Abstract

Polishing can be more uniform if the polishing path provides uniform coverage of the surface. It is known that Peano paths can provide uniform coverage of planar surfaces. Peano paths also contain short path segments and turns: (1) all path segments have the same length, (2) path segments are mutually orthogonal at the turns, and (3) path segments and turns are uniformity distributed over the domain surface. These make Peano paths an attractive candidate among polishing tool paths because they enhance multidirectional approaches of the tool to each surface location. A method for constructing Peano paths for uniform coverage of aspherical surfaces is proposed in this paper. When mapped to the aspherical surface, the path also contains short path segments and turns, and the above attributes are approximately preserved. Attention is paid so that the path segments are still well distributed near the vertex of the surface. The proposed tool path was used in the polishing of a number of parabolic BK7 specimens using magnetorheological finishing (MRF) and pitch with cerium oxide. The results were rather good for optical lenses and confirm that a Peano-like path was useful for polishing, for MRF, and for pitch polishing. In the latter case, the surface roughness achieved was 0.91 nm according to WYKO measurement.

© 2013 Optical Society of America

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References

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  1. R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra34–39 (1963).
  2. H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
    [CrossRef]

2010 (2)

H. Hu, Y. F. Dai, and X. Q. Peng, “Restraint of tool path ripple based on surface error distribution and process parameters in deterministic finishing,” Opt. Express 18, 22973–22981 (2010).
[CrossRef]

H. Y. Tam and H. B. Cheng, “An investigation of the effects of the tool path on the removal of material in polishing,” J. Mater. Process. Technol. 210, 807–818 (2010).
[CrossRef]

2009 (3)

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

W. J. Deng, X. J. Zhang, X. K. Wang, and X. Wang, “Novel method for optimizing polishing tool-path in CCOS based on weighted-iterative algorithm,” Proc. SPIE 7282, 728214 (2009).
[CrossRef]

X. Pessoles and C. Tournier, “Automatic polishing process of plastic injection molds on a 5-axis milling center,” J. Mater. Process. Technol. 209, 3665–3673 (2009).
[CrossRef]

2008 (1)

2006 (1)

M. J. Tsai and J. F. Huang, “Efficient automatic polishing process with a new compliant abrasive tool,” Int. J. Adv. Manuf. Technol. 30, 817–827 (2006).
[CrossRef]

2005 (1)

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

2004 (1)

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5l80, 107–114 (2004).
[CrossRef]

2003 (1)

2000 (1)

1999 (1)

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

1992 (1)

Y. Mizugaki, M. Sakamoto, and T. Sata, “Fractal path generation for a metal-mold polishing robot system and its evaluation by the operability,” CIRP Ann. 41, 531–534 (1992).
[CrossRef]

Bifano, T. G.

Brooks, D.

Cheng, H. B.

H. Y. Tam and H. B. Cheng, “An investigation of the effects of the tool path on the removal of material in polishing,” J. Mater. Process. Technol. 210, 807–818 (2010).
[CrossRef]

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Cheng, K.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Cho, U.

U. Cho, D. G. Eom, D. Y. Lee, and J. O. Park, “A flexible polishing robot system for die and mould,” in Proceedings of the 23rd International Symposium on Industrial Robots (1992), pp. 449–456.

Dai, Y. F.

Deng, W. J.

W. J. Deng, X. J. Zhang, X. K. Wang, and X. Wang, “Novel method for optimizing polishing tool-path in CCOS based on weighted-iterative algorithm,” Proc. SPIE 7282, 728214 (2009).
[CrossRef]

Dunn, C. R.

Eom, D. G.

U. Cho, D. G. Eom, D. Y. Lee, and J. O. Park, “A flexible polishing robot system for die and mould,” in Proceedings of the 23rd International Symposium on Industrial Robots (1992), pp. 449–456.

Feinberg, M. R.

Feng, Z. J.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Freeman, R.

Golini, D.

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

Horenstein, M. N.

Hu, H.

Huang, J. F.

M. J. Tsai and J. F. Huang, “Efficient automatic polishing process with a new compliant abrasive tool,” Int. J. Adv. Manuf. Technol. 30, 817–827 (2006).
[CrossRef]

Jones, R. A.

R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra34–39 (1963).

Kim, S. W.

King, A.

Kordonski, W.

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5l80, 107–114 (2004).
[CrossRef]

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

Lee, D. Y.

U. Cho, D. G. Eom, D. Y. Lee, and J. O. Park, “A flexible polishing robot system for die and mould,” in Proceedings of the 23rd International Symposium on Industrial Robots (1992), pp. 449–456.

McCavana, G.

Mizugaki, Y.

Y. Mizugaki, M. Sakamoto, and T. Sata, “Fractal path generation for a metal-mold polishing robot system and its evaluation by the operability,” CIRP Ann. 41, 531–534 (1992).
[CrossRef]

Morton, R.

Park, J. O.

U. Cho, D. G. Eom, D. Y. Lee, and J. O. Park, “A flexible polishing robot system for die and mould,” in Proceedings of the 23rd International Symposium on Industrial Robots (1992), pp. 449–456.

Peng, X. Q.

Pessoles, X.

X. Pessoles and C. Tournier, “Automatic polishing process of plastic injection molds on a 5-axis milling center,” J. Mater. Process. Technol. 209, 3665–3673 (2009).
[CrossRef]

Sakamoto, M.

Y. Mizugaki, M. Sakamoto, and T. Sata, “Fractal path generation for a metal-mold polishing robot system and its evaluation by the operability,” CIRP Ann. 41, 531–534 (1992).
[CrossRef]

Sandri, G.

Sata, T.

Y. Mizugaki, M. Sakamoto, and T. Sata, “Fractal path generation for a metal-mold polishing robot system and its evaluation by the operability,” CIRP Ann. 41, 531–534 (1992).
[CrossRef]

Sekeres, A.

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5l80, 107–114 (2004).
[CrossRef]

Shanbhag, P. M.

Shorey, A.

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5l80, 107–114 (2004).
[CrossRef]

Tam, H. Y.

H. Y. Tam and H. B. Cheng, “An investigation of the effects of the tool path on the removal of material in polishing,” J. Mater. Process. Technol. 210, 807–818 (2010).
[CrossRef]

Tournier, C.

X. Pessoles and C. Tournier, “Automatic polishing process of plastic injection molds on a 5-axis milling center,” J. Mater. Process. Technol. 209, 3665–3673 (2009).
[CrossRef]

Tsai, M. J.

M. J. Tsai and J. F. Huang, “Efficient automatic polishing process with a new compliant abrasive tool,” Int. J. Adv. Manuf. Technol. 30, 817–827 (2006).
[CrossRef]

Walker, D. D.

Wang, X.

W. J. Deng, X. J. Zhang, X. K. Wang, and X. Wang, “Novel method for optimizing polishing tool-path in CCOS based on weighted-iterative algorithm,” Proc. SPIE 7282, 728214 (2009).
[CrossRef]

Wang, X. K.

W. J. Deng, X. J. Zhang, X. K. Wang, and X. Wang, “Novel method for optimizing polishing tool-path in CCOS based on weighted-iterative algorithm,” Proc. SPIE 7282, 728214 (2009).
[CrossRef]

Wang, Y. T.

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

Wang, Y. W.

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Yam, Y.

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

Zhang, X. J.

W. J. Deng, X. J. Zhang, X. K. Wang, and X. Wang, “Novel method for optimizing polishing tool-path in CCOS based on weighted-iterative algorithm,” Proc. SPIE 7282, 728214 (2009).
[CrossRef]

Appl. Opt. (1)

CIRP Ann. (1)

Y. Mizugaki, M. Sakamoto, and T. Sata, “Fractal path generation for a metal-mold polishing robot system and its evaluation by the operability,” CIRP Ann. 41, 531–534 (1992).
[CrossRef]

Int. J. Adv. Manuf. Technol. (1)

M. J. Tsai and J. F. Huang, “Efficient automatic polishing process with a new compliant abrasive tool,” Int. J. Adv. Manuf. Technol. 30, 817–827 (2006).
[CrossRef]

Int. J. Mach. Tools Manuf. (1)

H. B. Cheng, Z. J. Feng, K. Cheng, and Y. W. Wang, “Design of a six-axis high precision machine tool and its application in machining aspherical optical mirrors,” Int. J. Mach. Tools Manuf. 45, 1085–1094 (2005).
[CrossRef]

Int. J. Mod. Phys. B (1)

W. Kordonski and D. Golini, “Progress update in magnetorheological finishing,” Int. J. Mod. Phys. B 13, 2205–2212 (1999).
[CrossRef]

J. Mater. Process. Technol. (3)

X. Pessoles and C. Tournier, “Automatic polishing process of plastic injection molds on a 5-axis milling center,” J. Mater. Process. Technol. 209, 3665–3673 (2009).
[CrossRef]

H. Y. Tam and H. B. Cheng, “An investigation of the effects of the tool path on the removal of material in polishing,” J. Mater. Process. Technol. 210, 807–818 (2010).
[CrossRef]

H. B. Cheng, Y. Yam, and Y. T. Wang, “Experimentation on MR fluid using a 2-axis wheel tool,” J. Mater. Process. Technol. 209, 5254–5261 (2009).
[CrossRef]

Opt. Express (3)

Proc. SPIE (2)

W. Kordonski, A. Shorey, and A. Sekeres, “New magnetically assisted finishing method: material removal with magnetorheological fluid jet,” Proc. SPIE 5l80, 107–114 (2004).
[CrossRef]

W. J. Deng, X. J. Zhang, X. K. Wang, and X. Wang, “Novel method for optimizing polishing tool-path in CCOS based on weighted-iterative algorithm,” Proc. SPIE 7282, 728214 (2009).
[CrossRef]

Other (2)

R. A. Jones, “Computer control for grinding and polishing,” Photon. Spectra34–39 (1963).

U. Cho, D. G. Eom, D. Y. Lee, and J. O. Park, “A flexible polishing robot system for die and mould,” in Proceedings of the 23rd International Symposium on Industrial Robots (1992), pp. 449–456.

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Figures (27)

Fig. 1.
Fig. 1.

Peano path, filleted: (a) first iteration and (b) second iteration.

Fig. 2.
Fig. 2.

Peano-like path.

Fig. 3.
Fig. 3.

Path and coverage on a square region.

Fig. 4.
Fig. 4.

Path on the uw domain: (a) partitioning of the domain and (b) path segments within a partition.

Fig. 5.
Fig. 5.

Path on a pair of u-strips: (a) ni old and (b) ni even.

Fig. 6.
Fig. 6.

Termination of u-direction increment.

Fig. 7.
Fig. 7.

Peano-like path on an off-axis spherical segment: (a) uw domain and (b) Cartesian domain.

Fig. 8.
Fig. 8.

Scan path on an off-axis spherical segment.

Fig. 9.
Fig. 9.

Peano-like paths on a spherical surface with full rotational symmetry: (a) without return path, (bi) with return path, Cartesian domain, and (bii) with return path, uw domain.

Fig. 10.
Fig. 10.

Including a return path from (u2I,wI0) back to (u0,w00).

Fig. 11.
Fig. 11.

Peano-like path on a parabolic surface.

Fig. 12.
Fig. 12.

Peano-like path on a parabolic surface with one edge, r<50mm.

Fig. 13.
Fig. 13.

Circular region covered by square elements.

Fig. 14.
Fig. 14.

Geometric entities with a circular region.

Fig. 15.
Fig. 15.

Coverage rate Ci increases with ni.

Fig. 16.
Fig. 16.

Vertex cap of an aspherical surface.

Fig. 17.
Fig. 17.

Directed path segments inside a circular region.

Fig. 18.
Fig. 18.

Peano-like path on the parabolic surface near the vertex: (a) without and (b) with the vertex cap/square elements.

Fig. 19.
Fig. 19.

Peano-like path on parabolic surface near the vertex cap, 3D view.

Fig. 20.
Fig. 20.

Residual error map of circular specimen before (upper) and after (lower) MRF polishing based on a Peano-like path.

Fig. 21.
Fig. 21.

Removal function used, with peak removal rate 0.38λ/min.

Fig. 22.
Fig. 22.

Polishing path: (a) a Peano-like path and (b) the raster path.

Fig. 23.
Fig. 23.

In-section profiles of the error map before (left) and after (right) polishing.

Fig. 24.
Fig. 24.

PSD results before (upper red curve) and after (lower green curve) polishing.

Fig. 25.
Fig. 25.

Residual error map of rectangular specimen (a) before and (b) after MRF polishing based on scanning path.

Fig. 26.
Fig. 26.

PSD results before (upper red curve) and after (lower green curve) polishing for the second experiment.

Fig. 27.
Fig. 27.

Roughness measurement of a circular specimen after pitch polishing based on a Peano-like path.

Tables (2)

Tables Icon

Table 1. Coverage Rate and Entities of the Circular Region with Optimized nis

Tables Icon

Table 2. Radius of the Central Region of the Vertex Cap for Various N

Equations (14)

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

S=[xyz]=[rcosθrsinθz(r)],
Su=[cosθsinθzr]ruandSw=[sinθcosθ0]rθw,
|Su|=|ru|1+zr2and|Sw|=r|dc|.
Su·Sw=[cosθsinθzr]ru·[sinθcosθ0]rθw=0,
C=lengtharea=2pp2=2/p.
P=time×task ratearea=lengthfeed rate×task ratearea=C×task ratefeed rate.
Δui=p/|Su|andΔwi=p/|Sw|,
{(u2i2,wi0),(u2i1,wi1),(u2i2,wi2),,(u2i1,wiJi),(u2i,wi(Ji1)),,(u2i,wi0)}
{(u0,w00),,(u1,w0J0),,(u2,w00),,(u2I2,wI0),,(u2I1,wIJI),,(u2I,wI0)}
{(u2I,wI0),(u2I1,wI1),(u2I2,wI0),,(u2,w00),(u1,w01),(u0,w00)}
wij=Ji1+Jiwijfori=0,,Iandj=1,,Ji.
ri=sinθisin(45°θi)·li,
l1=r,li=li1+2ri1fori>1
(li+2ri,(ni+1j)×360°ni),(li2+2liri+ri2,(ni+1/2j)×360°ni),(li,(ni+1j)×360°ni),and(li2+2liri+ri2,(ni+3/2j)×360°ni),

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