## Cylindrical coordinate machining of optical freeform surfaces

Optics Express, Vol. 16, Issue 10, pp. 7323-7329 (2008)

http://dx.doi.org/10.1364/OE.16.007323

Acrobat PDF (1056 KB)

### 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

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

## 2. Cylindrical coordinates machining

*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 (

*X*,

_{s}*Y*,

_{s}*Z*) is based on the Cartesian coordinate system. The conversion between the two systems is made as follows,

_{s}*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

*z*=

_{s}*f*(

*x*,

_{s}*y*) using the conversion Eq. (1). Cutting tool compensation (Δ

_{s}*x*,Δ

_{s0}*y*,Δ

_{s0}*z*) should be considered in surface modeling coordinates, which is then converted to the machining system (Δ

_{s0}*x*,Δ

_{0}*φ*,Δ

_{0}*z*). The position (

_{0}*x*,

_{t0}*φ*,

_{t0}*z*) of the cutting tool is computed with the following offsetting method.

_{t0}## 3. Cutting path compensation

### 3.1 Mathematic basis

*z*=

_{s}*f*(

*x*,

_{s}*y*), the normal vector

_{s}*of a given point can be obtained using partial derivative of model,*

**n***f*and

_{x}*f*are the partial derivatives; and

_{y}*α*,

*β*,

*γ*are directional cosine angles, which are employed to compute the

*d*offset point

*p*for a given point

_{1}*p*of the model in the

_{0}*direction.*

**n**### 3.2 Tool nose radius compensation

*R*and zero rake angle of cutting tool are assumed, the tool model can be schematically shown in Fig. 2. The model point

_{0}*p*

_{0}(

*x*,

_{s0}*y*,

_{s0}*z*) is the one to be machined. The cutting plane normal

_{s0}*can be given using the corresponding machining polar angle*

**n**_{t0}*φ*.

_{0}*o*(

_{t}*x*,

_{st}*y*,

_{st}*z*) is the

_{st}*R*offset of

_{0}*p*, calculated with the projection vector

_{0}*of surface normal*

**n**_{p}*in the rake plane.*

**n**### 3.3 Clearance angle compensation

*α*

_{0}of the cutting tool. The normal

*of the cutting plane is obtained using polar angle*

**n**_{t1}*φ*and rake angle

_{0}*α*.

_{0}*o*’

_{t}(

*x*’

_{st},

*y*’

_{st},

*z*’

_{st}) of the rake face is deduced using a method similar to the zero rake. The contact point

*p*(

_{1}*x*,

_{s1}*y*,

_{s1}*z*) of the cutting tool is the

_{s1}*R*offset point of

_{0}*o*’

_{t}in

*. The position*

**n**_{2}*o*(

_{t}*x*,

_{st}*y*,

_{st}*z*) of the cutting tool is the corresponding point of

_{st}*o*’

_{t}(

*x*’

_{st},

*y*’

_{st},

*z*’

_{st}) rotating angle

*α*along the axis

_{0}*=*

**B***p*=(-

_{1}s*x*,-

_{s1}*y*,0) shown as the following,

_{s1}## 4. NURBS description of freeform

### 4.1 NURBS fitting and normal vector

*u*,

*ν*). One point

*P*(

*x*,

_{s}*y*,

_{s}*z*) is expressed as the following,

_{s}*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]

### 4.2 NURBS implement of freeform cutting

*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 (

*x*,

_{0}*y*) is given, the point inversion algorithm[11

_{0}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]

*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

*φ*and polar radius

_{0}*r*are given by.

_{0}## 5. Sectional curve method

*z*=

_{s}*f*(

*x*,

_{s}*y*) when the polar angle

_{s}*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

*can be derived from this expression.*

**n**### 5.1 Optimized included angle

*θ*between curve normal vector

*and the*

**n***z*axis determines the included angle

*A*. Assuming one cutting point, whose cross angle is

*θ*, the included angle of cutting tool must meet

_{i}*A*≥2

*θ*, so that this point can be cut with this tool. Over the whole surface, the included angle

_{i}*A*of cutting tool meets the following requirement.

*θ*

_{max}=max

*θ*

_{i}.

### 5.2 Optimized tool nose radius

*R*needs to be less than the minimal radius of the circle fitted to the concave portion of the sectional curves, shown in Fig. 6. What the important thing of this step is how to search the concave parts.

*θ*has a sign, if the cross angle between

*and*

**n***Z*-axis is sharp angle, the sign of cos

*θ*is negative. In contrast, the sign of cos

*θ*for a blunt angle is positive. Figure 6 is the plot of cos

*θ*to a given sectional curve. The procedure is to search the concave part shown in Fig. 6 as follows: (1) Search the inflection point, at which the sign of cos

*θ*changes from negative to positive, and call this point as a jumping point

*J*. (2) Search near the inflection points further at the two edges of

*J*, which change the value trend of cos

*θ*, and label the nearer one as

*e*. (3) Calculate the distance

_{1}*d*between

_{h}*J*and

*e*in x direction, and find the point as

_{1}*e*, which has the same distance with

_{2}*e*and is at different edge of

_{1}*J*. Then the part between

*e*and

_{1}*e*is the concave part to be found. Finally, the data of concave part is fitted to circle by the least square procedure, whose radius is used to judge the tool nose radius according to the above method.

_{2}## 6. Experimental verification

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

*mm*and the clearance angle of 10°. One workpiece with a diameter of 12.7

*mm*is machined using the generated path. The sag depth of cutting is 3

*mm*, and the amplitude of the micro lens array is 0.2

*mm*. The final cutting result is shown in Fig. 7(c).

*mm*.

*A*≥62.32° and

*R*≤0.51

*mm*.

## 7. Conclusion

*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

## References and links

1. | X. Jiang, P. Scott, and D. Whitehouse, “Freeform surface characterisation - A fresh strategy,” Ann. CIRP |

2. | H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, “Dimensional micro and nano metrology,” Ann. CIRP |

3. | L. De Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” Ann. CIRP |

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 |

5. | C. C. Chen, C. M. Chen, and J. R. Chen, “Toolpath generation of diamond shaping of aspheric lens array,” J. Mate. Proc. Tech. |

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 |

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. |

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. |

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. |

11. | Y. L. Ma and W. T. Hewitt, “Point inversion and projection for NURBS curve and surface: control polygon approach,” Comp. Aide. Geom. Desi. |

12. | K. H. Jeong, J. Kim, and L. P. Lee, “Biological inspired artificial compound eyes,” Science |

**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

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### References

- X. Jiang, P. Scott, and D. Whitehouse, "Freeform surface characterisation - A fresh strategy," Ann. CIRP 56, 553-556 (2007). [CrossRef]
- H. N. Hansen, K. Carneiro, H. Haitjema, and L. De Chiffre, "Dimensional micro and nano metrology," Ann. CIRP 55, 721-743 (2006). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- L. Piegl and W. Tiller, "The NURBS Book," (New York: Springer-Verlag, 1997).
- Q8. W. Ma, J. P. Kruth, "NURBS curve and surface fitting for reverse engineering," J. Adv. Manuf. Technol. 14, 918-927 (2005). [CrossRef]
- 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]
- K. H. Jeong, J. Kim, L. P. Lee, "Biological inspired artificial compound eyes," Science 312, 557-561 (2006). [CrossRef] [PubMed]

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