## Parametric modeling of edge effects for polishing tool influence functions

Optics Express, Vol. 17, Issue 7, pp. 5656-5665 (2009)

http://dx.doi.org/10.1364/OE.17.005656

Acrobat PDF (935 KB)

### Abstract

Computer controlled polishing requires accurate knowledge of the tool influence function (TIF) for the polishing tool (i.e. lap). While a linear Preston’s model for material removal allows the TIF to be determined for most cases, nonlinear removal behavior as the tool runs over the edge of the part introduces a difficulty in modeling the edge TIF. We provide a new parametric model that fits 5 parameters to measured data to accurately predict the edge TIF for cases of a polishing tool that is either spinning or orbiting over the edge of the workpiece.

© 2009 Optical Society of America

## 1. Introduction

1. M. Johns, “The Giant Magellan Telescope (GMT),” Proc. SPIE **6986**, 698603 (2008). [CrossRef]

2. M. Clampin, “Status of the James Webb Space Telescope (JWST),” Proc. SPIE **7010**, 70100L (2008). [CrossRef]

3. R. Aspden, R. McDonough, and F. R. Nitchie Jr., “Computer assisted optical surfacing,” Appl. Opt . **11**, 2739–2747 (1972). [CrossRef] [PubMed]

11. D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express . **13**, 910–917 (2005). [CrossRef] [PubMed]

3. R. Aspden, R. McDonough, and F. R. Nitchie Jr., “Computer assisted optical surfacing,” Appl. Opt . **11**, 2739–2747 (1972). [CrossRef] [PubMed]

12. D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. Cassie, C. Dunn, R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, “New results from the Precessions polishing process scaled to larger sizes,” Proc. SPIE **5494**, 71–80 (2004). [CrossRef]

*z*, which is known as the Preston’s equation [11

11. D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express . **13**, 910–917 (2005). [CrossRef] [PubMed]

*z*is the integrated material removal from the workpiece surface,

*κ*the Preston coefficient (i.e. removal rate),

*P*pressure on the tool-workpiece contact position,

*V*magnitude of relative speed between the tool and workpiece surface and ∆

_{T}*t*dwell time. It assumes that the integrated material removal, ∆

*z*, depends on

*P*,

*V*and ∆

_{T}*t*linearly.

11. D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express . **13**, 910–917 (2005). [CrossRef] [PubMed]

13. E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, and M. H. Pedrayes-Lopez, “Edge effects with Preston equation,” Proc. SPIE **4840**, 598–603 (2003). [CrossRef]

14. A. Cordero-Davila, J. Gonzalez-Garcia, M. Pedrayes-Lopez, L. A. Aguilar-Chiu, J. Cuautle-Cortes, and C. Robledo-Sanchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt . **43**, 1250–1254 (2004). [CrossRef] [PubMed]

*p*(

*x,y*), approaches [8, 13

13. E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, and M. H. Pedrayes-Lopez, “Edge effects with Preston equation,” Proc. SPIE **4840**, 598–603 (2003). [CrossRef]

14. A. Cordero-Davila, J. Gonzalez-Garcia, M. Pedrayes-Lopez, L. A. Aguilar-Chiu, J. Cuautle-Cortes, and C. Robledo-Sanchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt . **43**, 1250–1254 (2004). [CrossRef] [PubMed]

*P*, values for a given polishing configuration [15].

*κ*, which has been regarded as a universal constant in the spatial domain as a function of position in the TIF via the parametric approach. By doing so, we can simulate the combined net effect of many complex factors without adding more terms to the original Preston’s equation, Eq. (1).

*κ*map, and show simulated parametric edge TIFs from the model in Section 3. The experimental demonstration and value of the parametric edge TIF model are summarized in Sections 4 and 5, respectively.

## 2. Theoretical background for the parametric edge TIF model

### 2.1 Linear pressure distribution model

*x, y*), centered at the workpiece edge with the

*x*axis in the overhang direction (i.e. the radial direction from the workpiece center). The pressure distribution under the tool-workpiece contact area should satisfy two conditions [14

14. A. Cordero-Davila, J. Gonzalez-Garcia, M. Pedrayes-Lopez, L. A. Aguilar-Chiu, J. Cuautle-Cortes, and C. Robledo-Sanchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt . **43**, 1250–1254 (2004). [CrossRef] [PubMed]

*f*, applied on the tool should be the same as the integral of the pressure distribution,

_{0}*p(x,y)*, over the tool-workpiece contact area,

*A*. ii) The total sum of the moment on the tool should be zero. It is assumed that the pressure distribution in

*y*direction is constant, and it is symmetric with respect to the

*x*axis. The moment needs to be calculated about the center of mass of the tool, (

*x′, y′*) [14

**43**, 1250–1254 (2004). [CrossRef] [PubMed]

*x′*is the

*x*coordinate of the center of mass of the tool.

*p(x,y)*, is determined by solving two equations, Eqs. (2) and (3), for two unknown coefficients,

*c*

_{1}and

*c*

_{2}. Even though this analytical solution yields negative pressures for large overhang cases [14

**43**, 1250–1254 (2004). [CrossRef] [PubMed]

*c*

_{1}and

*c*

_{2}by iteration. Some examples of the linear pressure distribution,

*p*(

*x*), are plotted in Fig. 1 (left) when a circular tool overhang ratio,

*S*, changes from 0 to 0.3.

_{tool}*S*is defined as the ratio of the overhang distance,

_{tool}*H*, to the tool width in the overhang direction,

*W*, in Fig. 1 (left).

_{tool}### 2.2 The first (edge-side) correction

^{15}Pa: extremely rigid tool and 0.7×10

^{11}Pa: typical Aluminum). The tool was deformed by gravity, and the pressure distribution in the gravity direction was calculated under the tool-workpiece contact area.

*S*, varies.

_{tool}*f*, described in detail later in Section 3.2 is formed to correct this edge-side phenomenon.

_{1}### 2.3 The second (workpiece-center-side) correction

*f*, to address this discrepancy in Section 3.2. It allows us to increase or decrease the workpiece-center-side removal without considering many factors, such as tool bending effect, non-linearity of the Preston’s equation, fluid dynamics of the polishing compound, etc.

_{2}## 3. Parametric edge TIF model

### 3.1 Generation of the basic edge TIF

**13**, 910–917 (2005). [CrossRef] [PubMed]

*R*, and does not rotate. ii) Spin: The tool rotates about the center of the tool. These tool motions are depicted in Fig. 2.

_{orbital}*S*, is fixed for the spin tool motion case, but varies as a function of tool position (A~F in Fig. 2 (left)) for the orbital case while the basic edge TIF calculation is being made.

_{tool}### 3.2 Spatially varying Preston coefficient (κ) map

*κ*map for the parametric edge TIF model is introduced. The

*κ*map represents the spatial distribution of the Preston coefficient,

*κ(x,y)*, on the basic edge TIF that already includes the linear pressure gradient. It changes as a function of TIF overhang ratio,

*S*, and five function control parameters (

_{TIF}*α, β, γ, δ*and

*ε*).

*S*is defined as the ratio of the overhang distance,

_{TIF}*H*, to the TIF width in the overhang direction,

*W*, in Fig. 3. The parametric edge TIF can be calculated by multiplying the basic edge TIF by the

_{TIF}*κ*map.

*R*).

_{orbital}*κ*map approach is that it does not require independent understanding of each and every factor affecting the material removal process. Instead, only the combined net effect of them is represented by the

*κ*map. The

*κ*map is defined by a local coordinate centered at the edge of the workpiece.

*x*represents the radial position from the workpiece edge.

*f*, with two parameters,

_{1}*α*and

*β*. The first parameter,

*α*, determines the range of the quadratic correction from the edge of the workpiece. The second parameter,

*β*, controls the magnitude of the correction. This degree of freedom using

*α*and

*β*is shown in Fig. 3. This correction is shown graphically in Fig. 3 and defined analytically as

*z*) is the step function; 1 for

*z*≥ 0 and 0 for

*z*<0.

*f*, to address the discrepancy between the simulated (i.e. predicted) edge removal using basic edge TIF and measured edge removal in the workpiece-center-side region (mentioned in Section 2.3) is defined by Eq. (6). Similar to

_{2}*f*, it has two parameters,

_{1}*γ*and

*δ*. The third parameter,

*γ*, determines the range of the second correction, and the fourth parameter,

*δ*, controls the magnitude of the correction as shown in Fig. 3.

*κ*map is defined in Eq. (7). It is a sum of the first and second correction terms, and includes a fifth parameter,

*ε*. The fifth parameter,

*ε*, was introduced to change the magnitude of the

*κ*map as a function of TIF overhang ratio,

*S*. Larger

_{TIF}*ε*means that required correction magnitude increases faster as overhang ratio increases.

*κ*

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

*x*-profiles of example

*κ*maps are plotted in Fig. 4. An arbitrary parameter set (

*α*=0.2,

*β*=2,

*γ*=0.2,

*δ*=1 and

*ε*=0.2) was used in the example.

### 3.3 Generation of the parametric edge TIF

*κ*map (i.e. the spatial distribution of the Preston’s coefficient) by the basic edge TIF (with

*κ*=1) introduced in Section 3.1. The overhang ratio,

*S*, was varied from 0 to 0.3. Five parameter values (

_{TIF}*α, β, γ, δ*, and

*ε*) were used to fit the experimental data in Section 4.1 and 4.2. The parametric edge TIFs are shown in Table 1. As we increase the overhang ratio,

*S*, non-linearly increasing removal near the workpiece edge is clearly shown as a result of the first correctional term for both the orbital and spin cases. The effects of the second correction are also observed. Due to the opposite signs of

_{TIF}*δ*for the orbital (

*δ*= 20) and spin (

*δ*= -3) cases, in the workpiece-center-side region, there is more and less removal than the basic edge TIF’s.

## 4. Experimental demonstration of the parametric edge TIF model

### 4.1 Experimental set 1: Orbital tool motion

*S*= 0.05, 0.14, 0.24 and 0.28. The measured removal profiles with RMS error bars are plotted in Fig. 5. The simulated removal profiles based on the parametric edge TIF model (

_{TIF}*α*=0.2,

*β*=4,

*γ*=0.4,

*δ*=20 and

*ε*=1.5) are also plotted. The five parameters were optimized to fit the experimental data. With one set of parameters, most of the simulated removal profiles for all overhang ratio cases are well fit to the measured removals within the RMS error bars. It means that we can predict all series of removal profiles with any overhang ratio for a given tool and tool motion as long as we perform a few edge runs to determine the tool’s characteristic parameter set initially.

### 4.2 Experimental set 2: Spin tool motion

*S*, was changed to 0.02, 0.17, 0.22 and 0.4. The measured removal profiles with RMS error bars are plotted in Fig. 6. The simulated removal profiles based on the parametric edge TIF model are plotted also. They are reasonably well matched with the measured removal profiles for all overhang ratio cases including very high overhang ratio case,

_{TIF}*S*= 0.4.

_{TIF}### 4.3 Performance of the parametric edge TIF model

*x*= 0 ~ -60mm). The computed removal profile using basic edge TIF model seems to have a closer overall slope to the measured removal. However, two mismatches between the measured and simulated removal are clearly observed in the edge-side and workpiece-center-side regions. The parametric edge TIF model using only the first correction allows us to correct the discrepancy in the edge-side removal. The removal profile based on the parametric edge TIF model using both the first and second correction is well matched with the experimental removal profile over the whole range of the removal profile.

*S*< 0.14 for orbital case and

_{TIF}*S*< 0.02 for spin case). It basically means that there is no difference between nominal and edge TIF models when the overhang effects are negligible.

_{TIF}*S*=0.28, the normalized fit residual, ∆, falls to 10% (parametric edge TIF using both corrections) from 52% (nominal TIF), or from 30% (basic edge TIF). For the spin tool motion case with

_{TIF}*S*=0.4, the normalized fit residual, ∆, is dramatically improved to 12% (parametric edge TIF using both corrections) from 87% (nominal TIF), or from 66% (basic edge TIF). The second correction is not really required for the spin tool motion case, in contrast to the orbital tool motion case, where the second correction brought significant improvement.

_{TIF}## 5. Concluding remarks

*κ*map, which represents the spatial distribution of the Preston coefficient. In this way, we were able to express the net effects of many entangled factors affecting the edge removal process in terms of a parametric

*κ*map. Then the parametric edge TIF was derived from a multiplication of the

*κ*map and the basic edge TIF.

## Acknowledgments

## References and links

1. | M. Johns, “The Giant Magellan Telescope (GMT),” Proc. SPIE |

2. | M. Clampin, “Status of the James Webb Space Telescope (JWST),” Proc. SPIE |

3. | R. Aspden, R. McDonough, and F. R. Nitchie Jr., “Computer assisted optical surfacing,” Appl. Opt . |

4. | R. E. Wagner and R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt . |

5. | D. J. Bajuk, “Computer controlled generation of rotationally symmetric aspheric surfaces,” Opt. Eng . |

6. | R. A. Jones, “Grinding and polishing with small tools under computer control,” Opt. Eng . |

7. | R. A. Jones, “Computer-controlled polishing of telescope mirror segments,” Opt. Eng . |

8. | R. A. Jones, “Computer-controlled optical surfacing with orbital tool motion,” Opt. Eng . |

9. | J. R. Johnson and E. Waluschka, “Optical fabrication-process modeling-analysis tool box,” Proc. SPIE |

10. | R. A. Jones and W. J. Rupp, “Rapid optical fabrication with CCOS,” Proc. SPIE |

11. | D. W. Kim and S. W. Kim, “Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes,” Opt. Express . |

12. | D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. Cassie, C. Dunn, R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, “New results from the Precessions polishing process scaled to larger sizes,” Proc. SPIE |

13. | E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, and M. H. Pedrayes-Lopez, “Edge effects with Preston equation,” Proc. SPIE |

14. | A. Cordero-Davila, J. Gonzalez-Garcia, M. Pedrayes-Lopez, L. A. Aguilar-Chiu, J. Cuautle-Cortes, and C. Robledo-Sanchez, “Edge effects with the Preston equation for a circular tool and workpiece,” Appl. Opt . |

15. | B. C. Crawford, D. Loomis, N. Schenck, and B. Anderson, Optical Engineering and Fabrication Facility, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, (personal communication, 2008). |

16. | D. W. Kim, College of Optical Sciences, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, W. H. Park and J. H. Burge, are preparing a manuscript to be called “Edge tool influence function model including tool stiffness and bending effects.” |

17. | D. W. Kim, College of Optical Sciences, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, and J. H. Burge are preparing a manuscript to be called “Time scale dependent conformable tool.” |

**OCIS Codes**

(220.0220) Optical design and fabrication : Optical design and fabrication

(220.4610) Optical design and fabrication : Optical fabrication

(220.5450) Optical design and fabrication : Polishing

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: January 13, 2009

Revised Manuscript: March 3, 2009

Manuscript Accepted: March 20, 2009

Published: March 25, 2009

**Citation**

Dae Wook Kim, Won Hyun Park, Sug-Whan Kim, and James H. Burge, "Parametric modeling of edge effects for polishing tool influence functions," Opt. Express **17**, 5656-5665 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5656

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### References

- M. Johns, "The Giant Magellan Telescope (GMT)," Proc. SPIE 6986, 698603 (2008). [CrossRef]
- M. Clampin, "Status of the James Webb Space Telescope (JWST)," Proc. SPIE 7010, 70100L (2008). [CrossRef]
- R. Aspden, R. McDonough, and F. R. Nitchie, Jr, "Computer assisted optical surfacing," Appl. Opt. 11, 2739-2747 (1972). [CrossRef] [PubMed]
- R. E. Wagner and R. R. Shannon, "Fabrication of aspherics using a mathematical model for material removal," Appl. Opt. 13, 1683-1689 (1974). [CrossRef] [PubMed]
- D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).
- R. A. Jones, "Grinding and polishing with small tools under computer control," Opt. Eng. 18, 390-393 (1979).
- R. A. Jones, "Computer-controlled polishing of telescope mirror segments," Opt. Eng. 22, 236-240 (1983).
- R. A. Jones, "Computer-controlled optical surfacing with orbital tool motion," Opt. Eng. 25, 785-790 (1986).
- J. R. Johnson and E. Waluschka, "Optical fabrication-process modeling-analysis tool box," Proc. SPIE 1333, 106-117 (1990). [CrossRef]
- R. A. Jones and W. J. Rupp, "Rapid optical fabrication with CCOS," Proc. SPIE 1333, 34-43 (1990). [CrossRef]
- D. W. Kim and S. W. Kim, "Static tool influence function for fabrication simulation of hexagonal mirror segments for extremely large telescopes," Opt. Express. 13, 910-917 (2005). [CrossRef] [PubMed]
- D. D. Walker, A. T. Beaucamp, D. Brooks, V. Doubrovski, M. Cassie, C. Dunn, R. Freeman, A. King, M. Libert, G. McCavana, R. Morton, D. Riley, and J. Simms, "New results from the Precessions polishing process scaled to larger sizes," Proc. SPIE 5494, 71-80 (2004). [CrossRef]
- E. Luna-Aguilar, A. Cordero-Davila, J. Gonzalez Garcia, M. Nunez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sanchez, J. Cuautle-Cortez, and M. H. Pedrayes-Lopez, "Edge effects with Preston equation," Proc. SPIE 4840, 598-603 (2003). [CrossRef]
- A. Cordero-Davila, J. Gonzalez-Garcia, M. Pedrayes-Lopez, L. A. Aguilar-Chiu, J. Cuautle-Cortes, and C. Robledo-Sanchez, "Edge effects with the Preston equation for a circular tool and workpiece," Appl. Opt. 43, 1250-1254 (2004). [CrossRef] [PubMed]
- B. C. Crawford, D. Loomis, N. Schenck, and B. Anderson, Optical Engineering and Fabrication Facility, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, (personal communication, 2008).
- D. W. Kim, College of Optical Sciences, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, W. H. Park, and J. H. Burge are preparing a manuscript to be called "Edge tool influence function model including tool stiffness and bending effects."
- D. W. Kim, College of Optical Sciences, University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, and J. H. Burge are preparing a manuscript to be called "Time scale dependent conformable tool."

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