1. Introduction

Fast tool servo (FTS) assisted diamond turning has been extensively regarded as a very promising technology for fabricating freeform optics and micro-structured functional surfaces due to its superior performances such as high efficiency of generating desired surface, high quality of machined surfaces, sufficient capacity of deterministically fabricating intricate surfaces and so on [1–3]. For decades, with the increasing complexity of machined surfaces and the scale shifts of their dominant features, more and more efforts have been devoted to enhancing performance of FTS systems to meet the ever-increasing fabrication requirements. Up to now, performance of FTS systems has been significantly improved, and various types of FTS systems with different motion configurations, different driving and guiding principles have been well developed. Taking advantage of the developed FTS systems, dominant featured sizes of the generated components can range from millimeter scales [4, 5] to micrometer scales [6, 7], and even to nanometer scales [1,8], and the operating bandwidth can even reach up to dozens of kilo-Hertz [9, 10]. By iteratively determining the maximum feedrate for a given surface, intricate surfaces with crack-free qualities can also be achieved on brittle materials, which shall extend the flexibility of FTS technique to a much wider spectrum of materials [11].

As for the FTS assisted diamond turning process, the cutting tool will follow a spatial spiral trajectory with constant radial spacing, resulting in the residual tool marks left on machined surfaces with strong periodicity in high spatial frequency domain. Unfortunately, the repetitive feature with high spatial frequency will give rise to undesired scattering effects on machined surfaces, leading to degradation performance of machined optics such as imagining distortions, breakup of high power laser systems and so on [12–14]. During the last two decades, the scattering behavior of diamond turned surfaces have been mentioned and characterized, and the relationship between the surface texture and the scattering effects has also been investigated. By means of experiments, Prof. Yi and his coauthors have studied the influences of turning parameters on scattering effects of obtained surfaces. According to their results, the first order diffraction intensity caused by the periodic tool marks can significantly decrease when the tool mark spacing is small enough [15, 16]. However, the machining efficiency should be disastrously decreased by adopting this method. For practical applications of these optics, certain laborious post-processing approaches, such as abrasive jet polishing, magneto-rheological finishing, bonnet polishing and so on, should be further employed to remove these periodic tool marks [17–20]. However, these post-processing approaches will greatly decrease the efficiency and increase the cost of optics. Moreover, it is generally difficult to perfectly implement these approaches on surfaces with intricate structures. Facing this dilemma, a much better solution should be to develop certain novel machining methods which can actively disturb the inherent periodicity of tool marks during the cutting process, accordingly achieving the scattering homogenization of machined surfaces.

Motivated by this, a novel pseudo-random diamond turning (PRDT) method is proposed in this paper to eliminate the periodicity of the residual tool marks during the turning process without loss of efficiency and accuracy, accordingly achieving scattering homogenization. The toolpath generation strategy (TGS) is introduced, and the corresponding theoretical surfaces are obtained and characterized by a novel surface topography generation algorithm (STGA). To implement the PRDT method, a novel two degree of freedom FTS (2-DOF FTS) is introduced, which possesses totally decoupled motions along the radial and axial directions of the workpiece. Machining experiments on flat and typical micro-structured surfaces are carried out to validate the effectiveness of the proposed PRDT method for fabricating surfaces with periodicity elimination and scattering homogenization.

2. Working principle of the PRDT method

Based on Harvey-Shack’s scattering theory, the scattering intensity of a rough surface will be proportional to the power spectral density (PSD) function of the micro-textures on the surfaces [12, 21]. It suggests that the scattering effects can be well homogenized by actively eliminating the periodicity of residual tool marks. Thus, one direct way is to actively adjust motions of the cutting tool with pseudo-random vibration modulation (PRVM) along the radial direction during the cutting process. As for conventional FTS, it has but one degree of freedom (DOF) of motion which is parallel to the C-axis of the lathe, without the capability to actively change motions of the tool in the XZ plane. Although a novel two DOF slow slide servo (S³) technique has recently been introduced [22], it will not be a good choice for modifying surface micro-textures in real time due to its relatively low operating bandwidth.

Motivated by this, a novel 2-DOF FTS based diamond turning technique is proposed and the configuration is illustrated in Fig. 1. As shown in Fig. 1, the machining system mainly consists of three key components, namely: a spindle to hold and rotate the workpiece, a X-axis and a Z-axis linear servo system to move the slide carriage of the lathe, and a 2-DOF FTS system mounted on the carriage serving as two auxiliary fast translation axes along the X- and Z-directions of the lathe. As for the fast motion of the 2-DOF FTS along the Z-axis, it can be regarded as the same one of conventional FTS, while the fast motion along the X-axis is especially adopted to actuate the cutting tool following pseudo-random motions along radial direction to break up the periodicity of the tool marks as shown in Fig. 1(b). Generally, the actual tool location position (TLP) is highly depended on tool contact position (TCP) for creating complex surfaces [23, 24]. So, as is illustrated in Fig. 1(b), the TCP variations induced by PRVM motions along the X-axis should be compensated along the Z-axis to avoid machining errors. Characterized by PRVM motions of the cutting tool, the proposed PRDT method can be constructed.

 

Fig. 1 Working principle of the PRDT method (a) Axis configuration of turning machine; (b) Schematic of the motions of the cutting tool.

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As discussed above, there are two key issues in constructing the PRDT method, namely actual generation of the pseudo-random toolpaths and the development of a 2-DOF FTS system with high performance, which will be detailed in next sections.

3. Actual TGS for the PRDT

3.1 Description of the cutting tool

The coordinate systems of the workpiece and cutting tool during turning process are illustrated in Fig. 2, where $o_{\text{W}}-x_{\text{W}}{y}_{\text{W}}{z}_{\text{W}}$ and $o_{\text{T}}-x_{\text{T}}{y}_{\text{T}}{z}_{\text{T}}$ denote the local Cartesian coordinate system of the workpiece and the cutting tool, respectively. In the local coordinate system of the workpiece, the desired surface can be expressed as $z_{\text{W}}=f{}_{\text{W}}(x{}_{\text{W}},y{}_{\text{W}})$. However, the turning process can be naturally expressed in cylindrical coordinates $(\rho ,\varphi ,z)$ [23–25]. Taking the following conversions: $x_W=\rho {}_{\text{W}}cos\phi {}_{\text{W}}$, $y_W=\rho {}_{\text{W}}sin\phi {}_{\text{W}}$, the desired surface can also be expressed as $z_{\text{W}}=g{}_{\text{W}}(\rho {}_{\text{W}},\phi {}_{\text{W}})$. In Cartesian coordinate system, the normal vector at any given point $(x_{\text{s}},y_{\text{s}},z_{\text{s}})$ can be expressed as

 

Fig. 2 Schematic of toolpath generation.

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Assume that an arc-edged cutting tool with nose radius R_T and rake angle γ₀ is used, as shown in Fig. 2, it can be better modeled in its local spherical coordinates $(R_{\text{t}},\gamma ,\theta )$, where it can be expressed as

To conduct numerical calculation, the cutting edge is uniformly discretized into N₀ + 1 points. The i-th point in its local coordinate is represented as $(x_i^{\text{T}},y_i^{\text{T}},z_i^{\text{T}})$, the corresponding tangent vector can be obtained by

3.2 Determination of the TCP

Similar to the discretization of the cutting edge, the rotational angle of the spindle is also uniformly discretized into N_s points. As for the l-th point in the k-th revolution of spindle, if ${\rho }_{k,l}\ge 0$, then it can be expressed in the cylindrical coordinate by

However, limited by dynamics of 2-DOF FTS system, the cutting tool cannot follow real pseudo-random motions. So, the generated pseudo-random sequence will be preprocessed by passing through a low-pass filter where the cut-off frequency is specially selected with consideration of the working bandwidth of the 2-DOF FTS system. In addition, the preprocessed sequence should be further scaled into the desired moving range

By means of the rotation transformation of vector, the point $P_i^{\text{T}}(x_i^{\text{T}},y_i^{\text{T}},z_i^{\text{T}})$ on the cutting edge in the local coordinate of the workpiece yields

The corresponding projection of this point on the desired surface $P_i^{\text{S}}(x_i^{(k,l)},y_i^{(k,l)},z_{\text{S},i}^{(k,l)})$ is

Substituting $(x_i^{(k,l)},y_i^{(k,l)},z_{\text{S},i}^{(k,l)})$ into Eq. (1), the normal vector can be obtained by

How to determine the TCP can be reduced to solving a position, at which the cutting edge is tangential to the desired freeform surface. Mathematically, it is to find a proper point on the cutting edge where its tangent vector is perpendicular to the normal vector of its projection point on the desired surface [25]. As for discrete calculation, the TCP can be determined by the following approximation:

3.3 An iteration strategy for tool path generation

As is evident from Eq. (10), the calculation accuracy will highly depend on the sample size of the cutting edge, namely N_s. Larger N_s will lead to more accurate TCPs, just at expense of much lower computation efficiency. As for the material removal of continuous surface, the arbitrary neighborhood TCPs will also possess continuous features. So, an effective zone containing the l-th TCP can be defined according to the (l-1)-th TCP. With the assumption that the proper index i of the (l-1)-th TCP is m^(k,l-1), an iteration strategy for the determination of the l-th TCP can be expressed by

As shown in Fig. 2, the TLP $P_L{}^{(k,l)}$ can further be obtained:

Following through all the k and l by repeating the aforementioned steps, the toolpath for the whole surface can be well generated.

4. Numerical investigation into the machined surface

4.1 Topography generation of the machined surface

Generally, there are two crucial folds determining performance of the developed TGS for the PRDT method, namely the forming accuracy and the periodicity elimination capacity. To theoretically examine the efficiency of the TGS, modeling the machined surface topography will be a crucial issue. In this paper, a novel STGA is proposed here, and the main procedure can be summarized as follows.

Step 1. Mesh the workpiece in the Cartesian coordinate plane $o_W-x_W{y}_W$ with resolutions ${\varsigma }_x$ and ${\varsigma }_y$, respectively. When the tool moves from the previous $P_L{}^{(k,l-1)}$ to the current $P_L{}^{(k,l)}$, the material removal zone, projected on the coordinate plane $o_W-x_W{y}_W$, is shown in Fig. 3. The grid nodes $P_{}^{\text{G}}$ in this zone swept by the tool are the points to be removed or to be remained.

 

Fig. 3 Schematic of the material removal.

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Step 2. Let $P_{m,n}^{\text{G}}$ be one of the grid nodes in this zone. The corresponding TLP $P_L{}^{\text{In}}$ can be obtained according to $P_{m,n}^{\text{G}}$ based on the linear interpolation method. Then the Euler distance $d=\Vert P_{m,n}^{\text{G}}-P_L{}^{\text{In}}\Vert$ between the two points can be obtained.

If $d\le R_{\text{T}}$, $P_{m,n}^{\text{G}}$ will be the point to be removed; otherwise, it should be reserved. If $P_{m,n}^{\text{G}}$ is the point to be removed, the corresponding z-coordinate should be adjusted to the z-coordinate on the tool nose.

Step 3. Following through all the TCPs by repeating the steps (1) and (2), the machined surface topography will be generated.

4.2 Characteristics of the machined surface

To examine the form accuracy and the periodicity of surface textures, a typical sinusoidal grid surface with z(x,y) = 0.005sin(2πx) + 0.005cos(2πy) mm is employed as the desired surface. The radius of the workpiece is 3 mm. The nose radius of the cutting tool is 0.5 mm, and the nominal rake angle is 0°. The feedrate of the carriage along the x-axis direction is set as 50 μm/rev. During the toolpath calculation, the discretization sampling numbers N_s is set as 720. As for the PRDT process, the pseudo-random vibrations along the x-axis are within ± 40 μm, and the cutoff frequency of the filter for preprocessing is set as 100 Hz. The obtained toolpaths for the conventional turning (CT) method and the PRDT method are illustrated in Figs. 4(a) and 4(b), respectively. By conducting the STGA, the machined surfaces with respect to the two generated toolpaths are obtained and illustrated in Figs. 5(a) and 5(b), and the corresponding error maps are shown in Figs. 5(c) and 5(d). To clarify the periodicity of machined surfaces, two-dimensional (2-D) profiles along the x-axis and through the origin are extracted and characterized by the PSD function. The extracted profiles and the corresponding PSDs of the two generated surfaces are illustrated in Figs. 5(e) and 5(f).

 

Fig. 4 Toolpaths of sinusoidal grid surfaces for (a) the CT method, and (b) the PRDT method.

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Fig. 5 Characteristics of machined surfaces (a) Surface topography generated by the CT method; (b) Surface topography generated by the PRDT method; (c) Error map generated by the CT method; (d) Error map generated by the PRDT method; (e) Features of the extracted 2-D profile of the surface generated by the CT method; (f) Features of the extracted 2-D profile of the surface generated by the PRDT method.

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As shown in Fig. 4(a), the toolpath of the CT method is the standard spatial spiral with constant spacing along the radial direction, while that for the PRDT can be regarded as the PRVM of the conventional toolpath, resulting in irregular spacing along the radial direction at any TLP as shown in Fig. 4(b). To generate the surface, the required TCPs of the CT method and the PRDT method are 51912 and 51516, respectively. The approximately identical numbers of TCPs indicate that there are no significant differences between the cutting efficiencies of the two methods.

From the generated surfaces shown in Figs. 5(a) and 5(b), the sinusoidal grid profile generated by the CT method is much smoother than that generated by the PRDT method, attributing to the constant feedrate of the CT process. The errors shown in Figs. 5(c) and 5(d) are just the residual tool marks, and no tendency form errors can be observed, well demonstrating the accuracy of the generated toolpaths. However, by adopting smaller feedrate and more discretization numbers N_s, the heights of the residuals can be further cut down. In addition, non-uniform distributions of residual heights can also be observed in Fig. 5(c), which may be caused by the non-uniform curvatures of the desired surfaces along both cutting and feeding directions. All these observations demonstrate the efficiency of the proposed TGS and the STGA.

From the resulted error topographies of machined surface, as shown in Figs. 5(c) and 5(d), regular tool marks are observed on the surface generated by the CT method, while irregular features with randomly distributed height and spacing can be observed on the surface generated by the PRDT method. More specified features of the two surfaces are characterized in Figs. 5(e) and 5(f). As shown in Fig. 5(e), the waviness of the residual tool marks along the x-axis is of strong periodicity with nearly constant amplitude, the location of the peak (f = 20 mm⁻¹) in the PSD diagram indicates that the periodicity is caused by the constant feedrate of the cutting tool along the radial direction. As for the profile shown in Fig. 5(f), the variable spacing between any two adjacent peaks and the variable values of the peaks depict a disordered profile. The corresponding PSD diagram shows that the peak in the location f = 20 mm⁻¹ is well eliminated, replaced by several chaotic peaks located in a wider domain with much lower frequencies. The results demonstrate that the PRDT method can effectively eliminate the inherent periodicity of the residual tool marks.

5. Development of the novel 2-DOF FTS system

A piezoelectrically actuated 2-DOF flexural mechanism with parallel configuration is developed for activating the cutting tool, schematic of the developed mechanism is illustrated in Fig. 6. As shown in Fig. 6, the mechanism consists of three key components, namely the two symmetric driving units (DUs) A and B, and the Z-shaped flexure guidance unit (ZFGU). The DU is a typical structure with a group of parallel and symmetric right circle flexure hinges. The ZFGU is constructed by a group of parallel and symmetric Z-shaped flexure hinges (ZFHs) and a platform holding the cutting tool, details of the structure and the moving principle of the ZFGU is illustrated in Fig. 7. As shown in Fig. 7, when the two DUs push the two ends of the ZFGU, long beams of the ZFHs cannot shrink straightly, rather, they will bend to accommodate the space decrease, resulting in motions of the platform along the z-axis. Simultaneously, the difference between the driving forces of the two DUs will induce motions of the platform along the x-axis direction, obeying the well-known differential moving principle (DMP). Thus, two-DOF decoupled motions of the platform can be achieved, accompanying with a superior advantage that each actuator may contribute to motions along both directions.

 

Fig. 6 Schematic of the designed 2-DOF FTS mechanism. 1. The base; 2. The right circle flexure hinge; 3. Piezoelectric actuator; 4. The cutting tool; 5. The Z-shaped flexure hinge.

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Fig. 7 Schematic of the ZFGU mechanism.

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Practically, the actuators are embedded into the linkage platforms of the ZFGU to achieve impact structure sizes. Capacity transducers (Micro-sense II 5300) are chosen for dynamic position measurements of the cutting tool. Two piezoelectric stack actuators (Polytec PI, Inc., Karlsruhe, Germany) are employed for the mechanism. The amplifier module PI E-617 with a nominal amplification factor 10 ± 0.1 is chosen to amplify the driving signal of the actuators. The measured displacement signals are gathered through the Power PMAC control card sampled at 0.45 ms, which is also used for closed-loop control of the system. Taking advantage of the decoupled motions in the two moving directions, the system is regarded as two single input and single output (SISO) sub-systems. A simple proportional, integral and derivative (PID) controller is developed for the SISO system. To reject vibrations and external noises with high frequencies, a simple low pass filter is also embedded in the feedback control loop. Additionally, a velocity feedforward compensation loop is also utilized to enhance tracking accuracies.

By constructing performance testing experiments, the stroke of the mechanism along the z-axis can reach up to 27.03 μm with a resolution of 14 nm, and bi-directional motions are achieved along the x-axis ranging from −8.379 μm to 7.544 μm with a resolution of 8 nm. The working bandwidth along the two directions can both reach up to 200 Hz by implementing sweep excitation tests. The obtained performance demonstrates that the developed 2-DOF FTS system is suitable for micro/nano machining of complicated surfaces.

6. Preliminary machining results and discussions

The newly developed 2-DOF FTS system is integrated into a Spinner SB/C-TMC precision lathe as shown in Fig. 8(a), front view and back view of the developed 2-DOF FTS mechanism are illustrated in Figs. 8(b) and 8(c), respectively. The cutting tool is a synthetic polycrystalline diamond (PCD) with the nose radius of 0.4 mm, nominal rake angle of 0°, clearance angle of 15°. The Zr-based bulk metallic glasses (BMG) is employed as the workpiece. During the cutting process, half of the region of the work surface is generated by conventional turning method and the other half is generated by the PRDT method to guarantee the identity of the cutting process. A flat surface and a typical micro-structured surface with sinusoidal wave along the radial direction (SWR) [11] are machined, and scattering features are examined by a simple testing method as shown in Fig. 9. During the scattering testing process, the workpiece is fixed on a holder and will be irradiated by a laser beam with proper incidence angles. The reflect lights will irradiate on a screen with certain scattering fringes. If energies of the scattering fringes are weaken or the fringes are gone, then it suggests that surfaces with scattering homogenizations are obtained.

 

Fig. 8 Configuration of the hardware of the PRDT system (a) The machining system; (b) Front view of the 2-DOF FTS; (c) Back view of the 2-DOF FTS. (1. The 2-DOF FTS mechanism; 2. The workpiece; 3. Spindle of the lathe; 4. Machine bed; 5. Slide carriage of the lathe; 6. The cover; 7. The PEA; 8. Height adjustment screws; 9. Flexure mechanism for height adjustment; 10. Preloading screws of the PEAs; 11. The diamond cutting tool; 12. Probes of capacity transducers; 13. Connection part for displacement measurement; 14. The base; 15. Fastening screws).

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Fig. 9 Principle of scattering tests.

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6.1 Flat surface machining

As for the flat surface machining, the spindle speed and the feedrate of the slide carriage are chosen to be r = 60 rev/min and f₀ = 5 μm/rev, respectively. The pseudo-random vibrations of the cutting tool along the X-axis are within ± 4.5 μm. The nominal cutting depth is 10 μm. Trajectories of the cutting tool along the two directions and the tracking performance of the 2-DOF FTS system are shown in Fig. 10. Overall, the cutting tool can well follow the desired pseudo-random trajectories. Simultaneously, small vibrations can also be observed along the Z-axis which may be excited by the cutting forces. To avoid the undesired vibrations, more sophisticated control strategies with high robustness and disturbance rejection capacities should be further developed in the future.

 

Fig. 10 Motions of the cutting tool. The upper and lower red lines denote the desired trajectories along the z and the x-axis, respectively. The black line and the blue line denote the practical motions of the cutting tool along the z and the x-axis, respectively.

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As for the scattering testing process, the wavelength of the laser beam is 632.8 nm and the diameter of the beam is about 700 μm. From the scattering results shown in Fig. 11, it can be obviously observed that the reflected light from zone A which is machined by the CT method will induce strong scattering fringes on the screen. While the reflected light from zone B which is machined by the developed PRDT method will well focus on a spot, verifying that the PRDT method can well homogenize the scattering effects of the turned surfaces.

 

Fig. 11 Observed scattering effects of the flat surface.

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6.2 Machining of typical micro-structured surfaces

A typical SWR surface with 500 μm wavelength and 4 μm amplitude is fabricated on the BMG material to investigate the generation and scattering homogenization capacity of the PRDT method for complex surfaces. The spindle speed and the feedrate of the slide carriage are chosen to be r = 30 rev/min and f₀ = 5 μm/rev, respectively. The pseudo-random vibrations are also within ± 4.5 μm, and the nominal cutting depth is 15 μm.

The obtained micro-topographies generated by the CT method and the PRDT method are captured by OLYMPUS OLS3000 and illustrated in Figs. 12(a) and 12(b), respectively. As shown in Fig. 12, surface generated by the CT method possesses regular structures with obvious periodicity, while the surface generated by the PRDT method presents non-regular behaviors along the radial direction. The roughness of the two machined surfaces are S_a = 28.2 nm and S_a = 49.1 nm, respectively. In view of the vibrations along the X-axis direction, the maximum feedrate between the l-th cutting point of the k-th and the (k + 1)-th revolution can reach up to 19 μm. If it happens, the height of local residual tool marks will be much higher than that of the CT method. Thus, it is reasonable that the roughness of the machined surface generated by the PRDT method is about double of that generated by the CT method. However, it should be noticed that the relative large roughness maybe partially caused by undesired vibrations along the z-axis. Thus, by using sophisticated tool positioning strategy with high robustness and disturbance rejection capacities, surfaces with optical qualities can be achieved.

 

Fig. 12 Micro-topographies on machined surfaces generated by (a) The CT method, and (b) The PRDT method.

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To have more specified view of machined surface properties, the scanned 2D profiles along three arbitrary lines, which are all parallel to the x-axis of the measurement system shown in Fig. 12, are extracted and then characterized by PSD function, as shown in Figs. 13(a) and 13(b). During the PSD analysis process, the trend components of the profiles are removed by polynomial fitting. As shown in Fig. 13(a), the profiles generated by the CT method appear to be regular with an identical spatial frequency about 0.2 μm⁻¹, which corresponds to the feeding of cutting tool. From the profiles shown in Fig. 13(b), irregular residuals can be observed. Comparing with the PSD results shown in Fig. 13(a), the frequency component caused by the tool feeding disappears, while the components with much lower frequencies are strengthened. The experiment results show a good agreement with the numerical simulation results, demonstrating the efficiency of the proposed PRDT method for actively eliminating the periodicity of surface textures.

 

Fig. 13 The extracted 2D profiles and the corresponding PSD features of surfaces generated by (a) The CT method, and (b) The PRDT method.

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The scattering effects of the obtained surface are further explored by the aforementioned testing method, the results obtained are shown in Fig. 14. It can be obviously observed that the reflected light from zone A which is machined by the CT method will induce strong scattering fringes on the screen. While the reflected light from zone B which is machined by the proposed PRDT method will well focus on a narrow ribbons with no dissipation fringes. These qualitative observations well demonstrate the efficiency of the developed PRDT method for fabricating complicated surfaces with scattering homogenizations.

 

Fig. 14 Scattering effects of machined surface.

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7. Conclusions

In this paper, a novel pseudo-random diamond turning (PRDT) method is proposed to fabricate freeform optics with scattering homogenization by means of actively eliminating the inherent periodicity of residual tool marks. The strategy for accurately determining the spiral toolpath with pseudo-random vibration modulation (PRVM) is deliberately explained. A novel two degree of freedom fast tool servo (2-DOF FTS) system with decoupled motions is introduced to implement the PRDT method. The main conclusions can be summarized as follows.