## Abstract

Fast tool servo/ slow tool servo (FTS/STS) diamond turning is a very promising technique for the generation of freeform optics. However, the currently adopted constant scheme for azimuth sampling and side-feeding motion possesses no adaptation to surface shape variation, leading to the non-uniform surface quality and low machining efficiency. To overcome this defect, this paper reports on a novel adaptive tool servo (ATS) diamond turning technique which is essentially based on the novel two-degree-of-freedom (2-DOF) FTS/STS. In the ATS, the sampling interval and the side-feeding motion are actively controlled at any cutting point to adapt the machining process to shape variation of the desired surface, making both the sampling induced interpolation error and the side-feeding induced residual tool mark be within the desired tolerances. Characteristic of the required cutting motion suggests that besides the conventional *z*-axis servo motion, another servo motion along the *x*-axis synthesizing by the *c*-axis is mandatory for implementing the ATS. Comparative studies of surface generation of typical micro-structured surfaces in FTS/STS and ATS are thoroughly conducted both theoretically and experimentally. The result demonstrates that the ATS outperforms the FTS/STS with improved surface quality while simultaneously enhanced machining efficiency.

© 2015 Optical Society of America

## 1. Introduction

Freeform optics, including smooth surfaces (e.g. F-theta surface) and micro-structured surfaces (e.g. microlens array), is increasingly being applied to both imagining and non-imaging optical systems [1–3 ]. To satisfy practical requirements in industries, fast tool servo (FTS) or slow tool servo (STS) diamond turning has been developed and is widely regarded as a very promising technique due to its capacity of efficiently generating complicated surfaces with sub-micron form accuracy and nanometric roughness [4–6 ]. Recently, to improve the quality of FTS/STS turned freeform surfaces, much effort has been devoted to achieving the optimal processing conditions, including optimal selection of geometries of the diamond tool (tool nose radius, rake angle and clearance angle) [7–10 ], optimal selection of machining parameters (feedrate, spindle speed and sampling) [11, 12 ] and determination of optimal toolpath (tool geometry compensation, static and dynamic error motion compensation) [9, 13, 14 ]. With aid of these state-of-the-art techniques, the machining capacity of FTS/STS has been significantly enhanced, leading to the ultra-fine generation of freeform surfaces on a wide spectrum of engineering materials [15, 16 ].

In FTS/STS diamond turning, the relative motions between the diamond tool and the workpiece are used to approximate the desired freeform surface. In principle, there are two intrinsic sources that will lead to the inevitable form errors in FTS/STS, namely the azimuth sampling and the side-feeding motion. With the azimuth sampling, the interpolation motion between any two successive cutting points leads to the deviation between the generated and the desired surface [17, 18 ]. Although the spline interpolation method, reported to be superior in generating more smooth and accurate surfaces, is embedded in certain computer aided manufacturing (CAM) systems for optics turning (e.g. DiffSys), the computation load increases by an order when comparing with the more common linear interpolation method [12, 18 ]. Generally, the interpolation error along the forward cutting direction is highly dependent on the distance between the corresponding two successive cutting points and the local surface profile [11, 17 ].

Currently, there are mainly two kinds of azimuth sampling methods, one is the constant-angle sampling strategy (CASS) and the other one is the constant-arc-length sampling strategy (CLSS). With the more common CASS, the interpolation error is highly dependent on the rotation radius at the cutting point. To guarantee an acceptable interpolation error over the whole surface, a large enough sampling number regarding the outer region is required for the whole surface. This leads to the redundant control points for the region close to the rotation center [12]. Deriving from this, another two fatal defects for the CASS occur: a) the volume of control point data for the optics with large apertures will be often too large to be easily operated by the computer numerical control (CNC) system [19]; and b) too many control points in one revolution will significantly reduce the spindle speed due to the required data transfer and servo times of the physical system, leading to the low machining efficiency [12, 17 ]. To reduce the number of control points and achieve a relatively uniform sampling, the CALS [20] as well as the hybrid sampling strategy [17] combining the CASS and CALS were developed. Recently, another space Archimedean spiral based toolpath planning strategy was also introduced for a special sort of freeform surfaces featuring quasi-revolution [21]. In terms of these improved strategies, another factor predominating the interpolation error was ignored, namely the surface profile. As for freeform optics, the local geometrical features may vary significantly with respect to different cutting points. Thus, sampling from a constant viewpoint will inevitably lead to a heterogeneous distribution of the interpolation error.

The other kind of principle form errors in FTS/STS is caused by side-feeding motion of the diamond tool, which is the residual tool mark (RTM). The RTM in the high-spatial frequency domain is usually regarded as surface roughness. However, the height is often at the same level of surface form error in ultra-precision machining, which is also highly dependent on the local geometry of the desired surface [22]. Generally, the currently adopted constant side-feeding motion in FTS/STS results in the heterogeneous distribution of the height of the RTM, especially for the freeform surfaces with strong varied curvatures. To ensure the peak-to-valley (PV) value of the RTM is within tolerance levels, a minimum feedrate for all the cutting points is required, suggesting a much lower machining efficiency [11]. However, little attention has been paid to the concept of adapting side-feeding in diamond turning of freeform optics due to the lack of another translational servo motion along the *x*-axis of the machine tool. Essentially, the STS system possesses three-axis servo motions (*X*, *Z*, and *C*), enabling it to directly follow the adaptive scheme [23]. As for conventional FTS, it has but one servo motion along the *z*-axis. Fortunately, the recently developed two-degree-of-freedom (2-DOF) and three-DOF FTS also provide a potential for adopting the adaptation concept [22, 24–26
].

With conventional multi-axis CNC milling of freeform surfaces, it is crucial to adapt the side-feeding and forward sampling of tool motions to surface shape variations to achieve uniform surface quality with maximum efficiency, effectively constructing a variety of adaptive multi-axis CNC machining methods [27–30
]. However, it is not popular for the multi-axis CNC machining to generate freeform optics due to limited form accuracy and rough surface quality of the resulting components [4, 28
]. Besides, it is also much more difficult to generate micro-structured functional surfaces limited by system dynamics and tool geometries. As a predominant technique for the generation of freeform optics including the intricate micro-nanostructured functional surfaces [4, 31
], adaptive strategy in terms of machining tolerance is ignored in FTS/STS diamond turning in both academic and industrial fields [4, 21, 26
]. For instance, the adaptive scheme is not included in the famous DiffSys CAM software which is especially developed for ultra-precision turning of optics [32]. Therefore, the current machining of freeform optics inherently suffers non-uniformity of surface quality and low machining efficiency. To improve machining performances, special attentions on machining tolerances with consideration of surface features in both forward cutting and side-feeding directions are urgently required. Thereby, on basis of the newly developed 2-DOF FTS/STS technique, a novel adaptive diamond turning method, known as the *Adaptive Tool Servo (ATS)*, with full consideration of desired surface features is proposed and demonstrated. The main contribution of the present study can be summarized as follows:

i) The concept of adaptive tool servo diamond turning is proposed for the generation of freeform optics which may significantly improve machining efficiency with simultaneously enhanced machining quality when comparing with state-of-the-art diamond machining techniques; ii) The adaptive toolpath generation strategy is proposed and verified for the ATS with consideration of its unique characteristics of cutting kinematics and tool geometries; iii) Characteristics of required motions for the ATS are discussed in detail, providing guidance for further development of mechatronic systems of the 2-DoF FTS/STS to better realize the adaptive scheme; iv) The necessity and superiority for adopting adaptive scheme in diamond turning of freeform optics are demonstrated both theoretically and experimentally; v) The present study provides a new application field for the newly developed 2-DoF FTS/STS technique, and it may in turn further promote the development of this innovative technique.

## 2. Basic principle of the adaptive tool servo

The FTS/STS diamond turning is essentially operated in the Cylindrical coordinate system (*ρ*,*θ*,*z*) as shown in Fig. 1(a)
. In turning, the diamond tool follows a spatial spiral motion, where the relative motions along the azimuth and polar axis directions are referred as the forward cutting and side-feeding directions, respectively. Translational servo motion along the *z*-axis is used to construct the complicated freeform surface. Along the forward cutting direction as illustrated in Fig. 1(b), there are two kinds of interpolation errors in terms of the local concavity and convexity of the freeform surface: the concavity induces excessive residuals (positive), while the convexity induces undercutting effects (negative). Along the side-feeding direction as illustrated in Fig. 1(c), the side-feeding motions will always result in the residual marks (positive). If the boundary of the interpolation error is ± *ε*
_{f}, and that of the PV valure of the RTM is *ε*
_{s}, the range of surface principle error will be *P*e$\in $[-*ε*
_{f}, *ε*
_{s} + *ε*
_{f}].

For a specified freeform surface with selected diamond tool, the controllable factors governing the interpolation error and the RTM are the sampling interval and the side-feeding motion, respectively. In the ATS diamond turning, both the sampling interval and the side-feeding motion in each cutting point is deliberately determined to adapt to shape variations of the desired surface as shown in Fig. 2
. With the adaptive sampling, the adaptive criterion is to make the interpolation error *ε*
_{f} always be the same as the preset tolerance for each cutting point, i.e. the constraint of constant interpolation error. Thus, a much larger sampling interval can be adopted for the cutting point with slight curvature variation as shown in Fig. 2(a). This operation will significantly reduce the volume of required control points with acceptable interpolation errors. Similarly, the side-feeding motion also follows the adaptation strategy to make the RTM *ε*
_{s} always be the same as the preset one at any cutting points, i.e. the constraint of constant scallop height. As shown in Fig. 2(b), the side-feeding updating will always lead to an adjustment for the servo motion along the *z*-axis to avoid the interference induced machining errors.

## 3. Adaptive toolpath determination for the ATS

As discussed above, each cutting point is required to simultaneously satisfy the two form error constraints in both forward cutting and side-feeding directions. Since the adaptive sampling of the azimuth will not change the orbit of the cutting points, the toolpath determination process is divided into three stages to enhance the calculation robustness, which are: a) determine the adaptive cutter contact positions (CCPs) with the constraint of constant scallop height; b) resample the afore-obtained CCPs with respect to the constraint of constant interpolation error; c) establish the geometrical mapping between the CCPs and the cutter location points (CLPs).

#### 3.1 Determination of the adaptive CCP

The coordinate system of the workpiece during turning is illustrated in Fig. 1(a), where *o*
_{w}-*x*
_{w}
*y*
_{w}
*z*
_{w} denotes the local Cartesian coordinate system of the workpiece. In the system, the desired surface can be expressed as *z*
_{w} = *f*
_{w}(*x*
_{w},*y*
_{w}). In the more natural Cylindrical coordinate system (*ρ*
_{w},*θ*
_{w},*z*
_{w}), the desired surface can also be expressed as *z*
_{w} = *g*
_{w}(*ρ*
_{w},*θ*
_{w}) by taking the following conversions: *x*
_{w} = *ρ*
_{w}cos*θ*
_{w}, *y*
_{w} = *ρ*
_{w}sin*θ*
_{w}.

### 3.1.1 Iterative determination of the adaptive CCP

Be similar with the FTS/STS diamond turning, each rotation of the spindle is uniformly discretized into *N*
_{s} points. With the *l*-th cutting point in the *k*-th revolution of the spindle, the corresponding azimuth position of the CCP *P*
^{(}
^{k,l}^{)} can be expressed by

The corresponding curvature of the cross-sectional profile passing through the rotation center can be estimated by

By approximating the local profile as a segment of circular arc, the distance *L*
_{s}
^{(}
^{k+}^{1}
^{,l}^{)} (Fig. 1(c)) between the two cutting points *P*
^{(}
^{k,l}^{)} and *P*
^{(}
^{k+}^{1}
^{,l}^{)} with respect to a given RTM error tolerance *ε*
_{s} yields [33]

*R*

_{t}denotes the nose radius of the diamond tool.

Thus, an equation is constructed as

where $\Vert \cdot \Vert $ denotes the Euler distance between any two points.Without loss of generality, *ρ*
_{w}
^{(}
^{k}^{+1,}
^{l}^{)} is assumed to be larger than *ρ*
_{w}
^{(}
^{k}^{,}
^{l}^{)} with consideration of the side-feeding motion. Then, the following relationship in the Cylindrical coordinate system (*ρ*
_{w},*θ*
_{w},*z*
_{w}) can be derived from Eqs. (3) and (4)

Since there is only one parameter *ρ*
_{w}
^{(}
^{k}^{+1,}
^{l}^{)} is unknown, it can be found by numerically solving the nonlinear Eq. (1), Eq. (2) and Eq. (5). Then, position coordinates of the corresponding CCP can be determined by

### 3.1.2 Determination of the initial CCPs in the first revolution

During the calculation, the first CCP *P*
^{(1,1)} is set at the rotation center with *ρ*
_{w}
^{(1,1)} = 0, *θ*
_{w}
^{(1,1)} = 0. Following the steps in the last section, the corresponding CCP *P*
^{(2,1)} for the first CCP in the second revolution can be obtained. As for the first revolution, side-feeding with constant feedrate is adopted, and the polar axis variation, namely *ρ*
_{w}
^{(2,1)}, serves as the corresponding feedrate. Thus, the azimuth and polar coordinates in the first revolution can be accordingly determined by

Taking the CCPs in the first revolution as the initial points, the CCPs for the whole surface can be obtained by following the iteration strategy developed in the last section.

#### 3.2 Adaptive resampling of the CCPs

After resampling, assume that the azimuth coordinate of a updated the *j*-th CCP in the *k*-th revolution is *φ*
_{w}
^{(}
^{k}^{,}
^{j}^{)}. The closet CCP determined by the CASS in last section can be identified by

By selecting a set of successive points form the (*g*-*m*)-th to the (*g* + *m*)-th CCPs, namely: ${\theta}_{g}=\left\{{\theta}_{\text{w}}^{(k,g-m)},{\theta}_{\text{w}}^{(k,g-m+1)},\cdots ,{\theta}_{\text{w}}^{(k,g+m-1)},{\theta}_{\text{w}}^{(k,g+m)}\right\}$, ${\rho}_{g}=\left\{{\rho}_{\text{w}}^{(k,g-m)},{\rho}_{\text{w}}^{(k,g-m+1)},\cdots ,{\rho}_{\text{w}}^{(k,g+m-1)},{\rho}_{\text{w}}^{(k,g+m)}\right\}$ the polar axis coordinate can be determined by adopting the cubic spline interpolation [34] as

*m*is the user-defined integrate number, and

*S*(⋅) denotes the cubic spline interpolation operator with respect to azimuth position ${\phi}_{\text{w}}^{(k,j)}$ on the curve $\left({\theta}_{g},{\rho}_{g}\right)|$.

The local curvature radius of the freeform surface along the forward cutting direction is approximated as the curvature radius of a curve, which is generated by the freeform surface *F*
_{w}(*x*
_{w},*y*
_{w}
*,z*
_{w}) = *z*
_{w}-*f*
_{w}(*x*
_{w},*y*
_{w}) = 0 intersecting the cylinder with a radius of the polar axis at the specified cutting point. With the cylinder, it can be mathematically described by

The tangent to the intersection curve is given by

The curvature radius for the given CCP can be accordingly expressed by [35]

By approximating the intersection curve between any two updated CCPs to be an arc, the corresponding distance as shown in Fig. 1(b) can be determined by ${L}_{\text{f}}=\sqrt{8{R}_{\text{f}}{\epsilon}_{\text{f}}}$. Thus, the following relationship can be constructed

Since there is only one parameter ${\phi}_{\text{w}}^{(k,j+1)}$ is unknown, it can be found by combining the nonlinear equations from Eq. (8) to Eq. (13). Then, position coordinates of the corresponding CCP $\left({x}_{\text{CCP}}^{(k,j)},{y}_{\text{CCP}}^{(k,j)},{z}_{\text{CCP}}^{(k,j)}\right)$ can also be obtained following Eq. (6). By conducting the iterative steps with respect to ${\phi}_{\text{w}}^{(1,1)}\text{=}0$, the whole toolpath can be adaptively resampled.

#### 3.3 Determination of the CLP from the adaptive CCP

The CLP is defined as the coordinate of the tool edge center. A distance offset from the CCP should be conducted to compensate for tool nose radius, with a value of the radius along the projected direction of surface normal in the rake plane as shown in Fig. (3) . The unite normal vector of the machined surface at the CCP can be obtained by [26]

Assume a round edged diamond tool with zero rake angle is adopted, the CLP can be determined by [9]

#### 3.4 Practical implementation of motions in ATS

In practice, there are three motion components to be implemented on the machine tool, namely the *φ*
_{w}, *z*
_{CLP} and ${\rho}_{\text{CLP}}=\sqrt{{x}_{\text{CLP}}^{2}+{y}_{\text{CLP}}^{2}}$. The *φ*
_{w} component is the motion of the spindle (*c*-axis), and it can serve as the reference to synthesize motions along the other two directions. Similar to FTS/STS diamond turning, the translational servo motions along the z-axis of the machine tool will implement the motion component *z*
_{CLP}.

The motion component *ρ*
_{CLP} is the side-feeding motion of the slide along the *x*-axis. It features a long distance movement with imposition of strong fluctuations to adapt to surface shape variations. Thus, it requires a long range stroke to cover the whole aperture. Meanwhile, it still requires the capacity of ultra-precision tracking of the fluctuations. With the STS, slide servo motion along the x-axis can directly realize the component *ρ*
_{CLP}. With the recently developed 2-DOF-FTS, the servo motion along the *x*-axis direction only has a very limited stroke to follow the fluctuations. Thus, the component *ρ*
_{CLP} can be decomposed into a best-fitted linear motion for the machine tool and a pure fluctuation motion for *x*-axial servo motion of the 2-DOF-FTS.

## 4. Theoretical investigation on characteristics of the ATS

To characterize the novel ATS, a typical sinusoidal grid micro-structured surface was employed for the investigation, which could be mathematically described by *z*
_{w}(*x*
_{w},*y*
_{w}) = *A*
_{x}sin(2π*f*
_{x}
*x*
_{w}) + *A*
_{y}sin(2π*f*
_{y}
*y*
_{w}). The amplitude and spatial frequency were set as *A*
_{x} = *A*
_{y} = 5 μm and *f*
_{x} = *f*
_{y} = 5 mm^{−1}, respectively. The tool nose radius was set as *R*
_{t} = 0.1 mm with a zero degree rake angle.

#### 4.1 Characteristics of the toolpath in the ATS

In order to have a clear view, tolerances of the interpolation error and the RTM were set as *ε*
_{f} = 100 nm and *ε*
_{s} = 200 nm when characterizing the adaptive toolpath for the ATS. The micro-structured surface and the corresponding updated CCP are illustrated in Fig. 4(a)
. After conducting the offsetting, the CLP for the surface is further illustrated in Fig. 4(b). To better describe features of the sampling and side-feeding motions, the projected toolpath for the FTS/STS and the ATS is presented in Figs. 4(c) and 4(d), respectively. As shown in Fig. 4(c), the trajectory for the FTS/STS featured regular spiral curves with constant angle interval. Meanwhile, the adaptive toolpath for the ATS possessed variable sampling intervals with more even arc-length at different positions as shown in Fig. 2(c). Besides, the side distance between any two successive CCPs in the cross-sectional plane also varied with strong dependence on surface shapes, resulting in the irregular distribution of the adaptive CLP.

To characterize the required motions for the machining, the sampling interval Δ*φ*
_{w} and the motion component *z*
_{CLP} and *ρ*
_{CLP} as defined in section 3.4 are extracted in Figs. 5(a), 5(b) and 5(c)
. It is to be noted that the sampling strategy in the first revolution followed the scheme of CASS, thus, a set of constant values are observed at the initial stage as shown in Fig. 5(a). An exponent decrease tendency with imposition of strong fluctuations is also observed, with the position varying from the center to the outer space. Moreover, the sampling variation becomes more and more intensive with the increase of the rotation radius. This is due to the reason that more shape variation occurs in one revolution at the outer space. With the motion of the *x*-slide shown in Fig. 5(c), it can be further decomposed into the best-fitted linear motion component and the fluctuations as shown in Fig. 5(d). As discussed above, the linear motion component can be implemented by the *x*-slide of the machine tool with constant feedrate, while servo motion along the *x*-axis direction realizes the fluctuations. Compared with the required servo motion along the *z*-axis, the *x*-axial servo motion possesses similar complexity. In detail, the amplitudes along the two directions remain at the same scale, while the actual values might be dependent on the user-set RTM tolerance and surface curvature variations. By combining the biaxial servo motions synthesized by the rotation axis (*c*-axis), the novel ATS diamond turning for freeform optics can be effectively operated. Inheriting from features of the servo motion in conventional FTS/STS, bi-axial servo motions with relatively high bandwidth along both *x*- and *z*-axis directions in the 2-DoF FTS/STS endows the cutting system with better acceleration and deceleration performance, making it more suitable for adopting the adaptive scheme when comparing with conventional 5-axis CNC machining.

#### 4.2 Characterization of surface generation in the ATS

Taking advantage of a novel surface generation algorithm [26], numerical simulation of surface generations in both the ATS and the FTS/STS was conducted to investigate the resulting form errors. The diamond tool and surface shape for simulation was the same as the one employed in last section, and the aperture of the workpiece was 6 mm. As for the ATS, the interpolation and RTM tolerances were set as *ε*
_{f} = 50 nm and *ε*
_{s} = 10 nm for the determination of the adaptive toolpath. The accurate toolpath determination strategy developed in Ref [26]. was employed for generating the toolpath for the FTS/STS. The feedrate for FTS/STS was set as *f* = 6.4μm/rev to have the same number of revolutions as that in the ATS. The sampling number for FTS/STS was set as *N*
_{s} = 1800 to have the same number of control points as that for the ATS.

By subtracting the desired surface, error maps of the theoretically generated surfaces for the ATS and FTS/STS are illustrated in Figs. 6(a) and 6(b)
, respectively. With the surface generated by the ATS, the difference of error distribution was only determined by the concavity and convexity of the desired surface. For the area with concavity, range of the principle form error was *P*e$\in $[*ε*
_{f}, *ε*
_{s} + *ε*
_{f}]. Meanwhile, that of the area with convexity was *P*e$\in $[-*ε*
_{f}, *ε*
_{s}-*ε*
_{f}]. From the error map shown in Fig. 6(a), there are two kinds of uniformly distributed errors. A uniform distribution of the principle error in areas with the same concavity or convexity was well observed, well showing the effectiveness of the proposed adaptive strategy. From the error map obtained in the FTS/STS as shown in Fig. 6(b), periodic change of the form error which may strongly vary with respect to surface shape variations was observed, showing an intensive non-uniformity of the errors.

To have a much clear comparison, the resulting form errors obtained in the ATS and FTS/STS were projected to the *xz*-plane as shown in Figs. 6(c) and 6(d), respectively. As shown in Fig. 6(c), the form error was constrained in a narrow band with two flat boundaries ranging from −14 nm to 62 nm. Both the distribution features and the corresponding boundary values were as expected. From the projected error in FTS/STS as illustrated in Fig. 6(d), the upper and lower boundaries were approximately two harmonic curves. The maximum value of the upper boundary was about 116nm, which was about twice of the minimum value of the upper boundary (54nm). This suggests that there is a strong variation of the form error for freeform surface with respect to the constant side-feeding motion. As for the lower boundary with negative values, it was the interpolation error deriving from the sampling operation that varied from −30nm to 0, also suggesting a high dependences of interpolation error on the local shapes of the desired surface.

On the other hand, to make the PV value of RTM be within the desired tolerance, the maximum feedrate for the FTS/STS was estimated to be *f* = 4.5 μm/rev according to the analytical formula in Ref [22, 33
]. Meanwhile, by taking the minimum angle interval in the ATS, the minimum sampling number was estimated to be 4680 for the FTS/STS to make the interpolation error be within the desired tolerance. Thus, the required number of revolution for the ATS was 467 which was about 70% of that for the FTS/STS (667). Meanwhile, the number of control points for the ATS (789,413) was only about 25% of that for the FTS/STS (3,121,560). Due to the accumulative effects of more revolutions and more control points, the required machining time for the FTS/STS would be much longer than that for the ATS to have similar machining accuracy.

## 5. Experimental results and discussion

#### 5.1 Experimental setup

The sinusoidal grid micro-structured surface with *A*
_{x} = *A*
_{y} = 1.25 μm and *f*
_{x} = *f*
_{y} = 2.5 mm^{−1} were employed for investigating the performances of the ATS diamond turning. To have a comparison, the surface was fabricated by both the ATS and the STS diamond turning. In practice, the interpolation and RTM tolerances were set as *ε*
_{f} = 30 nm and *ε*
_{s} = 10 nm for the ATS. The aperture for both cutting was 5 mm. Similarly, the feedrate for the STS was set as *f* = 5 μm/rev to have a same number of revolutions as that of the ATS. The sampling number for the STS was set as *N*
_{s} = 1200 to meet the same tolerance requirements as that in ATS.

The cutting experiments were performed on a CNC ultra-precision lathe (Moore Nanotech 350FG, USA). The hardware configuration of experiment setup is shown in Fig. 7
. A commercial natural single crystal diamond (Contour Fine Tooling, UK) was used in the cutting. It had a nose radius of 0.104 mm and a rake angle of 0°. After cutting, the surfaces were cleaned with alcohol to remove the attached chips. The Optical Surface Profiler (Zygo Nexview) was employed to capture the topographies of machined structures with effective magnifications. To map a large area beyond the measurement range, a small group of images were stitched together by using the software system Mx^{TM}. After the measurement, a novel robust metaheuristic method was adopted for characterizing the resulting form error of the practical surfaces. More details of the method will be discussed in our future work.

#### 5.2 Results and discussion

The ATS was realized on the ultra-precision lathe by adopting the servo motions along the *x*-axis, namely the 2-DOF STS technique. The following error for the two servo motions was observed from the control panel to be less than 10nm during the machining. The obtained surfaces by using the ATS and STS were captured by the Optical Surface Profiler with an amplification of 20 × and further illustrated in Figs. 8(a) and 8(b)
. By subtracting the desired surface, the obtained error maps for the two surfaces are shown in Figs. 8(c) and 8(d). As shown in Fig. 8(c), a relatively uniform distribution of the form error was achieved which agree well with the theoretical one as illustrated in Fig. 6(a). The PV value of the form error was much higher than the preset theoretical one. This is because the preset tolerance is simply the constraint for the principle error, without consideration of practical error motions of the hardware system. From the error map obtained by the STS as shown in Fig. 8(d), four regions (marked by *A*, *B*, *C* and *D*) were observed with obviously different heights of the form error, which was similar with that obtained in simulations as shown in Fig. 6(b) and might correspond to the four valleys of the original surface as shown in Fig. 8(b). This demonstrates that the error distribution of the surface obtained by the STS is highly dependent on the shape of the desired surface. Moreover, the comparison also verifies the effectiveness of the ATS for achieving uniform surface quality which might be free from surface shapes.

More details of the experimental results are summarized in Table 1 . The total number of control points for the ATS was 383,000, while that for the STS was 600,000. This suggests that conducting the adaptive scheme could reduce the volume of data points by one third. Furthermore, the required machining times for the two surfaces also suggested that almost one third of the cutting time could be saved by adopting the ATS. In addition, the PV and RMS values of the form error as well as the surface roughness obtained by the ATS were all slightly smaller than that obtained by the STS. The results suggest that the ATS is a novel superior machining method, which could not only improve surface quality, but also enhance the machining efficiency. It is to be noted that the differences of surface roughness between the two methods obtained practical experiments are not large enough compared to that in simulations. In practical experiments, the desired surface was much smoother with smaller curvature changes, and the maximum sampling number was employed in STS to meet the interpolation tolerance. Thereby, the two factors may jointly contribute to the deviation.

## 6. Conclusions

By means of the newly developed two-degree-of-freedom (2-DOF) slow tool servo (STS) or fast tool servo (FTS) systems, this study proposes a novel adaptive tool servo (ATS) for the generation of complex freeform optics to improve surface quality and simultaneously enhance machining efficiency. The main conclusions are as follows:

- a) By computing the local curvature along both forward cutting and side-feeding directions at a given cutting point, the optimal sampling and side-feeding motion adapting to surface shapes can be determined. By iterating the process for all the cutting points, adaptive cutting motion for the whole surface can be obtained.
- b) To adapt to surface shapes, the side-feeding motions can be decomposed into linearly feeding and rapid fluctuation components. The fluctuation requires another servo motion along the
*x*-axis of the machine tool. Thereby, 2-DOF STS/FTS systems are required to practically realize the ATS diamond turning method. - c) A theoretical comparison of surface generation in both the ATS and FTS/STS is conducted. The results indicate that the uniformity of principle error distribution can be significantly improved by adopting the ATS. Moreover, with acceptable form error, less number of spindle revolutions and toolpath control points are required for the ATS when compared with the FTS/STS, which will significantly increase the machining efficiency.
- d) By employing the 2-DOF STS diamond turning, a comparison investigation on surface generation in ATS and STS was experimentally conducted on a typical micro-structured surface. By adopting the ATS, about one third of both the number of toolpath control points and the machining time can be saved. Simultaneously, surface form error as well as roughness can also be slightly improved.

## Acknowledgments

The work described in this paper was supported by the Research Committee of The Hong Kong Polytechnic University (RTJZ) and the Guangdong Innovative Research Team Program (grant number 201001G0104781202).

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