## 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]. 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]. Lens arrays are competent for light integration and image improvement [5]. F-theta lenses are widely used in scanning systems due to their precision location characteristics [6]. Freeform optics has become the key element of quantitative light technology, which is becoming increasingly important in various fields [7].

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]. 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 (*X _{s}*,

*Y*,

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

_{s}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 *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}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 *z _{s}*=

*f*(

*x*,

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

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

**n**where *f _{x}* and

*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

When tool nose radius *R _{0}* and zero rake angle of cutting tool are assumed, the tool model can be schematically shown in Fig. 2. The model point

*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}The cutting tool position *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

Figure 3 shows the cutting model incorporating the rake angle *α*
_{0} of the cutting tool. The normal * n_{t1}* of the cutting plane is obtained using polar angle

*φ*and rake angle

_{0}*α*.

_{0}The nose center *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}$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-\left(B\times {p}_{1}o\prime \right)+\mathrm{sin}{\alpha}_{0}+\left({p}_{1}o\prime \xb7B\right)\xb7B+{p}_{1}$$

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

## 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]. 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*(*x _{s}*,

*y*,

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

_{s}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]. 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.

$\mathrm{where}\{\begin{array}{c}{S}_{u}=\frac{\partial}{\partial u}S(u,v)=\sum _{i=0}^{n}\sum _{j=0}^{m}{N}_{i,p}^{\left(1\right)}{N}_{j,q}{Q}_{i,j}\\ {S}_{v}=\frac{\partial}{\partial v}S(u,v)=\sum _{i=0}^{n}\sum _{j=0}^{m}{N}_{i,p}{N}_{j,q}^{\left(1\right)}{Q}_{i,j}\end{array}.$

#### 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 (*x _{0}*,

*y*) is given, the point inversion algorithm[11] can calculate its corresponding point (

_{0}*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}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 *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

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

*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.

where *θ*
_{max}=max*θ*
_{i}.

#### 5.2 Optimized tool nose radius

The 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.

Assumed cos*θ* has a sign, if the cross angle between * n* and

*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

Compound eye (insect analogy) freeform surfaces have multi-aperture, field view of 360° and high-speed capture ability [12]. 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 0.5*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).

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 0.59*mm*.

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

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

## 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).

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