail:
[email protected] C. Wang e-mail:
[email protected] S. Ye e-mail:
[email protected] Z. Wang e-mail:
[email protected] P. Yang e-mail:
[email protected] Y. Peng e-mail:
[email protected] Y. Guo e-mail:
[email protected] C. Wang : Q. Xu Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China Q. Xu e-mail:
[email protected]
turns out that both peak-to-valley (PV) and root-mean-square (RMS) values of the TPR_error become larger with the increase of the tool step size, and the variation tendency likes a reversed “L” shape curve. And, the method adopted in the simulation was further validated by the experiment. Therefore, the tool step size at the inflection point would be optimal to restrain the TPR_error together with saving the polishing time to a certain extent. This method could be used to determine the best-suited tool step size in SDP whose typical TIF is a Gaussian or Gaussian-like shape. Keywords Tool path ripple . Tool step size . Sub-aperture deterministic polishing . Finishing . Aspheric optics
1 Introduction When used in an optical system, aspheric optics can increase the degrees of freedom in a design, reduce the system’s weight and size, and provide a significant advantage for correcting system aberrations [1]. And, this stimulates the demand for aspheric optics, especially for high-precision aspheric optics. To acquire extremely high accuracy aspheric optics, many computer-controlled sub-aperture deterministic polishing (SDP) technologies have been developed to date, such as magnetorheological finishing (MRF) [2, 3], ion beam finishing (IBF) [4, 5], bonnet polishing (BP) [6–9], and abrasive jet polishing (AJP) [10–14]. In SDP, the material is accurately removed, and the surface error correction would be attained quickly based on the theory of computer-controlled optics surfacing (CCOS) firstly proposed by Itek Inc. in 1970s [15]. However, the microfabrication errors called “fragmentary errors” are increasing in the process of fast surface error removal, due to a reason that the “tool” size is less than the workpiece in SDP technologies [16, 17]. These medium-high frequency errors
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2 Theoretical background For SDP, the amount of material removed H(x, y) is equal to the two-dimensional convolution between the TIF per unit time R(x, y) and the dwell time function D(x, y), along with the motion track [11, 26, 27]: H ðx; yÞ ¼ Rðx; yÞDðx; yÞ
ð1Þ
The convolution between R(x, y) and D(x, y) leads to the tool path ripple (TPR) error caused by the superposition of R(x, y). Therefore, the TPR_error E(x, y) after the polishing process can be expressed as E ðx; yÞ ¼ H 0 ðx; yÞ−Rðx; yÞDðx; yÞ
ð2Þ
where H0(x, y) is the amount of material to be removed. In order to demonstrate the formation of TPR_error, its schematic diagram has been made as shown in Fig. 1. The blue line in Fig. 1 signifies R(x, y), and the convolution result H(x, y) between R(x, y) and D(x, y) is shown as the red line. The amount of material to be removed H0(x, y) is shown as the black line, and it subtract H(x, y) leading to a result E(x, y) as the green line shown. It can be noted that when the TIF of the polishing tool is a Gaussian-like shape, the TPR_error cannot be eliminated and the larger distance between each dwell point would lead to larger TPR_error.
3 Experiments designed to show the formation of TPR In order to demonstrate the TPR_error experimentally, two uniform removal experiments using BP technology are conducted, respectively, adopting raster path and circular path. The experimental device is as shown in Fig. 2 which is newly developed by our group. The bonnet tool comprises a spinning inflated bulging rubber membrane with a spherical form and covered with a polishing cloth (often composed of polyurethane) and operates in the presence of a cerium oxide polishing slurry. The bonnet is brought into contact with the surface to be polished and then compresses to create a defined polishing spot. The orientation of the tool rotation axis can be controlled at a defined inclination angle with respect to the local normal surface over the entire workpiece through controlling A-axis and B-axis as shown in Fig. 2. The tool axis is then precessed in (typically four) discrete steps about the local normal to the part’s surface. This process generates an accumulated influence function which is near-Gaussian. [7] H0
1.0
Material removal/um
would be serious in some applications such as in intense laser systems and high-resolution image formation systems which have the strict requirements for both of the surface error and medium-high frequency [18, 19]. This error is mainly affected by the initial surface error distribution (spatial and frequency domain), the removal function characters (profile, removal efficiency, and stability), path design, and residual error induced by the convolution effect [20, 21]. Yu et al. [22] presented a grolishing process that can remove mid-spatial grinding features by BoX™ ultraprecision grinding machine. Dai et al. [20] designed a partrandom path based on the maximum entropy method to restrain the mid-spatial frequency error. Hu et al. [21] certificated that the random pitch tool path based on the surface error distribution and parameters can restrict the tool path ripple height while the convergence efficiency would not be reduced. Dunn et al. [23] successfully adopted a pseudorandom tool path in BP to improve a surface with a strong pre-existing periodic structure and suppress the mid-spatial frequency error. Much attention has been paid to the effect of path style and tool removal characters to this error, whereas little attention has been paid to the effect of the tool step size and its relationship with the tool path ripple (TPR). Walker et al. [24] proposed that the peak-to-valley (PV) value of the TPR tends to be deterministic and depends on full width at half maximum (FWHM) of tool influence function (TIF), tool step size, and depth of material removed. They also determined the appropriate ratios of spot size to step size for polishing two different depths, but the principle has not been disclosed specifically. Hence, a definitive method to determine the optimal tool step size for different polishing depths is necessary and meaningful. In this paper, we proposed a method to determine the optimal tool step size to minimize the TPR_error in SDP technologies, including BP, IBF, and AJP whose typical TIF is a Gaussian or Gaussian-like shape [7, 11, 25]. Firstly, we demonstrate the formation of the TPR_error theoretically. Then, the TPR_error is presented using two uniform removal experiments. After that, three groups of simulation experiments were conducted to analyze the effect of tool step size to the TPR_error together with the method to determine the optimal tool step size. At last, the simulation method is experimentally validated and follows with the conclusion.
H
0.8
H0 H
0.6
x 10
E
0.4
R(TIF)
0.2 0.0
0
50
PV
0 E 100 x/mm
150
Fig. 1 Schematic diagram of the formation of TPR_error
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lists the other experimental conditions. An R80 bonnet tool was used in this experiment. The results were measured using QED SSI™. Figure 3 shows the surface profile after uniform removal using a raster path. Figure 3a shows the whole surface profile and partial
A-axis
H-axis
B-axis
a
mm
101
Bonnet tool
Work-piece
nm
1000
80
500
60
0
40
-500
20
-912 0 0
20
40
60
mm 101 nm
80
mm
3.7
y direction
There are two modes of BP tool motion: position dwell (PD) mode and velocity dwell (VD) mode [28]. The PD mode refers to the machine tool that moves between the neighboring dwell points with the maximum speed and dwells at the dwell point exactly at the calculated dwell time. For a raster path, this mode will lead to the TPR_error in the x and y directions. And, for a circular path, the TPR_error along the radial and circumferential directions would be generated. In the VD mode, the dwell time at each point is implemented by varying the speed with which the tool traverses. It can eliminate the TPR_error in one direction and reduce feeding time between each dwell point in the PD mode. However, TPR_error remains along raster space direction and circumferential direction. And, this would be demonstrated in the following experiments. Experiments were conducted on two plane BK7 glasses which have been fine polished before. In order to retain unpolished surfaces from which the absolute removal depth could be established, an 80-mm-diameter sub-area of each part was polished. The VD mode was used herein. Table 1
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1.5
10.0
x direction
-25.5 mm
0.0 0
2
4
6
b 20
33.2
8
10
12
14
x profile
nm
3mm
31.8nm
3mm
10
3mm
0 -10
Table 1 Experimental conditions Conditions
34.6nm
-20
Experiment number 1
0
2
4
27.1nm 6
2
a
Raster Φ100 3 23 500 600 0.7 0.25 24a 1.96
Circular Φ120 1.5 23 500 600 0.7 0.25 5.5 2.09
The polishing process was executed four times and 6 min for each time
10
12
14 mm
y profile
nm
Path type Workpiece size (mm) Path space distance (mm) Precession angle (deg) H-axis speed (rpm) Feed speed (mm/min) Z-offset (mm) Inner pressure (MPa) Total polishing time (min) wt% of CeO2 in the polishing slurry (%)
8
18 16 14 12 10 8 0.0
0.5
1.0
2.0
1.5
2.5
3.0
3.5 mm
(b) Fig. 3 Surface profile after raster path uniform removal. a The whole surface form profile. b Partial surface form profile
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a
mm
nm
371
120 200
100 80
0 60 -200 40 20 -457
0 0 8.3
20
40
60
80
100
120
mm
mm nm 36.7
y direction
surface profile magnified from the rectangle area. The section profiles of the partial enlarged surface along the red and blue lines as shown in the bottom part of Fig. 3a are demonstrated in Fig. 3b which is processed using the software WYKO Vision32. It is noted that the TPR_error only appears to be in the x direction and the PV value of the TPR_error varies from 25 to 35 nm over the entire surface. The variation of the PV value may be caused by the initial surface error and the instability of TIF. It is also found that the surface profile in the y direction is obviously tilted and the reason for this could be the holding error of the workpiece. Figure 4 shows the surface profile after uniform removal using a circular path. Figure 4a shows the whole surface profile and partial surface profile magnified from the rectangle area. There exists a groove along the radial direction. This may be induced by the vibration in the z direction when changing the circular radius each time and over removed of material when feeding the bonnet tool between each circular path. The section profiles of the partial enlarged surface along the red and blue lines as shown in the bottom part of Fig. 4a are demonstrated in Fig. 4b. TPR_error appears to be along the radius direction. The PV value of the TPR_error varies a lot over the entire surface as shown in Fig. 4b. Comparing these two results, the TPR_error is more obvious in the result of experiment 1. Both of them are interfered by some practical factors, such as the initial surface error, the instability of TIF, and the holding error of the workpiece. With this in mind, the TPR_error generated from the actual experiment would be unstable. Therefore, the actual experiment should preferably not be used to analyze how the tool step size affecting the TPR_error. Simulation experiment should be adopted instead in this study.
Int J Adv Manuf Technol (2014) 75:1431–1438
6.0
20.0
x direction
4.0
0.0 2.0 -20.3 0.0 0
2
4
b
6
8
10
mm
x profile
nm 4 2 0
4 Simulation analysis and results 4.1 Simulation experiment design In order to analyze the effect of the tool step size to the TPR_error, three groups of simulation experiments are conducted using three different TIFs. Considering that the shapes of TIF used in BP, IBF, and AJP are mostly Gaussian or Gaussian-like, three Gaussian functions with different FWHM as shown in Fig. 5 are adopted as TIF for these simulation experiments. Without loss of generality, all of TIFs are normalized in that the maximum value of the TIF is 1 μm. The left part of Fig. 5 is the three-dimensional shape of each TIF, and section profiles of them are in the right part. Three 400× 400 mm plane surfaces are assumed as the workpiece. The material removal is simulated using Eqs. (1) and (2) aforementioned, and the amount of material prospectively to be
-2 -4 -6 -8 0
30
4
2
6
8
10 mm
y profile
nm
20 10
1.5mm 4.7nm
0
6.2nm
-10 0
2
4
1.5mm 1.5mm 3.0nm 6
8 mm
Fig. 4 Surface profile after circular path uniform removal. a The whole surface form profile. b Partial surface form profile
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a
a 1.0 0.8
z/ um
z/ um
1.0 0.6 0.2 20
0.6 0.5 0.4
11.77
0.2
10 10 20 0 x/ m -10 -10 0 -20 -20 m m y/ m
0.0 -20
-10
0 x/ mm
10
20
10
20
5
10
b 1.0 0.8
z/ um
z/ um
1.0
0.2
0.6 0.5 0.4
20
0.2
0.6
10 10 20 x/ m 0 -10 -10 0 m m -20 -20 y/ m
0.0
7.06
-20
-10
0 x/ mm
b
c 1.0 1.0
z/ um
0.8
z/ um
0.6 0.2 20
0.6 0.5 0.4
3.53
0.2
10 10 20 x/ m0 -10 0 m -20 -20 -10 y/ mm
0.0 -10
-5
0 x/ mm
Fig. 5 Three TIFs with different FWHM for the simulation experiment. a TIF1: FWHM=11.77 mm. b TIF2: FWHM=7.06 mm. c TIF3: FWHM= 3.53 mm
removed is uniformed as 5 μm. The simulation condition is as shown in Table 2. All of the simulation conditions are the same except for the tool step size in each group of simulation experiments. PV value and root-mean-square (RMS) value of the TPR_error are adopted to evaluate the simulation results.
Fig. 6 Simulation results using three different TIFs. a Relationship between PV value of TPR_error and tool step size. b Relationship between RMS value of TPR_error and tool step size
vary with the tool step size as shown in Fig. 6a, b, respectively. The PV and RMS are calculated using the following equation:
4.2 Results and analysis Figure 6 shows the results of these three groups of simulation experiments, which are PVand RMS values of TPR_error that
PV ¼ Max TPR errorÞ−MinðffiTPR errorÞ sðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN TPR errori 2 i¼1 RMS ¼ N
ð3Þ
Table 2 Simulation experiment condition Group number
1
2
3
FWHM of TIF 11.77 7.06 3.53 (mm) Removed depth 5 5 5 (μm) Tool step size 0.5~10, Δda =0.5 0.5~6.5, Δd=0.5 0.2~3.4, Δd=0.2 (mm) a
Δd signifies the increment of each time
Table 3 The optimal tool step size and corresponding PV and RMS values Group number
Optimal tool step size (mm)
PV value (nm)
RMS value (nm)
1 2 3
~7.5 ~4.5 ~2.4
2.91 3.1 11.2
1.89 1.9 9.2
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a
20
mm 12
wv
10
0.85 0.50 0.00
6
1
2
3
4
5
6
60
8
-0.50
4 2 0
Fig. 7 Diagram of the experiment design (unit: mm)
-1.00 -1.25 0
2
4
6
8
10
12 mm
b where N denotes the total error points number of TPR_error. It is interesting to note that both PVand RMS values become larger with the increase of the tool step size and the variation tendency likes a reversed “L” shape curve. When the tool step size varies toward zero, the generated TPR_error will become extremely small; however, a long time it would take to perform a polishing cycle. When the tool step size becomes larger, both PV and RMS values increase. The growth rate at the beginning is rather slow; however, a sharp increase comes out once the tool step size exceeds the inflection point. As shown in Fig. 6, the positions of the inflection point in the PV curve and RMS curve are almost the same, and the results at this point in the reversed L shape curve restrain the TPR_error well. Hence, the tool step size at this point is the best suited to restrain the TPR_error and without taking much polishing time at the meantime. The optimal tool step size of these three situations, together with their corresponding PV and RMS values, is summarized in Table 3. This method can also be adopted to determine the optimal tool step size in other situations. Both the tool step sizes at the inflection point in Fig. 6a, b become smaller from group 1 to group 3. This implies that the larger the FWHM of the TIF, the smaller tool step size at the inflection point would be, which also can be seen in Table 3. In other words, relatively large tool step size could be employed when using the TIF with relatively large FWHM.
mm d=1mm d=2mm d=3mm d=4mm d=5mm d=6mm
0.00 -0.50
40
-1.00
20
-1.58 0 0
20
40
60
Conditions
Value
FWHM of TIF (mm) Dwell time at each point (s) Tool step size (mm)
~7.9 (averaged value) 0.5, 1, 1.5, 2, 2.5, 3 1, 2, 3, 4, 5, 6
80
100
mm
Fig. 8 Experiment results. a Extracted TIF used in these experiments. b Six material removal lines with different tool step size (d denotes the tool step size, wv=632.8 nm)
Experiments were designed to generate six material removal lines with the same removal depth adopting different tool step size as shown in Fig. 7. A plane BK7 sample was used, and ~2 wt% CeO2 was used as the polishing slurry. Each material removal line was designed as 60 mm, and the distance between each of them was 20 mm. The tool step sizes were 1, 2, 3, 4, 5, and 6 mm, respectively. Considering that the removal depth of them must be the same as described in the simulation experiment, the dwell time needs to be precisely controlled at each dwell point. The details of the experiment conditions are summarized in Table 4. The TIF as shown in Fig. 8a was derived using an R80 bonnet tool polishing with a tilted angle
PV value/nm
5 Experimental validation
Table 4 Experiment conditions
0.58
60
500
In order to validate that the simulation method mentioned above is effective, a group of experiments was also conducted.
wv
Measured data Trend curve
400 300 200 100 0
1
2
3 4 5 6 Tool step-size/mm
7
Fig. 9 The relationship between the tool step size and PV value of the TPR_error extracted from the result
Int J Adv Manuf Technol (2014) 75:1431–1438
dwelling 6 s and without the precession motion. It turns out that its shape is almost ellipse [29]. The length of the spot is ~12 mm, and the width is ~9 mm. The peak removal depth is 1,012.5 nm, and its FWHM is 9.5 and 6.3 mm in horizontal and vertical directions. Figure 8b shows the experiment results with six material removal lines on the workpiece surface. The result accorded with our expectancy which exists regular TPR_error obviously. It is noted that there is a scallop in the middle line when the tool step size is 5 mm. It is induced by an error in the NC code which makes the tool dwell time there more than the other 2.5 s. To analyze the effect of the tool step size to the PV value of the TPR_error, their relationship based on the results was extracted as shown Fig. 9. Its shape is nearly a reverse L shape which validates that the simulation method is correct. It is also noted that the PV value is not stable which is just the same as analyzed in Sect. 3.
6 Conclusion This paper demonstrates the formation of the TPR_error in SDP both theoretically and experimentally. And, simulation experiments have been conducted to analyze the effect of the tool step size to the TPR_error. The results show that both PV and RMS values of the TPR_error become larger with the increase of the tool step size and the variation tendency likes a reversed L shape curve. Hence, the tool step size at the inflection point is the best suited to restrain the TPR_error together with saving the polishing time to a certain extent. This method could be used to determine the optimal tool step size in SDP whose typical TIF is a Gaussian or Gaussian-like shape, and it is implemented according to the following steps: 1. Extracting the TIF which is about to be used in the polishing process, 2. Determining the polishing depth of material, 3. Executing the simulated polishing adopting a series of tool step size, 4. Drawing the curve of the relationship between the PV or RMS value of the final surfaces and the tool step size, and 5. Selecting the inflection point of the curve and the corresponding tool step size would be the optimal one to restrain the TPR_error. Acknowledgments This work was financially supported by major national science and technology projects (Grant No. 2013ZX04006011206).
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