Abstract

To pass from a spherical surface to a conic one, it is possible to use a petal tool or a small solid tool that is placed at different time intervals at several radial zones of the glass. Genetic algorithms are applied to calculate the angular sizes of the incomplete annular tools that make up the petal tools. We also present the desired wear results carried out with the petal tool that was designed on the basis of the dwell times of complete annular tools. These dwell times are calculated by using base functions that are generated with annular tools and by applying the genetic algorithms.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. Aguilar-Chiu, J. Cuautle-Cortés, and C. Robledo-Sánchez, "Edge effects with Preston equation for a circular tool and workpiece," Appl. Opt. 43, 1250-1254 (2004).
    [CrossRef] [PubMed]
  2. F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).
  3. M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).
  4. A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).
  5. R.-S. Chang and P.-Y. Lee, "Computer simulation of loose abrasive grinding aspherical optical surface by local figuring pitch," in Advanced Optical Manufacturing and Testing II, V.J.Doherty, ed., Proc. SPIE 1531,312-317 (1991).
  6. A. Cordero-Dávila, V. Cabrera-Peláez, J. Cuautle-Cortés, J. González-García, C. Robledo-Sánchez, and N. Bautista-Elivar, "Experimental results and wear predictions of petal tools that freely rotate," Appl. Opt. 44, 1434-1441 (2005).
    [CrossRef] [PubMed]
  7. R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).
  8. W. D. Dong, E. S. Putilin, and Y. V. Rudin, "Modeling the velocity and trajectory of the relative motion of a zone of a workpiece during surface lapping," J. Opt. Technol. 70, 573-575 (2003).
    [CrossRef]
  9. N. J. Brown, "Computationally directed axisymmetric aspheric figuring," Opt. Eng. 17, 602-620 (1978).
  10. Z. Michalewicz, Genetic Algorithms + Date Structures = Evolution Programs (Springer, 1992).
  11. F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
    [CrossRef]
  12. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison Wesley, 1992).
  13. 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]
  14. D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).
  15. A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).
  16. R. Aspden, R. McDonough, and F. R. Nitchie, Jr., "Computer assisted optical surfacing," Appl. Opt. 11, 2739-2747 (1972).
    [CrossRef] [PubMed]
  17. R. A. Jones, "Optimization of computer controlled polishing," Appl. Opt. 16, 218-224 (1977).
    [CrossRef] [PubMed]
  18. J. R. Johnson and E. Waluschka, "Optical fabrication-process modeling-analysis tool box," in Advanced Optical Manufacturing and Testing, G.M. Sanger, P.B. Reid, and L.R. Baker, eds., Proc. SPIE 1333,106-117 (1990).

2005 (1)

2004 (1)

2003 (3)

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

W. D. Dong, E. S. Putilin, and Y. V. Rudin, "Modeling the velocity and trajectory of the relative motion of a zone of a workpiece during surface lapping," J. Opt. Technol. 70, 573-575 (2003).
[CrossRef]

2002 (1)

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

1985 (1)

A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).

1978 (1)

N. J. Brown, "Computationally directed axisymmetric aspheric figuring," Opt. Eng. 17, 602-620 (1978).

1977 (1)

1976 (1)

D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).

1974 (1)

1972 (1)

1968 (1)

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

1927 (1)

F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).

Aguilar-Chiu, L.

Aspden, R.

Bajuk, D. J.

D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).

Bautista-Elivar, N.

Bogdanov, A. P.

A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).

Brown, N. J.

N. J. Brown, "Computationally directed axisymmetric aspheric figuring," Opt. Eng. 17, 602-620 (1978).

Cabrera-Peláez, V.

A. Cordero-Dávila, V. Cabrera-Peláez, J. Cuautle-Cortés, J. González-García, C. Robledo-Sánchez, and N. Bautista-Elivar, "Experimental results and wear predictions of petal tools that freely rotate," Appl. Opt. 44, 1434-1441 (2005).
[CrossRef] [PubMed]

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

Chang, R.-S.

R.-S. Chang and P.-Y. Lee, "Computer simulation of loose abrasive grinding aspherical optical surface by local figuring pitch," in Advanced Optical Manufacturing and Testing II, V.J.Doherty, ed., Proc. SPIE 1531,312-317 (1991).

Cordero-Dávila, A.

A. Cordero-Dávila, V. Cabrera-Peláez, J. Cuautle-Cortés, J. González-García, C. Robledo-Sánchez, and N. Bautista-Elivar, "Experimental results and wear predictions of petal tools that freely rotate," Appl. Opt. 44, 1434-1441 (2005).
[CrossRef] [PubMed]

A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. Aguilar-Chiu, J. Cuautle-Cortés, and C. Robledo-Sánchez, "Edge effects with Preston equation for a circular tool and workpiece," Appl. Opt. 43, 1250-1254 (2004).
[CrossRef] [PubMed]

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

Cortés, J. Cuautle

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

Cuautle-Cortés, J.

Cuevas, F. J.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

Dong, W. D.

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison Wesley, 1992).

Golovanova, M. N.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

González-Castillo, R.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

González-García, J.

A. Cordero-Dávila, V. Cabrera-Peláez, J. Cuautle-Cortés, J. González-García, C. Robledo-Sánchez, and N. Bautista-Elivar, "Experimental results and wear predictions of petal tools that freely rotate," Appl. Opt. 44, 1434-1441 (2005).
[CrossRef] [PubMed]

A. Cordero-Dávila, J. González-García, M. Pedrayes-López, L. Aguilar-Chiu, J. Cuautle-Cortés, and C. Robledo-Sánchez, "Edge effects with Preston equation for a circular tool and workpiece," Appl. Opt. 43, 1250-1254 (2004).
[CrossRef] [PubMed]

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

Johnson, J. R.

J. R. Johnson and E. Waluschka, "Optical fabrication-process modeling-analysis tool box," in Advanced Optical Manufacturing and Testing, G.M. Sanger, P.B. Reid, and L.R. Baker, eds., Proc. SPIE 1333,106-117 (1990).

Jones, R. A.

Kachkin, S. S.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

Krylova, Ye. I.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

Lee, P.-Y.

R.-S. Chang and P.-Y. Lee, "Computer simulation of loose abrasive grinding aspherical optical surface by local figuring pitch," in Advanced Optical Manufacturing and Testing II, V.J.Doherty, ed., Proc. SPIE 1531,312-317 (1991).

McDonough, R.

Michalewicz, Z.

Z. Michalewicz, Genetic Algorithms + Date Structures = Evolution Programs (Springer, 1992).

Nitchie, F. R.

Parra-Flores, A.

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

Pedrayes-López, M.

Preston, F. W.

F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).

Putilin, E. S.

Robledo-Sánchez, C.

Rudin, Y. V.

Sánchez, C. Robledo

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

Servin, M.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

Shannon, R. R.

Shevel'kova, L. I.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

Sossa-Azuela, J. H.

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

Tsesnek, L. S.

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

Venegas-Pérez, L.

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

Wagner, R. E.

Waluschka, E.

J. R. Johnson and E. Waluschka, "Optical fabrication-process modeling-analysis tool box," in Advanced Optical Manufacturing and Testing, G.M. Sanger, P.B. Reid, and L.R. Baker, eds., Proc. SPIE 1333,106-117 (1990).

Appl. Opt. (5)

J. Opt. Technol. (1)

J. Soc. Glass Technol. (1)

F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).

Opt. Commun. (1)

F. J. Cuevas, J. H. Sossa-Azuela, and M. Servin, "A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm," Opt. Commun. 203, 213-223 (2002).
[CrossRef]

Opt. Eng. (2)

D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).

N. J. Brown, "Computationally directed axisymmetric aspheric figuring," Opt. Eng. 17, 602-620 (1978).

Opt.-Mekh. Prom-st. (1)

A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).

Sov. J. Opt. Technol. (1)

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

Other (6)

A. Parra-Flores, A. Cordero-Dávila, J. Cuautle Cortés, C. Robledo Sánchez, J. González-García, and V. Cabrera-Peláez, "Simulación de desgastes en el pulido de superficies con la ecuación de Preston," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,138 (2003).

R.-S. Chang and P.-Y. Lee, "Computer simulation of loose abrasive grinding aspherical optical surface by local figuring pitch," in Advanced Optical Manufacturing and Testing II, V.J.Doherty, ed., Proc. SPIE 1531,312-317 (1991).

Z. Michalewicz, Genetic Algorithms + Date Structures = Evolution Programs (Springer, 1992).

R. González-Castillo, L. Venegas-Pérez, J. González-García, A. Parra-Flores, and A. Cordero-Dávila, "Análisis cinemático de una máquina pulidora comercial para superficies ópticas," inProgram of the 46th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 49,2 (2003).

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison Wesley, 1992).

J. R. Johnson and E. Waluschka, "Optical fabrication-process modeling-analysis tool box," in Advanced Optical Manufacturing and Testing, G.M. Sanger, P.B. Reid, and L.R. Baker, eds., Proc. SPIE 1333,106-117 (1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (25)

Fig. 1
Fig. 1

Functioning of a commercial polishing machine. Glass and tool rotate at angular velocities of ω G and ω T . The tool oscillates in a simple harmonic motion at an angular velocity of ω T O .

Fig. 2
Fig. 2

Solid tool.

Fig. 3
Fig. 3

Plot of the wear obtained with a solid tool that is oscillating around the rotation axis of the glass and has the same angular velocity as that of the glass.

Fig. 4
Fig. 4

Annular tool.

Fig. 5
Fig. 5

Plot of the wear obtained with an annular tool oscillating around the rotation axis of the glass and has the same angular velocity as that of the glass.

Fig. 6
Fig. 6

Tool made up of three incomplete rings.

Fig. 7
Fig. 7

Wear produced by a tool made up of three incomplete rings.

Fig. 8
Fig. 8

Petal tool made up of ten incomplete rings.

Fig. 9
Fig. 9

Wear produced by a petal tool made up of ten incomplete rings.

Fig. 10
Fig. 10

Construction of a concave aspherical surface beginning with an initial sphere.

Fig. 11
Fig. 11

Plot of the difference between the concave aspherical surface and the closest sphere.

Fig. 12
Fig. 12

Construction of a convex aspherical surface beginning with an initial sphere.

Fig. 13
Fig. 13

Plot of the difference between the convex aspherical surface and the closest sphere.

Fig. 14
Fig. 14

Plot of the difference between the convex aspherical surface and a sphere whose curvature radius is smaller than that of the closest sphere.

Fig. 15
Fig. 15

Design of the petal tools using T&E. (a) Tool used to generate the concave wear in (b). (c) Tool used to generate the convex wear in (d).

Fig. 16
Fig. 16

Design of petal tools using GAs. (a) Tool used to generate the concave wear in (b). (c) Tool used to generate the convex wear in (d).

Fig. 17
Fig. 17

Plot of S 2 as a function of the number of generations used in the genetic algorithm in the case of a concave surface.

Fig. 18
Fig. 18

Plot of S 2 as a function of the generations used in the genetic algorithm in the case of a convex surface.

Fig. 19
Fig. 19

Base function generated with annular tool number 50.

Fig. 20
Fig. 20

Plots of the 101 generated base functions.

Fig. 21
Fig. 21

Wear generated with dwell times calculated by (a) T&E and (b) GA, as well as their time distributions with (c) T&E and (d) GA.

Fig. 22
Fig. 22

Plot of S 2 as a function of the number of generations used in the GA to calculate a case of time intervals.

Fig. 23
Fig. 23

Angular values of the 101 incomplete annular tools in the T&E case. Each angular size is plotted on the glass in the center position of each ring width.

Fig. 24
Fig. 24

Angular values of the 101 incomplete annular tools for the GA case. Each angular size is plotted on the glass in the center position of each ring width.

Fig. 25
Fig. 25

Petal tools generated by converting times to angular sizes for (a) T&E and (b) GA; wear generated with these tools for (c) T&E and (d) GA.

Tables (4)

Tables Icon

Table 1 Parameters Used to Calculate Simulated Wear

Tables Icon

Table 2 m i Calculated by Trial and Error (T&E), and by Using Genetic Algorithms (GA)

Tables Icon

Table 3 Values of the α i Angular Sizes of the First Incomplete Ring in Each One of the Ten Sets of Ten Rings and the Corresponding Squares of the Errors and the rms

Tables Icon

Table 4 Δt i Calculated by T&E, and by Using GA

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

S 2 = j = 1 j = N P [ h P ( x j , y j , α 1 , α 2 ,   , α N A ) h D ( x j , y j ) ] 2 ,
0 α i π 2 .
α i + 1 = α i + Δ α i ,
Δ α i = [ m 1 if 1 i 10 m 2 if  11 i 20 m 10 if  91 i 100 ] .
S 2 = j = 1 j = N P [ h P ( x j , y j , m 1 , m 2 ,   , m 10 ) h D ( x j , y j ) ] 2 ,
f M = S 2 + C ,
h D = i = 1 M Δ τ i h b i .
α i = ( Δ t i T ) ( 90 ° ) ,
Δ τ i = [ m 1 = τ 11 τ 01 10 m 2 = τ 21 τ 11 10 m 10 = τ 101 τ 91 10 ] .

Metrics