1. Introduction

The primary mirrors for next-generation, large, ground-based telescopes are likely to be mosaic patterns of individual hexagonal segments of about 1.4-2 m in diameters [1–4] with few exceptions – including the Giant Magellan Telescope [5]. Examples of such large telescopes may include the Thirty Meter Telescope (TMT) primary mirror array consisting of 492 individual segments [4]. The rapid production requirements of such a large quantity of high precision and large, segmented mirrors pose significant technical challenges to the world-wide optics fabrication community today.

The fabrication process for precision astronomical and space optics includes bound abrasive grinding, loose abrasive lapping, polishing, and figuring. Classical bound abrasive grinding, though widely practiced, uses fixed abrasives of about 10 µm in size to generate approximated target surface form, and tends to leave a sub-surface damage of a few microns to be removed by the subsequent processes. Typically, polishing is used to remove about 1-2 µm of material from the substrate while the form is preserved. Then, figuring is applied for production of the final surface form while preserving the surface texture [6–7].

Over the last few decades, technology development for polishing and figuring has received a great deal of academic attention as they take up the largest portion of fabrication time. Examples of polishing and figuring technology include stressed lap polishing [8–9], dwell time polishing [10], ion beam figuring [11–12], laser ablation [13] and so forth. In the mean time, the optics fabrication communities have regarded bound and/or loose abrasive grinding as merely an initial process for generation of approximated surface form and hence have given relatively little attention to the process development.

However, the aforementioned rapid fabrication requirement for mass production of precision hexagonal segmented mirror leads us to re-consider bound abrasive grinding as an alternative process capable of generating a “mirror like” surface finish, as well as the precision form approximated to 1-2 microns, peak-to-valley, to the target. If developed, such grinding process would bring down considerably the time required for the loose abrasive lapping as well as for polishing.

Grinding is a complex process of material removal involving cutting, ploughing, and rubbing depending on the interaction between abrasive grains and substrate materials [14–16]. Grinding models for various non-optical materials [17–21] have been developed for prediction of resulting surface roughness from grinding runs and for controlling the Computer Numerical Control (CNC) grinding. They include a probabilistic un-deformed chip thickness model [17], a combined neuro-fuzzy and regression model [18], a truncated Gaussian distribution model [19], a probabilistic prediction model [20] and a life cycle model between wheel wear and bond fracture [21]. However, these “current state-of-the-art prediction models” demonstrate only limited successes. The limitations of these models include i) low surface prediction accuracy in excess of 20 nm (Ra); ii) non-specular surfaces of well above 200 nm (Ra) in target surface roughness; and iii) no active compensation technique for errors between predicted and measured surface roughness employed while consecutive grinding runs were made.

For optical materials, recent studies of precision grinding include the relationship between the surface roughness and grinding conditions [22], resulting surface roughness from ductile grinding [23], machining characteristics and material removal mechanism [24], optimization of grinding factors [25], and the relationship between surface roughness and subsurface damage during the machine run [26–28]. However, these studies were not concerned with compensation for machining error in surface roughness, let alone the controlled improvement of surface roughness in consecutive machine runs (i.e., “grinding process”).

For error compensation in precision CNC grinding, Yin et al. [29] used a tool path compensation model that varies both initial tool diameter and tip radius, and the trial run resulted in an improved form accuracy of 1.5 µm peak to valley over 180 mm in work-piece size in Electrolytic In-process Dressing (ELID) grinding. Okafor and Ertekin [30] reported a machine tool error model capable of compensating for tool inaccuracy. They used Vertical Machining Center (VMC) to compensate the measured error to 130 µm in length over 460 mm in linear motion. While these studies [29,30] showed the usefulness of error compensation techniques for reduction of machine tooling, form and tool inaccuracies, they were not concerned with surface roughness improvement and the techniques used do not have evolutionary self-learning capabilities.

Having witnessed the trend of the aforementioned investigations, this study reports a new evolutionary grinding process control model that i) has self learning ability to improve the process convergence, as well as the grinding accuracy; ii) can generate a set of optimum machining parameter inputs for a given target surface roughness prior to the machine run; iii) is computationally simple enough to be applied easily in optics shop-floor environments; and iv) can offer the optics fabrication community a nanometric controllability of material removal for “mirror-like surface finish.” The process model is introduced in Section 2 in detail. The initial grinding experiment for the model input data and the trial grinding runs for the model performance check are presented in Section 3. We then discuss the implications to the integrated machining time, to the relationship between surface roughness and linear speed of work-piece rotation and, finally, to the consideration for scaling the process model up to grinding run for the 1 m class optics in Section 4. This is followed by conclusions in Section 5.

2. Evolutionary grinding model

The grinding machine configuration is shown in Fig. 1. A rotating work-piece moves simultaneously in two orthogonal directions (X and Y), and the wheel moves along the Z axis. In a typical grinding run, the Y directional motion called the feed rate (F) is used while the X directional motion is not used. The Z directional motion controls the depth of cut (D), and therefore the surface figure. The grinding wheels used contain hard abrasive grains joined together with epoxy bonds. The average grain size, rotation speeds of both work-piece and wheel are expressed by G, Vp, and Vw respectively.

 

Fig. 1. Schematic illustration of the grinding configuration

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The evolutionary grinding process control model starts with a grinding database comprising grinding parameters used and the resulting surface roughness data for the chosen grinding parameters. The grinding parameters used are tool grain size, feed rate and linear speed of the work-piece rotation (V_PL). The linear speed is derived from rotation speed and position of work-piece. The initial database was constructed from a number of trial machine runs, as explained in subsection 3.2, with all the grinding parameters pre-determined and their resulting surface roughness measured. No evolutionary process control was implemented at this stage. For measured surface roughness data (Ra_M), we measured at 6 different radial locations on the work-piece surface. The grinding database can then be represented by the following expression (1), where ‘i’ is used to identify “i-th grinding run,” ‘N’ to express the total number of grinding, and ‘j’ to mean the measurement locations on the work-piece surface.

This process database starts with the initial database, but grows with the addition of new data sets as the machine run keeps stepping forward. The serial number ‘i’ is added by 1 each time, as expressed in Eq. (2), whenever a new grinding run (and hence model run) commences.

The true target surface roughness (Ra_TRij) for the i-th grinding run is defined as nominal target surface roughness (Ra_Tij) subtracted by a target correction factor (Δ_i-1) as expressed in Eq. (3). Ra_Tij is an approximated target surface roughness that is predetermined by opticians or machining operators. It is the single and representative surface roughness over the entire work-piece surface. Once fixed, it tends to generate 6 surface roughness data for 6 different control (i.e., “measurement”) locations on the work-piece surface (j = 6 at Eq. (1)). Δ_i-1 is the difference between the nominal target and the measured surface roughness data of the previous (i-1th) grinding run.

Tonshoff et al. [31] categorized the existing grinding models into two different groups i.e. physical and empirical models. The empirical models are typically constructed with the relationships between the input grinding parameters and the resulting (i.e. measured) performance indicator such as surface roughness. As a fair example of such empirical models, Kim [18] demonstrated the effectiveness of the multi variable regression analysis, relating wheel velocity, work-piece velocity, wheel depth of cut, dressing depth of cut and dressing feed rate to the surface roughness Ra. For this reason, our evolutionary grinding model adapts the regression Eq. (4), as it falls into the definition of empirical grinding model. The exponents (α, β and γ) and coefficient (δ) for the given target surface roughness are computed using MULTREGR routine [32] and LINEST function in Excel for each and every grinding run. The results of this multi-variable regression computation are a set of possible values for the input grinding parameters for the nominal target surface roughness Ra.

Because this computation using Eq. (4) tends to yield numerous combinations of possible values for the selected grinding parameters that could yield the given target surface roughness, we derive the final input values of the grinding parameters from the required machining time (T_i) expressed in Eq. (5), where L is feeding lengths and F_i is feed rate as expressed in Eq. (1). The minimum machining time within the envelope of machining parameter limits described in subsection 3.2 must be observed when applying Eq. (5).

The simultaneous application of Eq. (5) and of the minimum machining time condition to the possible values of the machining parameters yields a set of final input values for the grinding parameters. The work-piece is then machined with the input values, and the resulting surface roughness is measured following the measurement standard [33]. The measured surface roughness is then deposited into the database and the i+1th machine run (hence the model run) begins.

3. Experimental performance of evolutionary grinding model

3.1 Material and machines

A Zerodur blank of 100 mm in diameter with -5.95 in aspheric coefficients and 0.0034 mm in asphericity was used for the actual machining experiment. The machine run was performed by a Pneumo-K001 Nanoform 600 Single Point Diamond Turning machine (from Precitech Ltd.) with a grinding module capable of handling a work-piece of 300 mm in diameter. The size of the grinding wheel (from Diagrind Ltd.) was 75 mm in diameter. The surface roughness of the work-piece was measured by a contact profilometer named Form Talysurf (from Rank Talyor Hobson Ltd.). The measurement length used is 5.6 mm following the standard measurement length recommended by the International Organization for Standardization [33]. The ECOCOOL S47 BF (from Fuchs Lubricants Ltd.) coolant was mixed with de-ionized water in 1:10 ratio and then supplied in between the work-piece and the grinding wheel at 19 °C with pH of 9.4.

3.2 Initial grinding for model input data

We performed preparatory machine runs prior to the construction of initial database, using D = 0.5 µm to D = 6 µm and Vw = 5500 rpm to Vw = 9500 rpm. The results indicated that, for the following machine runs, D and Vw should be fixed to 1 µm and 8500 rpm respectively, since they produced the repeatable and best surface roughness data.

Employing fixed D and Vw, thirty-seven initial grinding experiments were then undertaken with the input grinding parameters and their values predetermined, considering the machine operational limit. The grinding machine limits include the feed rate ranging from 0.5 to 30 mm/min and the work-piece rotation of 100 to 400 rpm. The feed rate interval of 2.5 mm/min and work-piece rotation interval of 80 rpm were used for determining the machine control inputs.

While undertaking the experiments, we used a microscope called “Camscope” to check if there were any scratch like events taking place on the sample surface for every machine run and for every measurement location. This was to ensure that the surface roughness measurement was not contaminated from scratch-like surface features.

The resulting surface roughness data are plotted against feed rate (Fig. 2(a)) and linear speed of work-piece rotation (Fig. 2(b)). The cross, triangle, and dot symbols correspond to 3 different grinding wheels of 16.5, 9.3, and 6.9 µm in grain size. The averaged surface roughness values for each grain size were 172.1, 103.8, and 58.0 nm respectively. We note that, from Fig. 2, it is evident that the resulting surface roughness is approximately proportional to the wheel grain size

 

Fig. 2. Surface roughness (Ra) and (a) Feed rate (crosses, triangles, and dots are corresponded to 16.5, 9.3, and 6.9 µm in grain size) and (b) linear speed of work-piece rotation; Note that the three fitted trend lines come from Eq. 6 and 7 in Table A1 in Appendix.

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The resulting surface roughness data were fitted with the 1^st, 2^nd and 3^rd order least square methods, exhibiting the distribution characteristics of surface roughness against varying grain size, feed rate (Eq. (6)) and linear speed of work-piece rotation (Eq. (7)) as expressed in Table A1 in Appendix. Among them, Fig. 2 shows the trend lines expressed by the 3rd order polynomial equations as they offer the best R² value across the data of all the grain sizes, let alone for those of the grain size of 6.9 µm. The definitions of α₀, β₀, γ₀, and δ₀ in Table A1 are identical to those of Eq. (4) for the wheel of 6.9 µm in grain size.

While the initial database includes the surface roughness for all three grain sizes, the grain size was fixed to 6.9 µm throughout the trial grinding runs for the evolutionary model performance check, as described in subsection 3.3. It has been well known that, in ductile grinding mode, a mean grain size of smaller than 10 micrometers is recommended for producing aspheric surfaces of high quality surface texture [34]. Furthermore, in our experiments shown in Fig. 2 and Table A1, it is evident that the standard deviation of Ra for 6.9 and 16.9 µm in grain size are smaller than that of 9.3 µm in grain size and the grain size of 6.9 µm offers the lowest surface roughness as well. For this reason, we fixed the grain size to 6.9 µm and the regression computation for Eq. (4) used feed rate (F₀) and linear speed of work-piece rotation (V_PL0) only. We note that, for the grain size of 6.9 microns, the standard deviations obtained from 3^rd order least square fit and regression method do not show much difference in Table A1. This implies that the straightforward application of multi-variable regression to the data may not be good enough to bring the standard deviation down to 1-2 nm level for the further experiment run and it strengthens the usefulness of the target correction factor expressed in Eq. (3).

3.3 Experiment for nanometric surface roughness control

The evolutionary model described in Section 2 was applied iteratively and seven trial grinding runs (N = 7 at Eq. (1)) were performed. The averaged (i.e. representing) nominal target surface roughness values used are from 115 nm Ra to 56 nm Ra. The resulting input values for the grinding parameters suggested by the model (and hence collected from each corresponding grinding run) are listed in Table 1.

Table 1. Resulting input values for the grinding variables collected after seven machine runs were completed

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3.4 Results

The exponents and coefficient of Eq. (4), nominal and true target surface roughness, measured surface roughness, and correction factor collected after the completion of the seven trial grinding runs are listed in Table 2. It shows changes in exponents and coefficient of the regression equation and in the target correction factor as the machine run progresses. These very clearly demonstrate an evolutionary and self-educating process control.

Table 2. Exponents and coefficient of regression (β, γ and δ), nominal (Ra_T) and true (Ra_TR) target surface roughness, measured surface roughness (Ra_M) and target correction factor (Δ)

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Figure 3(a) shows the average trend lines and standard deviations from two different evolutionary controls for comparison. The open squares connected by a dotted line form a trend line of average surface roughness difference between the nominal target surface roughness and measured surface roughness. The error bars represent standard deviations obtained from six measurement locations on the work-piece surface as stated in the previous section and listed in Table 2. We note that the averaged difference converges from +15.4 nm down to -8.1 nm as well as the reduction of standard deviation (σ) from ±12.2 nm to ±6.5 nm. It is also worth mentioning that even setting a nominal target for each machine run incorporates a degree of “evolutionary process control” as it uses the multi-variable regression equation with the machining database that grows continuously with each grinding run. The results of this evolutionary process bring about gradual improvement in both the average surface roughness difference and the standard deviation.

The solid circles are the average difference between the true target (i.e. nominal target corrected by the correction factor) and measured surface roughness. The average difference and their standard deviation rapidly converged from an initial value of 15.4 ± 12.2(σ) nm to the final value of -0.2 ± 2.3(σ) nm toward the 7^th trial grinding run. This suggests that the integrated evolutionary process control, by incorporating the growth of machining database, the usage of multivariable regression equation, and the target surface roughness correction, has the potential to be almost deterministic in controlling surface roughness in precision optical grinding.

In Fig. 3(b), the performance difference of two different evolutionary controls in standard deviations of the averaged differences between target and measured surface roughness is shown. These two plots clearly exhibit a striking difference in performance improvement brought about by the target correction factor based on the machining error of the previous grinding run.

 

Fig. 3. Comparison between (a) grinding errors; and (b) standard deviation of residuals with (solid circles and bars) and without (open squares and bars) target correction factor of evolutionary process control

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4. Discussion

4.1 Feed rate and integrated machining time

Figure 4 shows the changing feed rate as the machine run progresses to achieve the true target surface roughness sequence as experimented. The solid triangle symbol indicates the feed rate obtained from the evolutionary process control, implemented in actual machine runs, using the growing database, evolving regression equation, and the target correction factor. The open rhombus symbol means the feed rate derived from the process of “non evolutionary control” characterized with the fixed initial database, non-evolving 3^rd order polynomial equation (Eq. (6) as in Table A1) and no application of the target correction factor. It is evident that the evolutionary process control offers the faster feed rates and therefore the resulting integrated machining time of 262.7 minutes. This shows the improvement of 55.9% in machining time, when compared to the total integrated machining time of 470.2 minutes with non-evolutional process control.

 

Fig. 4. Feed rates to achieve the experimented sequence of true target surface roughness with (solid triangles representing the actual experiment data) and without (open rhombuses indicating the computed data using Eq. (6)) evolutionary process control

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4.2 Surface roughness versus linear speed of work-piece rotation

Figure 5(a) shows the measured surface roughness plotted against the linear speed of work-piece rotation fitted by 3^rd order least square fitting. The same surface roughness data are re-plotted against the 6 measurement locations on the work-piece surface in Fig. 5(b). These figures suggest that the averaged surface roughness over the six measurement locations decreases with the linear speed of the work-piece rotation during the seven trial machine runs. This finding is supported even more strongly by the similar characteristics of the initial grinding database shown in Fig. 2(b).

However, for each single machine run, the surface roughness at the six different measurement locations (i.e. six different linear speeds of the work-piece rotation) shows more complex features that can be described with either 2^nd or 3^rd order polynomial fits. In particular, the data from the 5^th, 6^th and 7^th machine runs show that the surface roughness seems to increase as the linear speed of the work-piece rotation decreases. This is rather contradictory to the aforementioned trend of the averaged surface roughness data from the first machine run to the 7^th machine run. This contradiction and the complex relationships between the surface roughness Ra and the linear speed of the work-piece rotation in single machine run tend to call for an independent study as it falls outside the main scope of the current study reported here.

 

Fig. 5. Measured surface roughness vs. (a) linear speed of work-piece rotation and (b) distance from the work-piece center for seven trial grinding runs

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4.3 Consideration for grinding machine run for 1 m class optical surface

For scaling up the evolutionary grinding process to the fabrication of optical surfaces of one meter in diameter, it is very important to maintain the constant volumetric material removal over the duration of many machine runs. The several factors influencing the volumetric material removal rate may include, but not limited to, tool loading conditions, tool wear, and subsurface damage [26–28]. Having said that, assuming that this constant volumetric material removal rate can be successfully maintained over a grinding machine run across the diameter of 1 m, the best combination of input values of the chosen grinding parameters obtained from this study (i.e. feed rate of 1.2 mm/min, linear speed of work-piece rotation of 21,363 mm/min corresponded by 25 mm in length from work-piece center) would result in the surface roughness of around 60 nm Ra with the machining time of 1,042 minutes. The details of evolutionary grinding process simulation for 1 m class optical surface will be presented in a separate study [35].

5. Conclusions

A new evolutionary process control model for precision optics grinding is reported. Unlike those reported in the previous studies [22–28] of precision grinding of optical materials, this process model is built upon a combination of unique process control elements such as a growing machining database, multivariable regression equation and target surface roughness correction factor. When combined, these control elements tend to give the process “evolutionary characteristics” based on the experience (i.e. surface roughness error) of the previous grinding run.

Using the process model, seven trial grinding runs were performed and successfully demonstrated the rapid improvement of surface roughness and the reduction of error between the intended target surface roughness and the measured surface roughness. The process also offers a capability to first determine the intended target surface roughness, and secondly to derive the necessary input values for the machining parameters.

It should also be noted that the process control model, particularly “evolutionary nature,” is conceptually simple and yet numerically efficient. It does not require a heavy computational load and therefore can be easily adapted to most optics shop floor environments. For these reasons mentioned above, we argue that the evolutionary process control reported here is radically different from other existing techniques reported elsewhere [17–30] and has the potential to form a unique and important subset of next generation CNC grinding techniques for efficient mass production of large precision aspheric optical surfaces for the next generation of extremely large telescopes.

Appendix

For characterization of the initial database, we performed the 1^st, 2^nd and 3^rd order polynomial equation fitting to the data. A multi-variable regression equation was also used for data fitting. Table A1 summarizes the results including coefficients and exponents, their standard error, standard deviation and R² value.

Table A1. Characteristics of initial grinding data

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Acknowledgements

We would like to express thanks to Ju-Whan Kim, Hyo-Sik Kim, Min-Gab Bok, and Sang-Bo Kim at the Korea Basic Science Institute, and Dae-Wook Kim at the University of Arizona for technical assistance. We also thank Myung K. Cho at the National Optical Astronomy Observatory, Clay Williams at the University of Arizona, and Moo-Young Chun and Gazinur Galazutdinov at the Korea Astronomy and Space Science Institute for comments.

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