Abstract

In this paper, a novel pseudo-random diamond turning (PRDT) method is proposed for the fabrication of freeform optics with scattering homogenization by means of actively eliminating the inherent periodicity of the residual tool marks. The strategy for accurately determining the spiral toolpath with pseudo-random vibration modulation is deliberately explained. Spatial geometric calculation method is employed to determine the toolpath in consideration of cutting tool geometries, and an iteration algorithm is further introduced to enhance the computation efficiency. Moreover, a novel two degree of freedom fast tool servo (2-DOF FTS) system with decoupled motions is developed to implement the PRDT method. Taking advantage of a novel surface topography generation algorithm, theoretical surfaces generated by using the calculated toolpaths are obtained, the accuracy of the toolpath generation and the efficiency of the PRDT method for breaking up the inherent periodicity of tool marks are examined. A series of preliminary cutting experiments are carried out to verify the efficiency of the proposed PRDT method, the experimental results obtained are in good agreement with the results obtained by numerical simulation. In addition, the results of scattering experiments indicate that the proposed PRDT method will be a very promising technique to achieve the scattering homogenization of machined surfaces with complicated shapes.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  6. Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. D. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng.21(9), 095011 (2011).
    [CrossRef]
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  23. F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express16(10), 7323–7329 (2008).
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  24. D. Yu, Y. Wong, and G. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol.57(1–4), 85–99 (2011).
    [CrossRef]
  25. H. Gong, F. Fang, and X. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non-zero rake angle using symbolic computation,” Int. J. Adv. Manuf. Technol.58(9–12), 841–847 (2012).
    [CrossRef]

2013 (2)

A. Beaucamp and Y. Namba, “Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing,” CIRP Ann.62(1), 315–318 (2013).
[CrossRef]

E. Brinksmeier and W. Preuss, “How to diamond turn an elliptic half-shell?” Precis. Eng.37(4), 944–947 (2013).
[CrossRef]

2012 (4)

J. Xu, F. Wang, Q. Shi, and Y. Deng, “Statistical measurement of mid-spatial frequency defects of large optics,” Meas. Sci. Technol.23(6), 065201 (2012).
[CrossRef]

H. Gong, F. Fang, and X. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non-zero rake angle using symbolic computation,” Int. J. Adv. Manuf. Technol.58(9–12), 841–847 (2012).
[CrossRef]

E. Brinksmeier and W. Preuss, “Micro-machining,” Philos. Trans. A Math. Phys. Eng. Sci.370(1973), 3973–3992 (2012).
[CrossRef] [PubMed]

Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
[CrossRef]

2011 (5)

2010 (3)

L. Li, S. A. Collins, and A. Y. Yi, “Optical effects of surface finish by ultraprecision single point diamond machining,” J. Manuf. Sci. Eng.132(2), 021002 (2010).
[CrossRef]

J. M. Tamkin and T. D. Milster, “Effects of structured mid-spatial frequency surface errors on image performance,” Appl. Opt.49(33), 6522–6536 (2010).
[CrossRef] [PubMed]

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

2009 (1)

S. Rakuff and J. F. Cuttino, “Design and testing of a long-range, precision fast tool servo system for diamond turning,” Precis. Eng.33(1), 18–25 (2009).
[CrossRef]

2008 (2)

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE7018, 701835 (2008).
[CrossRef]

F. Z. Fang, X. D. Zhang, and X. T. Hu, “Cylindrical coordinate machining of optical freeform surfaces,” Opt. Express16(10), 7323–7329 (2008).
[CrossRef] [PubMed]

2006 (1)

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

2005 (2)

X.-D. Lu and D. L. Trumper, “Ultrafast tool servos for diamond turning,” CIRP Ann.54(1), 383–388 (2005).
[CrossRef]

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing (MRF),” Proc. SPIE5786, 296–304 (2005).
[CrossRef]

2003 (1)

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng.27(3), 289–298 (2003).
[CrossRef]

1999 (1)

S. J. Ludwick, D. A. Chargin, J. A. Calzaretta, and D. L. Trumper, “Design of a rotary fast tool servo for ophthalmic lens fabrication,” Precis. Eng.23(4), 253–259 (1999).
[CrossRef]

1995 (1)

A. Kotha and J. E. Harvey, “Scattering effects of machined optical surfaces,” Proc. SPIE2541, 54–65 (1995).
[CrossRef]

Araki, T.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng.27(3), 289–298 (2003).
[CrossRef]

Beaucamp, A.

A. Beaucamp and Y. Namba, “Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing,” CIRP Ann.62(1), 315–318 (2013).
[CrossRef]

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE7018, 701835 (2008).
[CrossRef]

Brecher, C.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Brinksmeier, E.

E. Brinksmeier and W. Preuss, “How to diamond turn an elliptic half-shell?” Precis. Eng.37(4), 944–947 (2013).
[CrossRef]

E. Brinksmeier and W. Preuss, “Micro-machining,” Philos. Trans. A Math. Phys. Eng. Sci.370(1973), 3973–3992 (2012).
[CrossRef] [PubMed]

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Calzaretta, J. A.

S. J. Ludwick, D. A. Chargin, J. A. Calzaretta, and D. L. Trumper, “Design of a rotary fast tool servo for ophthalmic lens fabrication,” Precis. Eng.23(4), 253–259 (1999).
[CrossRef]

Chargin, D. A.

S. J. Ludwick, D. A. Chargin, J. A. Calzaretta, and D. L. Trumper, “Design of a rotary fast tool servo for ophthalmic lens fabrication,” Precis. Eng.23(4), 253–259 (1999).
[CrossRef]

Cheung, C. F.

Choi, N.

Collins, S. A.

L. Li, S. A. Collins, and A. Y. Yi, “Optical effects of surface finish by ultraprecision single point diamond machining,” J. Manuf. Sci. Eng.132(2), 021002 (2010).
[CrossRef]

Cuttino, J. F.

S. Rakuff and J. F. Cuttino, “Design and testing of a long-range, precision fast tool servo system for diamond turning,” Precis. Eng.33(1), 18–25 (2009).
[CrossRef]

Dankwart, C.

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Deng, Y.

J. Xu, F. Wang, Q. Shi, and Y. Deng, “Statistical measurement of mid-spatial frequency defects of large optics,” Meas. Sci. Technol.23(6), 065201 (2012).
[CrossRef]

Dumas, P.

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing (MRF),” Proc. SPIE5786, 296–304 (2005).
[CrossRef]

Eberhardt, R.

Fang, F.

H. Gong, F. Fang, and X. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non-zero rake angle using symbolic computation,” Int. J. Adv. Manuf. Technol.58(9–12), 841–847 (2012).
[CrossRef]

Y. Nie, F. Fang, and X. Zhang, “System design of Maxwell force driving fast tool servos based on model analysis,” Int. J. Adv. Manuf. Technol., doi: (2013).
[CrossRef]

Fang, F. Z.

Freeman, R.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE7018, 701835 (2008).
[CrossRef]

Gao, W.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng.27(3), 289–298 (2003).
[CrossRef]

Gebhardt, A.

Gläbe, R.

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Golini, D.

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing (MRF),” Proc. SPIE5786, 296–304 (2005).
[CrossRef]

Gong, H.

H. Gong, F. Fang, and X. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non-zero rake angle using symbolic computation,” Int. J. Adv. Manuf. Technol.58(9–12), 841–847 (2012).
[CrossRef]

Harvey, J. E.

Ho, L. T.

Hong, G.

D. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng.21(9), 095011 (2011).
[CrossRef]

D. Yu, Y. Wong, and G. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol.57(1–4), 85–99 (2011).
[CrossRef]

Hu, X.

H. Gong, F. Fang, and X. Hu, “Accurate spiral tool path generation of ultraprecision three-axis turning for non-zero rake angle using symbolic computation,” Int. J. Adv. Manuf. Technol.58(9–12), 841–847 (2012).
[CrossRef]

Hu, X. T.

Kiyono, S.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng.27(3), 289–298 (2003).
[CrossRef]

Kopylow, C.

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Kotha, A.

A. Kotha and J. E. Harvey, “Scattering effects of machined optical surfaces,” Proc. SPIE2541, 54–65 (1995).
[CrossRef]

Krywonos, A.

Lange, S.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Li, L.

Li, S. Y.

Li, Z. Z.

Lin, J.

Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
[CrossRef]

Liu, Q.

Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
[CrossRef]

Lu, X.-D.

X.-D. Lu and D. L. Trumper, “Ultrafast tool servos for diamond turning,” CIRP Ann.54(1), 383–388 (2005).
[CrossRef]

Ludwick, S. J.

S. J. Ludwick, D. A. Chargin, J. A. Calzaretta, and D. L. Trumper, “Design of a rotary fast tool servo for ophthalmic lens fabrication,” Precis. Eng.23(4), 253–259 (1999).
[CrossRef]

Lünemann, B.

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Meier, A.

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Merz, M.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Milster, T. D.

Morton, R.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE7018, 701835 (2008).
[CrossRef]

Namba, Y.

A. Beaucamp and Y. Namba, “Super-smooth finishing of diamond turned hard X-ray molding dies by combined fluid jet and bonnet polishing,” CIRP Ann.62(1), 315–318 (2013).
[CrossRef]

Nie, Y.

Y. Nie, F. Fang, and X. Zhang, “System design of Maxwell force driving fast tool servos based on model analysis,” Int. J. Adv. Manuf. Technol., doi: (2013).
[CrossRef]

Niehaus, F.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Okazaki, Y.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng.27(3), 289–298 (2003).
[CrossRef]

Peng, X. Q.

Ponudurai, K.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE7018, 701835 (2008).
[CrossRef]

Preuss, W.

E. Brinksmeier and W. Preuss, “How to diamond turn an elliptic half-shell?” Precis. Eng.37(4), 944–947 (2013).
[CrossRef]

E. Brinksmeier and W. Preuss, “Micro-machining,” Philos. Trans. A Math. Phys. Eng. Sci.370(1973), 3973–3992 (2012).
[CrossRef] [PubMed]

Rakuff, S.

S. Rakuff and J. F. Cuttino, “Design and testing of a long-range, precision fast tool servo system for diamond turning,” Precis. Eng.33(1), 18–25 (2009).
[CrossRef]

Riemer, O.

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

Risse, S.

Scheiding, S.

Shi, Q.

J. Xu, F. Wang, Q. Shi, and Y. Deng, “Statistical measurement of mid-spatial frequency defects of large optics,” Meas. Sci. Technol.23(6), 065201 (2012).
[CrossRef]

Tamkin, J. M.

Tricard, M.

P. Dumas, D. Golini, and M. Tricard, “Improvement of figure and finish of diamond turned surfaces with magneto-rheological finishing (MRF),” Proc. SPIE5786, 296–304 (2005).
[CrossRef]

Trumper, D. L.

X.-D. Lu and D. L. Trumper, “Ultrafast tool servos for diamond turning,” CIRP Ann.54(1), 383–388 (2005).
[CrossRef]

S. J. Ludwick, D. A. Chargin, J. A. Calzaretta, and D. L. Trumper, “Design of a rotary fast tool servo for ophthalmic lens fabrication,” Precis. Eng.23(4), 253–259 (1999).
[CrossRef]

Tünnermann, A.

Walker, D.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE7018, 701835 (2008).
[CrossRef]

Wang, F.

J. Xu, F. Wang, Q. Shi, and Y. Deng, “Statistical measurement of mid-spatial frequency defects of large optics,” Meas. Sci. Technol.23(6), 065201 (2012).
[CrossRef]

Wang, J. M.

Weck, M.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Wenzel, C.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Winterschladen, M.

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

Wong, Y.

D. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng.21(9), 095011 (2011).
[CrossRef]

D. Yu, Y. Wong, and G. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol.57(1–4), 85–99 (2011).
[CrossRef]

Xu, J.

J. Xu, F. Wang, Q. Shi, and Y. Deng, “Statistical measurement of mid-spatial frequency defects of large optics,” Meas. Sci. Technol.23(6), 065201 (2012).
[CrossRef]

Yamanaka, M.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng.27(3), 289–298 (2003).
[CrossRef]

Yi, A. Y.

Yin, Z. Q.

Yu, D.

D. Yu, Y. Wong, and G. Hong, “Ultraprecision machining of micro-structured functional surfaces on brittle materials,” J. Micromech. Microeng.21(9), 095011 (2011).
[CrossRef]

D. Yu, Y. Wong, and G. Hong, “Optimal selection of machining parameters for fast tool servo diamond turning,” Int. J. Adv. Manuf. Technol.57(1–4), 85–99 (2011).
[CrossRef]

Zhang, X.

Y. Nie, F. Fang, and X. Zhang, “System design of Maxwell force driving fast tool servos based on model analysis,” Int. J. Adv. Manuf. Technol., doi: (2013).
[CrossRef]

Zhang, X. D.

Zhao, S.

Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
[CrossRef]

Zhou, X.

Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
[CrossRef]

Zhu, Z.

Z. Zhu, X. Zhou, Q. Liu, J. Lin, and S. Zhao, “Fabrication of micro-structured surfaces on bulk metallic glasses based on fast tool servo assisted diamond turning,” Sci. Adv. Mater.4(9), 906–911 (2012).
[CrossRef]

Appl. Opt. (2)

CIRP Ann. (4)

C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann.55(1), 547–550 (2006).
[CrossRef]

E. Brinksmeier, O. Riemer, R. Gläbe, B. Lünemann, C. Kopylow, C. Dankwart, and A. Meier, “Submicron functional surfaces generated by diamond machining,” CIRP Ann.59(1), 535–538 (2010).
[CrossRef]

X.-D. Lu and D. L. Trumper, “Ultrafast tool servos for diamond turning,” CIRP Ann.54(1), 383–388 (2005).
[CrossRef]

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Int. J. Adv. Manuf. Technol. (2)

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Figures (14)

Fig. 1
Fig. 1

Working principle of the PRDT method (a) Axis configuration of turning machine; (b) Schematic of the motions of the cutting tool.

Fig. 2
Fig. 2

Schematic of toolpath generation.

Fig. 3
Fig. 3

Schematic of the material removal.

Fig. 4
Fig. 4

Toolpaths of sinusoidal grid surfaces for (a) the CT method, and (b) the PRDT method.

Fig. 5
Fig. 5

Characteristics of machined surfaces (a) Surface topography generated by the CT method; (b) Surface topography generated by the PRDT method; (c) Error map generated by the CT method; (d) Error map generated by the PRDT method; (e) Features of the extracted 2-D profile of the surface generated by the CT method; (f) Features of the extracted 2-D profile of the surface generated by the PRDT method.

Fig. 6
Fig. 6

Schematic of the designed 2-DOF FTS mechanism. 1. The base; 2. The right circle flexure hinge; 3. Piezoelectric actuator; 4. The cutting tool; 5. The Z-shaped flexure hinge.

Fig. 7
Fig. 7

Schematic of the ZFGU mechanism.

Fig. 8
Fig. 8

Configuration of the hardware of the PRDT system (a) The machining system; (b) Front view of the 2-DOF FTS; (c) Back view of the 2-DOF FTS. (1. The 2-DOF FTS mechanism; 2. The workpiece; 3. Spindle of the lathe; 4. Machine bed; 5. Slide carriage of the lathe; 6. The cover; 7. The PEA; 8. Height adjustment screws; 9. Flexure mechanism for height adjustment; 10. Preloading screws of the PEAs; 11. The diamond cutting tool; 12. Probes of capacity transducers; 13. Connection part for displacement measurement; 14. The base; 15. Fastening screws).

Fig. 9
Fig. 9

Principle of scattering tests.

Fig. 10
Fig. 10

Motions of the cutting tool. The upper and lower red lines denote the desired trajectories along the z and the x-axis, respectively. The black line and the blue line denote the practical motions of the cutting tool along the z and the x-axis, respectively.

Fig. 11
Fig. 11

Observed scattering effects of the flat surface.

Fig. 12
Fig. 12

Micro-topographies on machined surfaces generated by (a) The CT method, and (b) The PRDT method.

Fig. 13
Fig. 13

The extracted 2D profiles and the corresponding PSD features of surfaces generated by (a) The CT method, and (b) The PRDT method.

Fig. 14
Fig. 14

Scattering effects of machined surface.

Equations (12)

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V W = ( f W x , f W y ,1) | x= x s ,y= y s
{ x T = R T cosθ y T = R T sinθsinγ z T = R T sinθcosγ , θ[ θ min , θ max ]
{ T i = ( x T θ , y T θ , z T θ ) | θ= θ i ,γ= γ 0 θ i = θ min + (i1)( θ max θ min ) N 0
{ φ k,l =2π(k+l/ N s ) ρ k,l = ρ max φ k,l f 0 / 2π + δ k,l
δ k,l =K f 0 (0.5 R k,l ' ), R k,l ' [0,1]
[ x i (k,l) y i (k,l) z i (k,l) ]=[ cos φ k,l sin φ k,l 0 sin φ k,l cos φ k,l 0 0 0 1 ][ x i T +Δ ρ k,l y i T z i T ]
Δ ρ k,l = ρ max ρ k,l
z S,i (k,l) =f ( x i (k,l) , y i (k,l) ) W
V i (k,l) = ( f W x , f W y ,1) | x= x i (k,l) ,y= y i (k,l)
P C (k,l) := arg P i T min{| V i (k,l) T i |,i}
{ P C (k,l) := arg P i T min{| V i (k,l) T i |} i[ m (k,l1) n 0 , m (k,l1) + n 0 ]
{ x (k,l) =ρ c k,l os k,l y (k,l) =ρ s k,l in k,l z (k,l) =| z m (k,l) T |+ z S, m (k,l) (k,l)

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