OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 10 — May. 12, 2008
  • pp: 7323–7329
« Show journal navigation

Cylindrical coordinate machining of optical freeform surfaces

F. Z. Fang, X. D. Zhang, and X. T. Hu  »View Author Affiliations


Optics Express, Vol. 16, Issue 10, pp. 7323-7329 (2008)
http://dx.doi.org/10.1364/OE.16.007323


View Full Text Article

Acrobat PDF (1056 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The cylindrical coordinate machining method (CCM) is systematically studied in generating optical freeform surfaces, in which the feature points are fitted to typical Non-Uniform Rational B-Splines (NURBS). The given points have the mapping coordinates in the variable space using the point inversion technique, while the other points have their NURBS coordinates due to the interpolation technique. The derivation and mathematical features are obtained using the fitting formula. The compensation and optimized values for tool geometry are studied using a proposed sectional curve method for fabricating designed surfaces. Typical freeform surfaces fabricated by the CCM method are presented.

© 2008 Optical Society of America

1. Introduction

Freeform surfaces can be used in optical systems to achieve novel functions, improve performances, reduce size, and decrease the cost of various products. Therefore, optical freeform surfaces find applications in the fields of optics, medicine, fiber communication, life science, aerospace etc. [1–3

1. X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation - A fresh strategy,” Ann. CIRP 56, 553–556 (2007). [CrossRef]

]. For instance, aspheric and Fresnel lenses can improve imaging quality and chromatic aberrations effectively, and can reduce the size of optical devices as well [4

4. Y. Takeuchia, S. Maedaa, T. Kawaib, and K. Sawadab, “Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving,” Ann. CIRP 51, 343–346 (2002). [CrossRef]

]. Lens arrays are competent for light integration and image improvement [5

5. C. C. Chen, C. M. Chen, and J. R. Chen, “Toolpath generation of diamond shaping of aspheric lens array,” J. Mate. Proc. Tech. 192–193, 194–199 (2007). [CrossRef]

]. F-theta lenses are widely used in scanning systems due to their precision location characteristics [6

6. H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, “Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens,” Optik 117, 271–276 (2006). [CrossRef]

]. Freeform optics has become the key element of quantitative light technology, which is becoming increasingly important in various fields [7

7. L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mode. Opti. 52, 1529–1536 (2005). [CrossRef]

].

Due to the geometrical complexity and optical property of materials, optical freeform surfaces are more difficult to make than the surfaces of conventional optical components. Presently, raster milling is employed to generate freeform surfaces though there are concerns of low machining efficiency and difficult tool-setting [8

8. C. F. Cheung, L. B. Kong, and W. B. Lee, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” J. Engi. Manu. 220, 1787–1801 (2006). [CrossRef]

]. A systematic study of cylindrical coordinate machining (CCM) is carried out in this paper, in which a higher machining accuracy and material removal rate can be achieved. The key issues in this method are discussed, such as freeform surface modeling, cutting path generation and tool geometric parameter design. A typical fabricated example of freeform optics, a compound eye structure, is presented to verify the theoretical approach.

2. Cylindrical coordinates machining

In conventional turning process, there are two coordinates, i.e., the X-axis and Z-axis. When the spindle rotation can be controlled or sensed accurately, the machining system can be expressed as cylindrical coordinate (X, Φ, Z), shown in Fig. 1(b). In cylindrical coordinates, the tool path is generated based on the polar angle, radius and a linear coordinate of Z-axis. The depth of cut varies with both the polar angle and radius according to the freeform surface, while the modeling system (Xs, Ys, Zs) is based on the Cartesian coordinate system. The conversion between the two systems is made as follows,

{xs=xcosφys=xsinφ{x=±xs2+ys2φ=d[tan1(ysxs)]
(1)

where the operator d[ ] determines the angle from [0,360).

In one cutting step, the machine position (x 0,φ 0) of the current cutting point is given as shown in Fig. 1(b). The third coordinate, z 0 of the cutting point can be derived by the freeform model zs=f(xs,ys) using the conversion Eq. (1). Cutting tool compensation (Δxs0ys0zs0) should be considered in surface modeling coordinates, which is then converted to the machining system (Δx0φ0z0). The position (xt0,φt0,zt0) of the cutting tool is computed with the following offsetting method.

Fig. 1. Cylindrical coordinate machining.
(xt,ϕt,zt)=(x+Δx,φ+Δφ,z+Δz)
(2)

The cutting path is generated when the cutting tool moves in a spiral polygon curve, whose points are calculated using the above method.

3. Cutting path compensation

3.1 Mathematic basis

As for cutting tool geometry, the compensation of tool nose radius and rake angle are studied using vector mathematics. For a typical freeform formula zs=f(xs,ys), the normal vector n of a given point can be obtained using partial derivative of model,

n=(fx,fy,1)=(cosα,cosβ,cosγ)
(3)

where fx and fy are the partial derivatives; and α, β, γ are directional cosine angles, which are employed to compute the d offset point p1 for a given point p0 of the model in the n direction.

p1=p0+d·n
(4)

3.2 Tool nose radius compensation

When tool nose radius R0 and zero rake angle of cutting tool are assumed, the tool model can be schematically shown in Fig. 2. The model point p 0(xs0,ys0,zs0) is the one to be machined. The cutting plane normal nt0 can be given using the corresponding machining polar angle φ0.

nt0=(sinφ0,cosφ0,0)
(5)

The cutting tool position ot (xst,yst,zst) is the R0 offset of p0, calculated with the projection vector np of surface normal n in the rake plane.

np=n(n·nt0)nt0
(6)

3.3 Clearance angle compensation

Figure 3 shows the cutting model incorporating the rake angle α 0 of the cutting tool. The normal nt1 of the cutting plane is obtained using polar angle φ0 and rake angle α0.

nt1=(sinφ0cosα0,cosφ0cosα0,sinα0)
(7)

The nose center ot(xst,yst,zst) of the rake face is deduced using a method similar to the zero rake. The contact point p1(xs1,ys1,zs1) of the cutting tool is the R0 offset point of ot in n2. The position ot(xst,yst,zst) of the cutting tool is the corresponding point of ot(xst,yst,zst) rotating angle α0 along the axis B=p1s=(-xs1,-ys1,0) shown as the following,

ot(xst,yst,zst)=(p1o(p1o·B)·B)cosα0
(B×p1o)+sinα0+(p1o·B)·B+p1
(8)

Equation (8) can be converted to cylindrical machine coordinates using Eq. (1).

Fig. 2. Model of tool nose radius compensation.
Fig. 3. Model of clearance angle compensation.

4. NURBS description of freeform

4.1 NURBS fitting and normal vector

The NURBS (Non-Uniform Rational B-Splines) method is one of the main approaches for modeling freeform surfaces [9

9. L. Piegl and W. Tiller, “The NURBS Book,” (New York: Springer-Verlag, 1997).

]. The NURBS expression is very helpful to describe the freeform model designed or undefined by a mathematical equation. The designed feature points can be fitted to typical NURBS formula by linear equation. The given points have the mapping coordinates in NURBS variable space using the point inversion technique, while the other points in the freeform surface get their NURBS coordinates from interpolation technique.

NURBS surfaces can be expressed as an integration of piecewise functions in different ranges of two dimensions (u,ν). One point P(xs,ys,zs) is expressed as the following,

P(xs,ys,zs)=S(u,v)=i=0nj=0mNi,p(u)Nj,q(v)Qi,j(xs,ys,zs)
(9)

where N is the basis function and Q is the control point; n and m are the numbers of control points array in (u,ν) dimensions; p and q are the orders of basis functions. The dual-cubic B-Spline method (p=3 and q=3) is used to do the fitting step, which deduces the unknown control points using the knot vectors according to designed points [10

10. W. Ma and J. P. Kruth, “NURBS curve and surface fitting for reverse engineering,” J. Adva. Manu. Tech. 14, 918–927 (2005). [CrossRef]

]. In addition, the normal vector calculation of NURBS formula is the key part for the cutting path generation. The following cross operation of two partial derivatives results normal vector of the model.

n=Su×SvSu×Sv
(10)

where{Su=uS(u,v)=i=0nj=0mNi,p(1)Nj,qQi,jSv=vS(u,v)=i=0nj=0mNi,pNj,q(1)Qi,j.

4.2 NURBS implement of freeform cutting

The (u,ν) coordinate system of NURBS model is established pointing the bottom left point as original point, which is different with the modeling system (x,y). When the rotating center point (x0,y0) is given, the point inversion algorithm[11

11. Y. L. Ma and W. T. Hewitt, “Point inversion and projection for NURBS curve and surface: control polygon approach,” Comp. Aide. Geom. Desi. 20, 79–99 (2003). [CrossRef]

] can calculate its corresponding point (u 0,ν 0) in NURBS. Making this rotating center point as polar original, the cylindrical coordinates are established easily. Firstly, the (u,ν) coordinates of one point to be cut are derived when the polar angle φ0 and polar radius r0 are given by.

{u=u0+r0cosφ0v=v0+r0sinφ0
(11)

Then the modeling coordinate of the cutting point and normal vector are calculated using Eqs. (9) and (10). Finally, the cutting path is generated by Eq. (8).

5. Sectional curve method

To avoid cutting interference, the defined tool geometry parameters are significant for freeform machining due to the geometrical complexity of freeform surfaces. The optimized tool parameters are studied using the sectional curve method, which decomposes the whole surface into two dimensional sectional curves, and integrates the tool parameters from curves as shown in Fig. 4. The parameters, such as tool nose radius and included angle, are defined using this proposed method.

One sectional curve is the typical expression of freeform formula zs=f(xs,ys) when the polar angle φ 0 is given. Using the conversion of Eq. (1), the curve has the function in machining system, assumed as equation z=g(x). Similarly, the normal vector n can be derived from this expression.

Fig. 4. Sectional curve method.

5.1 Optimized included angle

In Fig. 5, the cross angle θ between curve normal vector n and the z axis determines the included angle A. Assuming one cutting point, whose cross angle is θi, the included angle of cutting tool must meet A≥2θi, so that this point can be cut with this tool. Over the whole surface, the included angle A of cutting tool meets the following requirement.

A2θmax
(12)

where θ max=maxθ i.

5.2 Optimized tool nose radius

Fig. 5. Definition of tool nose radius and included angle.
Fig. 6. Sketch of searching concave of sectional curve.

6. Experimental verification

Compound eye (insect analogy) freeform surfaces have multi-aperture, field view of 360° and high-speed capture ability [12

12. K. H. Jeong, J. Kim, and L. P. Lee, “Biological inspired artificial compound eyes,” Science 312, 557–561 (2006). [CrossRef] [PubMed]

]. It finds wide applications in various fields due to its superior characteristics. A compound eye lens comprised of asphere and micro lens array was machined to achieve the required accuracy using the above-mentioned method by an ultra-precision precision machine (UPL250).

In the experiments, there were 4096 designed points used to simulate a compound eye. The NURBS model fitted by these points is shown in Fig. 7(a). The spiral cutting path, as shown in Fig. 7(b), is generated in the conditions of the nose radius of 0.5mm and the clearance angle of 10°. One workpiece with a diameter of 12.7mm is machined using the generated path. The sag depth of cutting is 3mm, and the amplitude of the micro lens array is 0.2mm. The final cutting result is shown in Fig. 7(c).

Fig. 7. Compound eye lens by CCM.
Fig. 8. Compound eye lens example.

Before machining, the fitted NURBS model was extracted to sectional curves and was analyzed to confirm if cutting tool parameters were optimized parameters according to the sectional curve method. A sectional curve and the included angle plot is shown, for example in Fig. 8(a), the maximum included angle is 60.11°. Figure 8(b) shows the fitting circle on concave segments which define the tool nose radius, and the minimum radius is 0.59mm.

Integrating all sectional curves, the tool parameters, which can be used to cut this compound eye lens, must satisfy A≥62.32° and R≤0.51mm.

Other typical freeform surfaces were also successfully achieved using the cylindrical coordinate machining method shown in Fig. 9.

Fig. 9. Typical freeform parts machined by CCM.

7. Conclusion

Application of freeform surfaces in various fields is a future trend. To machine this type of surfaces using ultra-precision machines with a three-axis of (X,Φ,Z), a cylindrical coordinate machining method is systematically studied in this paper. In addition to the freeform surfaces described by mathematical formulae, it can also be established using NURBS description, which can be applied to the proposed machining method. The compensation of cutting tool is discussed to achieve the required freeform profile. To avoid cutting interference, the sectional curve method is proposed to define the optimized cutting tool parameters including tool nose radius and included angle. Using the cylindrical coordinate machining method, various typical freeform surfaces are successfully machined.

Acknowledgments

The authors would like to thank Mr. X. Zhao for the preparation of the experiments. Support is also acknowledged by the 111 project (B07014).

References and links

1.

X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation - A fresh strategy,” Ann. CIRP 56, 553–556 (2007). [CrossRef]

2.

H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, “Dimensional micro and nano metrology,” Ann. CIRP 55, 721–743 (2006). [CrossRef]

3.

L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” Ann. CIRP 52, 561–578 (2003). [CrossRef]

4.

Y. Takeuchia, S. Maedaa, T. Kawaib, and K. Sawadab, “Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving,” Ann. CIRP 51, 343–346 (2002). [CrossRef]

5.

C. C. Chen, C. M. Chen, and J. R. Chen, “Toolpath generation of diamond shaping of aspheric lens array,” J. Mate. Proc. Tech. 192–193, 194–199 (2007). [CrossRef]

6.

H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, “Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens,” Optik 117, 271–276 (2006). [CrossRef]

7.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mode. Opti. 52, 1529–1536 (2005). [CrossRef]

8.

C. F. Cheung, L. B. Kong, and W. B. Lee, “Modelling and simulation of freeform surface generation in ultra-precision raster milling,” J. Engi. Manu. 220, 1787–1801 (2006). [CrossRef]

9.

L. Piegl and W. Tiller, “The NURBS Book,” (New York: Springer-Verlag, 1997).

10.

W. Ma and J. P. Kruth, “NURBS curve and surface fitting for reverse engineering,” J. Adva. Manu. Tech. 14, 918–927 (2005). [CrossRef]

11.

Y. L. Ma and W. T. Hewitt, “Point inversion and projection for NURBS curve and surface: control polygon approach,” Comp. Aide. Geom. Desi. 20, 79–99 (2003). [CrossRef]

12.

K. H. Jeong, J. Kim, and L. P. Lee, “Biological inspired artificial compound eyes,” Science 312, 557–561 (2006). [CrossRef] [PubMed]

OCIS Codes
(220.1250) Optical design and fabrication : Aspherics
(220.1920) Optical design and fabrication : Diamond machining

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: February 14, 2008
Revised Manuscript: April 8, 2008
Manuscript Accepted: May 3, 2008
Published: May 6, 2008

Citation
F. Z. Fang, X. D. Zhang, and X. T. Hu, "Cylindrical coordinate machining of optical freeform surfaces," Opt. Express 16, 7323-7329 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7323


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007). [CrossRef]
  2. H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006). [CrossRef]
  3. L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, "Surfaces in precision engineering, microengineering and nanotechnology," Ann. CIRP 52, 561-578 (2003). [CrossRef]
  4. Y. Takeuchia, S. Maedaa, T. Kawaib, K. Sawadab, "Manufacture of multiple-focus micro Fresnel lenses by means of nonrotational diamond grooving," Ann. CIRP 51, 343-346 (2002). [CrossRef]
  5. C. C. Chen, C. M. Chen, and J. R. Chen, "Toolpath generation of diamond shaping of aspheric lens array," J. Mater. Proc. Technol. 192-193, 194-199 (2007). [CrossRef]
  6. H. B. Wu, Z. Q. Wang, R. L. Fu, and J. Liu, "Design of a hybrid diffractive/refractive achromatized telecentric f-θ lens," Optik 117, 271-276 (2006). [CrossRef]
  7. L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, "Designing reflectors to generate a line-shaped directivity diagram," J. Mod. Opt. 52, 1529-1536 (2005). [CrossRef]
  8. C. F. Cheung, L. B. Kong, and W. B. Lee, "Modelling and simulation of freeform surface generation in ultra-precision raster milling," J. Eng. Manuf. 220, 1787-1801 (2006). [CrossRef]
  9. L. Piegl and W. Tiller, "The NURBS Book," (New York: Springer-Verlag, 1997).
  10. Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005). [CrossRef]
  11. Y. L. Ma, W. T. Hewitt, "Point inversion and projection for NURBS curve and surface: control polygon approach," Comput. Aided Geom. Des. 20, 79-99 (2003). [CrossRef]
  12. K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited