## Abstract

Edge effect is regarded as one of the most difficult technical issues for fabricating large primary mirrors, especially for large polishing tools. Computer controlled active lap (CCAL) uses a large size pad (e.g., 1/3 to 1/5 workpiece diameters) to grind and polish the primary mirror. Edge effect also exists in the CCAL process in our previous fabrication. In this paper the material removal rules when edge effects happen (i.e. edge tool influence functions (TIFs)) are obtained through experiments, which are carried out on a Φ1090-mm circular flat mirror with a 375-mm-diameter lap. Two methods are proposed to model the edge TIFs for CCAL. One is adopting the pressure distribution which is calculated based on the finite element analysis method. The other is building up a parametric equivalent pressure model to fit the removed material curve directly. Experimental results show that these two methods both effectively model the edge TIF of CCAL.

© 2014 Optical Society of America

## 1. Introduction

Edge effects are very common in many contact optical fabricating methods and are regarded as one of the most difficult technical issues, which can dominate the performance of segmented-mirror telescopes such as GMT [1], E-ELT [2], and JWST [3]. On the other hand, the edge effects will make the mirror edge “roll down” (except the magnetorheological finishing); to rectify this, more material needs to be removed over the bulk surface, and much more time will be consumed. To satisfied the demand of large mosaic primary mirror telescopes, fine edge figure controlling and a high-efficiency fabricating process are both needed urgently.

There are many studies that have been done regarding the edge effects based on computer controlled optical surfacing (CCOS) technology. The early studies tried to build a simplified pressure model to describe the pressure distribution when the tool overhangs the edge of the mirror. Jones [4] suggested a linear pressure model in 1986. The skin model, which was proposed by Luna-Aguilar *et al.* [5] and Cordero-Davila *et al*. [6], supposes that there are two different pressure regions: high- and low-pressure regions on the contact area. Combining these two models, a constant-linear (C-L) model was proposed by Han *et al.* [7], which divides the contact region into two parts: constant pressure region C and linear region L. Kim *et al.* [8] provided a new idea which defines a parametric model based on measured data, rather than assigning the edge effects to a certain type of analytical pressure model, which matches the experimental data very well. Hu *et al.* [9] researched reducing the edge effect in magnetorheological finishing by using small size removal function and utilizing the removal function compensation. Walker *et al.* [10] developed the edge-control technology on hexagonal parts for bonnet polishing by utilizing variable size influence functions through tool-lift method. Li *et al.* [11,12] reported the edge tool influence functions (TIFs) modeling for the tool-lift method based on measured TIF data [11], theoretic pressure, and velocity analysis [12] and got good results on a 200-mm across-corners hexagonal witness part. All these studies were based on CCOS technology.

Compared with the CCOS, the computer controlled active lap (CCAL) has larger size polishing pad,
different structure, and different motion type [13].
There are quite a few reports on the edge effect of active lap or stressed lap. The skin
pressure model was used to describe the edge effect for active lap processing before [6]. This model is not enough to compensate the pressure
variation and would cause the mirror edge roll-down in our previous fabrication. Figure 1 shows the surface shape error profiles of a 1.8-m
primary mirror. The edge roll-down caused by skin pressure model in active lap grinding process was
observed. There are eight profiles which are across the whole mirror surface with equal angle
space between each of the two neighboring lines in Fig.
1. Lines are separated along the *y* axis for better reading. The surface
profile data were measured by a coordinates-measuring machine (CMM) with 1-μm
accuracy.

In order to improve the efficiency of active lap processing and obtain the fine controlled edge figure on a large primary mirror, the edge effect of active lap first must be eliminated. To accomplish that, first, the edge ring TIFs, which mean the material removal rules of active lap when it overhangs the mirror edge, must be known. Precise and stable edge ring TIF modeling is also important. Second, the ability of active lap in edge figure control must be studied, and a good edge control strategy also must be developed. In this paper, we primarily focus on the former problem.

The theoretical background of the edge ring TIF modeling for active lap is shown in Section 2. Two methods are presented in Section 3 to model the edge ring TIFs. The experiments are carried out on a Φ1090 circular flat mirror. A 375-mm-diameter flat lap is used in the experiments. The experiment results and the comparisons between model TIFs and measured TIFs are reported in Section 4. The improvement of these two models and the discussions are presented in Section 5.

## 2. Theoretical background

#### 2.1 The Preston equation

In active lap grinding or polishing processes, the material removal on the mirror surface can be calculated based on the Preston equation, which was presented by Preston [14] in 1927.

where*dz*is the integrated removal material during

*dt*time,

*k*is the Preston coefficient,

*P*is the pressure on the lap–mirror contact region, and

*V*is the magnitude of relative speed between the lap and mirror.

#### 2.2 Methods to calculate the ring TIF

In rotational symmetric fabricating process, the workpiece will rotate about its symmetrical axis with speed *ω*_{1}, and the active lap will dwell at different radial positions with rotate speed *ω*_{2}. The values of *ω*_{1} and *ω*_{2} are at the same level, so the workpiece motion cannot be ignored when someone tries to calculate the TIFs for the active lap [13]. The ring TIF was proposed to describe the material removal rule in this situation [15]. The ring TIF is the rotational symmetric removal profile and looks like a ring on the workpiece. The ring TIF can be represented by a 1D profile, its *x* axis is the radial position on the workpiece, and its *y* axis is the removal depth after one unit time.

Figure 2 is a sketch of CCAL in edge figuring when the lap
dwells on radial position *e*. We adopt the polar coordinates system in which the polar axis is along the line
*O*_{1}*O*_{2}. *O*_{1}
is the center of the workpiece, and *O*_{2} is the center of the lap.
The edge ring TIF can be calculated by summing the removal material during one workpiece
rotation. For example, the ring TIF on workpiece surface point *A* is the sum of
the removal material when *A* moves from *M* to
*N* along the arc *MN*, which can be expressed as

*ω*

_{1}/2π is the factor to scale the removed material in one unit time. According to our previous work [13], Eq. (2) can be written as

*φ*is the polar angle of the point

*N*. In Eq. (3), the lap–workpiece relative velocity

*V*is known [13]. The pressure distribution on the lap–workpiece contact region is uniform (i.e.,

*P*does not vary with the coordinates

*ρ*and

*θ*) when the active lap stays inside the mirror. Once the lap overhangs the mirror edge, this status will be changed, and the pressure distribution will no longer be uniform as well. To simplify the pressure model, we use a 1D equivalent pressure curve ${P}_{E}(\rho ,e)$ to replace the 2D pressure distribution$P(\rho ,\theta ,e)$. A good equivalent pressure curve will satisfy the condition that

There are two ways to get the edge ring TIF. One is to find out the pressure distribution, $P(\rho ,\theta ,e)$, and the other is to construct an equivalent pressure curve, ${P}_{E}(\rho ,e)$. The linearity model, skin model, and C-L model belong to the first method, and they all hypothesize a simple pressure distribution model to calculate the removal material. Instead of assuming a simple pressure distribution model, one can use the finite element analysis (FEA) to solve the pressure distribution for the first way, and this will be reported in Section 3. The parametric model proposed by Kim *et al*. [8] does not aim to model the pressure distribution but to fit a parametric Preston coefficient map, $\kappa (x,y)$, based on the linearity pressure model. Based on this idea and Eq. (5), we can build up a parametric pressure model ${P}_{E}(\rho ,e)$ for the active lap, and this will be discussed in Section 3.

#### 2.3 The lap overhang ratio

The lap overhang ratio, *S*_{lap}, is a dimensional irrelevant parameter
that is used to describe the lap position status in edge figuring, as shown in Fig. 3.

## 3. Edge effect modeling

#### 3.1 Parametric pressure model for edge ring TIFs

The parametric model proposed by Kim *et al.* [8] suggests that the Preston coefficient, *κ*, is a function of the distance from the mirror edge, *x*, and the lap overhang ratio, *S*_{lap}. It is a five-parameter function, as follows:

*κ*

_{0}is the Preston coefficient when there is no overhang,

*f*

_{1}is the edge-side correction, and

*f*

_{2}is the workpiece-center-side correction.

Based on this idea, we build up a parametric equivalent pressure curve,
*P _{E}*(

*ρ*,

*e*), and put it into Eq. (5) to calculate the edge ring TIF. Considering that the active lap is working like the spin tool motion, the workpiece-center-side correction,

*f*

_{2}, is not necessary. So, when we build up

*P*(

_{E}*ρ*,

*e*), only the edge-side correction,

*f*

_{1}, is considered. Also, the variables

*ρ*and

*e*were replaced by

*x*and

*S*

_{lap}in

*P*(

_{E}*ρ*,

*e*) for convenience. The basic form of the parametric equivalent pressure curve is

*p*is the pressure scale factor, which scales the whole pressure to a reasonable level. The parameter

_{s}*ε*represents how fast the pressure increases in edge area. The basic idea of the parametric equivalent pressure curve is shown in Fig. 4.

The function *f* in Eq. (7) is used to describe the nonlinear pressure distribution in edge effects. It is a second-order function of variable *x*, and its influence width is equal to *W*_{lap}·*α*. The maximum value of this function can be controlled by the parameter *β*.

*x*≥ 0 and 0 for

*x*< 0. Consider that the width of edge effects may be different at each overhang ratio, so the parameter

*α*is a function of

*S*

_{lap}, $\alpha =g({S}_{\text{lap}})$. The parameters (i.e.,

*α*,

*β*, and

*ε*) are determined by using the measured edge ring TIF data to fit Eq. (5), and the least square method is used to solve this problem. This will be discussed in Section 4.

Example curves of the parametric pressure model for a 300-mm-diameter lap are plotted in Fig. 5.An arbitrary parameter set was used in the example, where *α* =
–2*S*_{lap} + 0.8, *β* = 3, and
*ε* = 0.2. The example shows that both width and slope of the nonlinear
pressure curve are different at each overhang ratio in this model.

#### 3.2 FEA model considering a soft layer for pressure analysis

The finite element method can be used to analyze the contact status between bodies and calculate the pressure distribution. The material of the active lap base plate and grind layer is aluminum alloy, and the material of the mirror is K4 glass. The material characteristics for modeling are listed in Table 1.

The workpiece is a circular flat mirror with 1090-mm diameter. To simplify, only part of the mirror is considered in the FEA model, which is a 375-mm-wide and 18-mm-thick block. Only the base plate and the grinding layer of the active lap are included in the model. The diameter is 375 mm for the base plate and 300 mm for the grinding layer. The thickness is 18 mm for the base plate and 2.5 mm for the grinding layer.

Our previous work [16] shows that the pressure will dramatically increase near the mirror edge if we only consider the lap and mirror and let them directly contact in the FEA model. The same FEA results can also be found in [8]. This phenomenon may be caused by the high rigidities of the lap and the mirror. So the mirror edge becomes a pivot to support the lap, and the pressure will be concentrated here. But the actual edge ring TIFs from experiments show that the slope of the edge area pressure is smaller than that from direct-contact FEA results. This difference may be caused by the medium between the lap and mirror, which changes the contact status. This medium consists of the abrasives or the slurry. The abrasives or the slurry are like a soft layer between the lap and mirror and will make the pressure dispersed over the contact region through its “deformation” (actually, particle redistribution or abrasion), instead of concentrated near the mirror edge. As the mirror and the lap are rotating, abrasive particles are moving around and filling the gaps between the lap and the mirror. The size and shape of abrasive particles will be changed because of the abrasion, especially in the edge area where more abrasion occurs. These two factors will change the contact status and affect the pressure distribution.

It is difficult to simulate the mechanical behavior of abrasives or slurry in FEA. So we add a
thin soft layer between the lap and mirror in our model to approximate it. The Young’s
modulus of the soft layer, *E _{s}*, is much smaller than that of the
mirror,

*E*. The soft layer is about 2 mm thick and with 1-mm mesh size in our model. The deformable to deformable separable surface contact is used in the model. Two symmetric contact pairs are created for lap–soft layer and soft layer–mirror contact. The FEA model is shown in Fig. 6.Fixed constraint is applied on the workpiece bottom, and only the gravity load is considered. The mesh size is 6 mm for the workpiece part and the lap and 1.25 mm for the grinding layer. The FEA modeling and analyzing are carried out on ANSYS Workbench 14.5.

_{m}The Young’s modulus of the soft layer, *E _{s}*, is essential to
this model. Different

*E*will produce different pressure distribution on the workpiece surface, as shown in Fig. 7.The width of the nonlinear pressure area is narrow, and the slope of the pressure along the radial line is high when

_{s}*E*is close to

_{s}*E*; otherwise, the width and slope are wide and low when

_{m}*E*is smaller than

_{s}*E*.

_{m}A best *E _{s}* which satisfies the condition that theoretical edge ring TIFs coincide with the measured TIFs may exist. But the proper

*E*is hard to find out. Our approach to solve this is to use one group of measured TIF data with the same overhang ratio to fit the

_{s}*E*and then use this

_{s}*E*to analyze the pressure distribution for different overhang ratios. After that, put the pressure in Eq. (3) to calculate the edge ring TIFs for those overhang ratios. Finally, we compared these TIFs with the measured data from experiments to verify

_{s}*E*. This will be presented in Section 4.

_{s}## 4. Edge ring TIF experiments and model fitting

#### 4.1 Experiment setup and measured TIFs

The active lap edge ring TIF experiments are carried out on a 1090-mm-diameter circular flat mirror. The lap is a 375-mm-diameter flat lap with a 300-mm-diameter grinding layer, as shown in Fig. 8(a).The W40 grit emery was used in the experiments, and the ratio between the emery and water is about 1–5 in volume. The pH of the slurry is about 7, and the temperature of the experimental environment is about 25 °C. The mirror rotate speed is 2.7 rpm, and the lap rotate speed is –6 rpm. The positive/negative indicates the rotation direction. The LEITZ PMM30-20-10 CMM is used to measure the mirror surface shape along four radial lines [see Fig. 8(b)]. The surface profile was measured with 1-μm height accuracy and 1-mm spatial sample space. The edge ring TIF is calculated by subtracting the measured data after grind from the data before grind.

Five group experiments are carried out to verify the models. The parameters for each group are shown in Table 2.Groups 1, 2, and 4 have same dwell time but different overhang ratios, and groups 3, 4, and 5 have same overhang ratio but different dwell times.

The edge ring TIFs from the experiments are shown in Fig. 9.The unit of the TIF value is μm/min. The shape of the edge ring TIF is changed with the overhang ratio but relatively stable with the dwell time. This is very important to the dwell time-based deterministic fabrication.

#### 4.2 Parametric pressure model fitting

Measured edge ring TIFs are used to fit the parametric pressure model,
*P _{E}* [see Eq. (7)].
There are five parameters that need to be fitted: parameter

*α*for each overhang ratio (i.e., for

*S*

_{lap}= 0.1, 0.2, and 0.3),

*β*, and

*ε*. We use the least square method to solve this problem. The target curves are the edge ring TIFs for

*S*

_{lap}= 0.1 and 0.2 and the average edge ring TIF for

*S*

_{lap}= 0.3. The fitted results are

*α*

_{0.1}= 1.559,

*α*

_{0.2}= 1.101,

*α*

_{0.3}= 0.540,

*β*= 18.511, and

*ε*= 0.642, where

*α*

_{0.1}represents

*α*for

*S*

_{lap}= 0.1 and so on. The fitted parametric pressure curves are shown in Fig. 10.The curves are normalized by the pressure when the active lap is inside the mirror. Edge ring TIFs predicted by the fitted parametric pressure model are shown in Fig. 11, where the solid lines represent the predicted TIFs and the dashed lines represent the experimental TIFs. The unit of the TIFs is μm/min.

The parametric pressure model can fit the edge ring TIFs reasonable well from the results in
Fig. 11. This proves that the parametric pressure model
has the ability to describe the edge effect for active lap processing. However, there is one
shortcoming of this method, which is that the parameter *α* is varying
along the *S*_{lap}. The
*α*-*S*_{lap} curve is needed in order to get all
the series of edge ring TIFs with any overhang ratio. The curve fitting method or the curve
interpolation method can be used to solve this problem. A simply curve fit has been done, and
the *α*-*S*_{lap} curve is plotted in Fig. 12(a). Figure 12(b) shows a set of the edge ring TIFs
calculated by using this *α*-*S*_{lap} curve. The
*β* and *ε* are as same as in Fig. 10. Because the Preston coefficient is related to workpiece material,
polishing slurry, and temperature, for simplicity, the model-predicted TIFs are normalized by
the maximum value of the TIF with 0.35 overhang ratio.

To improve the model accuracy, more edge ring TIF experiments with different overhang ratios are needed. Once the model is built up, it can be used in the entire fabricating process.

#### 4.3 FEA model fitting

Another way is analyzing the pressure distribution directly based on the FEA model which was
proposed in Section 3.2. A series of finite element analyses are carried out in order to find
the best fitting Young’s modulus of the soft layer, *E _{s}*, in
Fig. 6. The average experimental edge ring TIFs with
0.3 overhang ratio are used as the target in the searching. Figure 13 shows the comparison between the experimental TIF and theoretical TIFs with
different scale

*E*.From the figure we can find out (1) the theoretical TIF when

_{s}*E*equal or close to

_{s}*E*is far away from the experimental TIF and (2) the best fitting

_{m}*E*exists between 0.0001

_{s}*E*and 0.001

_{m}*E*. Through the searching we found that the best fitting modulus for this case is about 20.78 MPa, which is approximately 0.0003

_{m}*E*. More FEA models are built up with this Young’s modulus soft layer, and the pressure distributions are analyzed. The results are shown in Fig. 14.The overhang ratio varies from 0.05 to 0.4 at regular intervals of 0.05.

_{m}Center pressure curves are calculated from the FEA results and shown in Fig. 15(a).We noticed that the pressure on the inner side area is close to zero when the lap overhang ratio is up to 0.4. Based on the analyzed pressure distribution, the edge ring TIF family is calculated and plotted in Fig. 15(b). The TIFs are normalized because the Preston coefficient is unknown. These TIF curves are similar to those from the fitted parametric pressure model [see Fig. 12(b)]. Due to the pressure, the material removal rate on the inner side area is also close to zero when the lap overhang ratio is up to 0.4. This may be the reason why the edge is rolling down deeply but the near-edge part remains relatively high in Fig. 1.

A comparison between FEA modeling TIFs and experimental TIFs is shown in Fig. 16.The model-predicted TIFs match well with the measured TIFs in our experiments. This means that the FEA model is working, and adding the soft layer can help us to get more accurate pressure distribution.

## 5. Model comparison and application

#### 5.1 Comparison and discussion

The parametric pressure model and the FEA model are two different methods. Both of them are trying to achieve more accurate edge ring TIF modeling. In order to compare these two methods, we define the normalized model fit error, Err, as follows:

It represents the ratio of fit residual material to total removed material. The ideal model fit error is zero based on Eq. (9). So the volume convergent efficiency of the edge effect model can be expressed as 1 – Err. The fit errors for both models are all below 16%, which means these two models have more than 84% volume convergent efficiency. The parametric model can achieve 6.9% fit error at 0.3 overhang ratio, better than the FEA model (10.7%). There is nearly no difference between uncompensated TIF and model TIF when the overhang ratio is small because the overhang effects can be ignored [8]. The model fit errors (15.8% for the parametric model and 14.1% for the FEA model) are slightly larger than uncompensated fit error (10.8%) at 0.1 overhang ratio in Fig. 17, and this difference may be caused by heterogeneity of the slurry during grinding. As the overhang ratio increases, obvious improvement can be observed for both models.

The parametric model is simple and direct but needs a series of edge ring TIFs to fit it before using. The FEA model reveals the pressure distribution on the workpiece surface. It allows us to calculate more edge TIFs, especially for uncircular tools and uncircular workpieces.

#### 5.2 Application in simulation process

Two groups of virtual fabrication are carried out to study the error profile fit capability for different edge pressure models. The first one considers the piston target removing process, which means to remove material equally at each position, and the second one considers an arbitrary error profile removing process. The fabricating parameters for the two groups are the same: workpiece rotating at 2.1 rpm and active lap starting at 200 mm, stopping at 830 mm, and rotating at 3.3 rpm when its position is less than 600 mm and –3.3 rpm when its position is beyond 600 mm. The effective radius of the active lap is 175 mm, the radius of the workpiece is 892 mm, and the radius of the workpiece center hole is 145 mm. Two hundred lap dwell positions and 200 error profile sample points are used in the simulation. The nonnegative least square method is used to fit the target removal profile and get the lap dwell time distribution. In FEA modeling, 16 group FEA results are used, and the overhang ratio varies from 0.025 to 0.4 at regular intervals of 0.025.

The results from the first group of simulations are shown in Fig.
18.Curves are normalized by the target removal profile. The target removal profile is a
horizontal line in which the *y* axis value is equal to 1. The parametric
pressure model can fit the target profile very well, and the maximum fit residual error is only
0.4%. Because the number of FEA results is limited, FEA modeling results are not as smooth as
the parametric modeling results, but they also fit the target profile reasonable well, and the
maximum fit residual error is 2.6%. The skin model cannot fit well at the edge area (i.e., the
right side), and the maximum fit residual error is 12%.

The results of the second group of simulations are shown in Fig. 19.The target removal profile is the error profile of a 1.8-m primary mirror during grinding. The parametric model and the FEA model fit the target removal profile better than the skin model at the edge area. The maximum fit residual error is 6.2 μm for the parametric model, 5.6 μm for the FEA model, and 10 μm for the skin model.

Simulation results indicate that one adopting parametric pressure model or FEA pressure model could achieve more accurate fabrication and reduce the edge effect. Both of these two models are better than the skin pressure model.

## 6. Concluding remarks

We presented a way to calculate the edge ring TIF for active lap processes and two methods (the parametric model and the FEA model) to describe the edge effects when lap overhangs the workpiece edge. Parametric model builds up a parametric equivalent pressure curve in order to calculate the edge ring TIF and uses the measured TIFs to fit this curve. The FEA model simulates the pressure distribution on the workpiece surface using the finite element method. A thin soft layer with a special Young’s modulus is considered in our model between the lap and the workpiece in order to simulate the effect of abrasives or slurry in the process. The Young’s modulus of this soft layer is determined by fitting the model-predicted TIFs to the measured TIFs.

Experimental results show that both of these methods are effective in edge effect modeling for the CCAL process. The normalized fit error was proposed to evaluate the fit ability of edge effect models. About 10% fit error can be achieved in these two models. This means about 90% of the materials in the edge ring TIF can be modeled by these models and makes engineering use possible. Further fabricating simulations indicate that both the parametric model and the FEA model are better than the skin model in error profile fitting, especially at the edge area. This may be helpful in edge effects controlling and compensating.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (61178043). We thank Yong Zhang, Kai Fang and Linhan Xian for help in the grinding process and Changqing Lin for support of CMM measurement.

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