Optimization of Tool Path for Uniform Scallop-Height in Ultra-precision Grinding of Free-form Surfaces

Free-form surfaces have been widely used in complex optical devices to improve the functional performance of imaging and illumination quality and reduce sizes. Ultra-precision grinding is a kind of ultra-precision machining technology for fabricating free-form surfaces with high form accuracy and good surface finish. However, the complexity and variation of curvature of the free-form surface impose a lot of challenges to make the process more predictable. Tool path as a critical factor directly determines the form error and surface quality in ultra-precision grinding of free-form surfaces. In conventional tool path planning, the constant angle method is widely used in machining free-form surfaces, which resulted in non-uniform scallop-height and degraded surface quality of the machined surfaces. In this paper, a theoretical scallop-height model is developed to relate the residual height and diverse curvature radius. Hence, a novel tool-path generation method is developed to achieve uniform scallop-height in ultra-precision grinding of free-form surfaces. Moreover, the iterative closest-point matching method, which is a well-known algorithm to register two surfaces, is exploited to make the two surfaces match closely through rotation and translation. The deviation of corresponding points between the theoretical and the measured surfaces is determined. Hence, an optimized tool-path generator is developed that is experimentally verified through a series of grinding experiments conducted on annular sinusoidal surface and single sinusoidal surface, which allows the realization of the achievement of uniform scallop-height in ultra-precision grinding of free-form surfaces.

List of Symbols N s Rotational speed of grinding wheel spindle (mm) N w Rotational speed of workpiece spindle (rpm) R s Wheel radius (mm) r s Nose radius of the grinding wheel (mm) V f Feed speed (mm/min)

R
Rotation transformation matrix R w Workpiece radius (mm) S Tool path interval (mm) ρ w Curvature radius of workpiece surface (mm) R t Scallop height (mm) H Depth of cut (μm) Q ij Points on the measured surface A Amplitude of sinusoidal surface (mm) φ Phase angle (rad) λ Wave length of sinusoidal surface (mm) M Machining transformation matrix T Translation matrix P ij Points on the measured surface

Introduction
With increasing demand of optical and photonic manufacturing industries, many types of high-resolution and compact structure of optical components are widely used for digital cameras, solar concentrators, aspectual illumination systems, and collimators [1][2][3][4]. Free-form surfaces can reduce wave front error and optical elements, which are frequently employed to fabricate the units of high-performance optical systems [5]. However, machining of the free-form surfaces involves great challenges resulting from the complex curvature variation. Among various ultra-precision machining processes such as single-point diamond turning (SPDT), diamond milling, fly cutting, micro-chiseling, and ultraprecision diamond grinding, it is interesting to note that the grinding operation is highly capable of machining optical components made of hard and brittle materials than other machining processes due to high efficiency and high accuracy [6].
In the machining of free-form surfaces, the CNC system controls sequential cutting points of the machine tool over the workpiece by an interpolation algorithm and tool trajectory [7,8]. The tool-path generation is vital for determining the surface quality and machining efficiency, which attracts a lot of research attention. For a three-axis machine tool, the tool path generally evolves as an Archimedean spiral from the outmost area to the rotational center, in which a series of interaction points on the spiral are represented by a polar coordinate system according to the rotational angle and feed speed [9,10]. In conventional machining, the constant angle is a widely used tool path-generation strategy in machining complex surfaces [11][12][13], which resulted in non-uniform surface scallop-height. The outer area of the machined surface is coarser than that of the central region of the surface due to a lager arc length on outer area. Zhou et al. [14] studied the influence of two different tool path-generation strategies based on constant rotational angle and constant arc length, respectively. It is found that constant arc length was a preferable method to achieve the higher form accuracy.
However, most research about the tool-path planning method is based on constant arc length or operation parameters to study the scallop-height generation, and the influence of curvature of the machined surface on the scallop-height generation received little attention. In fact, for machining free-form surfaces, the variation of surface curvature resulted in different scallop heights and form errors of the machined surface. As a result, it is vital to develop a toolpath generation strategy to achieve constant scallop height so as to improve surface accuracy. In this study, the relationship between the curvature and scallop height is analyzed theoretically and a new control strategy for a tool path with variable feed speed is proposed, which can be used to achieve uniform scallop height in ultra-precision grinding of free-form surfaces.

Theoretical Modeling of Scallop Height in Ultra-precision Grinding
In ultra-precision grinding, the rotational workpiece traverses across the high-speed spinning wheel to remove redundant materials and create desired surfaces by changing the tool position with respect to the part and feed speed, as shown in Fig. 1a. The ground surface generation is directly related to the tool path, in which the grinding wheel moves in an Archimedes spiral in X-Y plane, as shown in Fig. 1b. In ultra-precision grinding of free-form surfaces, the tool trajectory is a spiral around the rotational center of the workpiece and scallop height on the workpiece surface producing between two adjacent paths is principally determined by the rotational speed of the workpiece, feed speed, and curvature radius of the machined surface. The surface curvature resulted in different contact points referring to the cutting profile of the grinding wheel, which caused the different scallop height on the ground surface. Figure 2 shows the different geometric relationships among flat surface, convex surface machining, and concave surface machining. For grinding flat surfaces, the tool-path interval is easy to determine, which keeps uniform spacing. However, for machining convex and concave surfaces, the calculation is more complex and the interval spacing is changeable according to the curvature radius.
For a given allowable tolerance for the scallop height in machining flat surface, the path interval can be determined according to the geometric relation as shown in Fig. 2a as follows: (1) where r s is the nose radius of the grinding wheel and S is the path interval.
In machining convex surfaces, the path interval can be derived as: where w is the curvature radius, S is the path interval, and R t is the scallop height.
In machining concave surfaces, the path interval can be obtained as follows: According to the geometric relationship between the machining curved surface as shown in Fig. 2b, the cutting profile of the grinding wheel (the center of the profile located in O 1 ) in machining convex surface can be derived as: To obtain the maximum scallop height, y equals 0, the intersecting point can be expressed as: the scallop height R t can be determined as: Feed speed (V f ) for machining convex surface can be calculated as: In the same way, for a concave surface as shown in Fig. 2b, the scallop height R t can be determined as: Feed speed (V f ) for machining concave surfaces can be calculated as: 1 3 In order to obtain uniform scallop height, the feed speed for the allowable residual error can be determined according to Eqs. (9) and (11) under different surface curvatures of the free-form surface in different contact points. This is due to the fact that the different curvatures of the free-form surface with respect to the different types of surfaces (convex, flat, and concave surface) pose a significant difference in scallop height. Figure 3 describes the influence of curvature variation on scallop height. Under constant feed speed, the scallop height is different, which is determined by the curvature. In order to discriminate the three different surfaces, the second partial derivative of the designed surface z = f (r, ) in different radical sections can be calculated as shown in Fig. 3.
First of all, the grid is generated on the free-form surface under the polar coordinate system ( r, ), the curvature in different position is then determined. Based on the curvature, the type of curvature surface (flat, convex, or concave surface) is determined according to Eq. (12). Hence, the variable feed speed can be calculated according to Eqs. (9) and (11). Finally, the grinding process completed when the last grid on the surface is ground ( r = 0, = 0 ). Figure 4 shows the flow chart of the algorithm for realizing the uniform scallop height in ultra-precision grinding free-form surfaces.

Experimental Design and Simulation
The machining of free-form surfaces is performed on an ultra-precision grinding machine (Moore Nanotech 450UPL). The grinding machine makes use of a rotational table (B axis) to hold the wheel spindle and the workpiece is mounted on an air bearing, in which the workpiece spindle is rotated and fed over a high-speed spinning wheel to remove excess materials and shape the desired form, as shown in Fig. 5. In the ultra-precision grinding experiments, tungsten (12) z �� = f �� (r, ) = 2 z r 2  Fig. 3 Computation of the scallop height and discrimination of concave, flat, and convex surfaces in machining carbide (WC) is used as the workpiece material, which is widely used in the fabrication of the optical mold; the machining conditions are summarized in Table 1. In order to avoid the disturbance of the original surface topography of the workpiece, all workpieces are processed through two steps of rough and fine grinding before grinding experiment, and then the grinding wheel is dressed before machining each workpiece so as to reduce the impact of grinding wheel wear. Finally, the non-contact Zygo Laser Interferometer Profiler is used to measure the ground surface topography, which is capable of measuring different types of surfaces including flat surfaces, spherical surfaces, aspherical surfaces, as well as free-form surfaces ranging from supersmooth to very rough surfaces.
In this experiment, two types of sinusoidal surface are machined. They are annular sinusoidal surface and single sinusoidal surface. In the Cartesian coordinate system, they can be expressed in Eqs. (13) and (14), respectively. According to Eqs. (13) and (14) and setting as: the amplitude of the sinusoidal surface A = 50 μm, the wave length of the   sinusoidal surface = 3 mm and phase angle = 90 • , the simulated sinusoidal surfaces are shown in Fig. 6a, b.
In polar coordinate system, the annular sinusoidal surface and single sinusoidal surface can be derived as: The allowable scallop height is set at 1 μm and 2 μm for single sinusoidal surface and annular sinusoidal surface, respectively. The grinding wheel nose radius is 0.5 mm. According to Eqs. (13) and (14), the tool path can be determined and Fig. 7 shows the estimated 3D tool path for machining the two sinusoidal surfaces. In order to observe the tool path clearly, the tool-path intervals are enlarged.
The machined sinusoidal surfaces were measured by using a non-contact laser interferometer profiler apparatus. Figure 8 shows the sinusoidal surfaces machined by threeaxis ultra-precision grinding machine. The result shows that the form error is larger for machining non-rotational  symmetric surfaces than that for rotational symmetric surfaces, which may be caused by the algorithm errors. For machining rotational symmetric surfaces, the radical cross section of the surface is the same and has lower sensitivity to the algorithm error as compared with that of the machining of the non-rotational symmetric surface, in which, the radical cross section of the surface is dependent on the angular position. The form accuracy in machining free-form surfaces is an important indicator to determine the functional performance in ultra-precision grinding. The deviation of whole ground surface and distribution of the form errors should be verified to evaluate machining performance. The main purpose of the surface matching between the measured surface and the designed surface is to make the two surfaces as close as possible. In order to improve the matching accuracy, it is necessary to continuously iterate and adjust again and again to find the optimal spatial position of the measured surface.
There are two parts in the matching process. One part is the translation operation, which can be represented by the matrix T(t x , t y , t z ) , the other part is the rotation process, which can be represented by the matrix R( , , ) . The whole transformation calculation process is The translation matrix can be expressed as: where t x , t y and t z represent translational transformation in x , y and z direction.
Rotation transformation matrix can be expressed as: cos cos cos sin sin − sin cos sin sin + cos sin cos 0 sin cos cos cos + sin sin sin sin sin cos − cos sin 0 − sin cos sin cos cos 0 where , and represent the rotation angle of the measured surface around x , y and z axis. According to Eqs. (15)-(17), the spatial coordinate transformation matrix can be expressed as: In Eq. (18), there are six parameters need to be solved, rotation angle , and and the translation distance t x , t y and t z . In order to figure out these six unknown parameters, the transformation matrix can satisfy the following equation: T is the point on the measured sur- is the closest point to the measured surface on the designed surface, j is the number of iterations.
Substitute the data points of the measured surface into Eq. (19), then take partial derivative and set it equal to 0 In order to further verify the deviation between the designed sinusoidal surfaces and measured surfaces cos cos cos sin sin − sin cos sin sin + cos sin cos t x sin cos cos cos + sin sin sin sin sin cos − cos sin t y − sin cos sin cos cos tion and translation. Figure 9 shows the evaluated matching errors for the machining of different sinusoidal surfaces. It shows larger errors for machining single sinusoidal surfaces. It is found that the scallop height is uneven in the conventional tool-path planning method, which resulted from the influence of the changed curvature of workpiece surface. On the relatively flat area, the scallop height is about 1 μm. However, on the area with small curvature radius, surface scallop height is larger, the scallop height increased to 2.5 μm, as shown in Fig. 10. Figure 11 shows that the measured cross-section profiles of sinusoidal surface and scallop height are about 1 μm in grinding annular sinusoidal surface and single sinusoidal surface (X = 0). The scallop height for both annular sinusoidal surface and single sinusoidal surface are kept uniform approximately. At the same time, the peaks on the maximum of the central sinusoid caused the tool setting error. Table 2 shows a comparison of the arithmetic roughness of these two types of free-form surfaces with respect to different areas corresponding to Fig. 10. Each workpiece is machined three times and then the average PV value is calculated. It was found that the PV value stays approximately uniform in machining those two free-form surfaces respectively and the matching error for machining a single sinusoidal surface in terms of root-mean-square value (RMS is 0.182 μm) is significantly larger than that of machining annual sinusoidal surfaces (i.e., RMS = 0.108 μm).

Conclusions
In this paper, a novel method for the optimization of tool path to achieve uniform scallop height in ultra-precision grinding free-form surface is presented. The theoretical models both for scallop height in grinding convex surface and concave surface are developed and the relationship between the scallop height and curvature radius is explained. The uniform scallop height is obtained by adopting variable feed speed in grinding of a free-form surface. The modelpredicted results for the machined surfaces agree well with that of the experimental results. For machining free-form surfaces, the scallop height is significantly different from machining a flat surface, which is more complex and has a strong correlation to the curvature of the surface. For different curvature radii in conventional grinding, the scallop height is ununiformed. The larger curvature radius results in higher scallop height, which adversely affects the surface quality. However, the scallop height is small for grinding of relative flat areas. This model provides an efficient way to overcome the uneven surface residual height by changing the feed speed. In addition, the ICP matching method is used to evaluate the performance of the theoretical model and it is found that there is the larger surface form errors for machining non-rotational symmetric surface than that of rotational symmetric surface.