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Abstract
Slow tool servo diamond turning has widespread application in fabricating freeform optics. Previous studies are focused on the methods of the tool path generation and verification of zero-rake-angle tools. However, these methods are unsuitable for non-zero-rake tools that are used for machining hard-and-brittle materials. This paper presents a universal location-point-drive tool path generation method, which caters to arbitrary rake angle tools and the steady X movement feature, and the corresponding universal tool interference check method. Systematic analysis and ultra-precision machining experiments confirmed the feasibility of our methods and present better surface quality and form accuracy compared to the traditional method.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With continuous developments in optoelectronics, optical communication, and photonics, there has been a dramatic increase in the application of optical components [1,2]. Owing to the critical significance of the properties of optical systems in these applications, there exists a need for the use of high-performance constituent optical elements [3,4]. Traditional optical systems usually comprise spherical or aspherical lenses, which not only increase the system size, but also limit the optical performance. Optical freeform surfaces, a revolutionary development in the optical field, offer excellent optical-imaging performance by significantly eliminating optical aberration, increasing depth of field and expanding field of view [5,6]. Moreover, freeform optics facilitate flexible design, and make the optical system more compact compared to traditional optics [6,7]. Owing to the non-rotational symmetry of freeform surfaces, the machining tool path for generating these optical surfaces is generally complex and requires the cooperation of three or more axes. Slow tool servo (STS) and fast tool servo (FTS) diamond turning [8] are widely applied in freeform machining for their easy configuration and ideal quality [9]. STS is frequently used in large-sag surface machining owing to the long Z-axis travel distance, whereas FTS has a prominent advantage in machining efficiency for smoothly changing freeform surfaces. There is a growing body of literature that concentrates on promoting the machining quality of STS/FTS through closed-loop control [10], online measurement [11], and adaptive turning [12]. Kinematic research, including tool path generation and tool interference check, has had considerable impact on the final machining result and has drawn increasing attention in the past few decades [13].
Machining tool paths are determined by the forms of the workpiece and the tool geometry; they are fundamental to STS/FTS diamond turning. The tool path is obtained and interpolated with a set of cutter location points (CLPs), which is coincident with the diamond tool tip or radius center by default. Moreover, the diamond tool contacts the workpiece at the cutter contact point (CCP), which varies along the cutting edge according to the variations in the workpiece surface morphology and does not overlap with CLPs normally. In recent years, a considerable amount of literature has focused on freeform tool path generation. According to whether CCPs or CLPs are directly obtained by spiral projection, the existing tool path generation methods can be divided into contact-point-drive tool path (CTP) and location-point-drive tool path (LTP) generation methods [14].
The CTP method assumes that the trajectory of the CCPs is the result of projecting a planar spiral onto a freeform surface. CCPs can be directly obtained by discretizing a continuous spatial spiral. However, CLPs must be computed by tool geometry compensation at CCPs in the surface normal direction [15]. Over the last few decades, researchers have explored applications of the CTP method by means of a rotating coordinate system [16], mesh segment turning [14], and adaptive feed speeds [17]. Various freeform surfaces have been successfully fabricated, including radial sinusoidal surfaces [18,19], off-axis aspheric surfaces [16,20], and compound eye arrays [21,22]. As normal compensation always has X/Z-direction components, the CLP trajectory has high-frequency oscillation and significant dynamic response errors along the X/Z-axis. This implies that the CTP method is only suitable for continuous smooth surfaces preferably with no corners or steps. Although X-Z dual-direction FTS systems show effective improvement over the traditional CTP systems, they also increase the cost and decrease the system stiffness [23,24]. Furthermore, previous studies on the CTP method treated all types of cutting edges as ideal circles [22]. This is only accurate for conical diamond tools with zero-rake-angle or special cylindrical tools with a specific negative rake and flank angle. The cutting edge of non-zero-rake angle conical tools and most cylindrical tools is, instead, an ellipse. Therefore, the traditional CTP method, inevitably, introduces systematic errors and deteriorates the shape form accuracy, as we prove in our experiment.
The LTP method, conversely, assumes that the trajectory of the CLPs is the projection result of the plane spiral. The tool geometry compensation process can be removed by utilizing the contact-based geometrical relationships between the cutting tool and the workpiece to determine the projection distance and calculate the coordinates of the final CLPs. Furthermore, all the tool oscillations are transferred to the Z-axis, which results in steady X-axis movement. In contrast to the CTP method, there is much less research concentrated on the LTP method. Yu et al. [25,26] and Ji et al. [20] studied the tool path generation of a zero-rake tool with normal installation. Gong et al. [27] discussed the LTP generation method of a zero-rake tool with tilted angle installation. However, no previous study has investigated the LTP generation method of built-in non-zero-rake tools, which have been widely used for machining hard, brittle and other hard-to-machine materials [28,29].
This paper proposes a novel universal location-point-drive tool path (U-LTP) generation method that can be applied to arbitrary rake angle tools and yields steady X movement features. The U-LTP method can obtain the tool path directly by symbolic calculation with freeform surface expression. This is more accurate than most commercial computer aided manufacturing software that can only be applied to non-uniform rational basis spline models [27]. Furthermore, the cutting edge of the diamond tool is designed based on the actual model. Thus, the U-LTP method is more accurate than the traditional CLP method, while avoiding systematic-compensation errors and guaranteeing machined-form accuracy. In this study, a cosinusoidal surface, was machined as an example of freeform optics to verify the superiority of the proposed U-LTP method.
In addition, the tool interference check is another core issue encountered during diamond turning. The choice of reasonable tool parameters can eliminate overcutting and undercutting during the machining process [30]. It is essential to check for the possibility of tool interference during the machining process. Previous research mostly utilized the sectional curve method [15] to check the tool radius and the included angle before machining. However, the same method cannot be applied to arbitrary rake angles tools because of their more complex and aggressive geometry, which tends to interfere at not only the tool curvature but also the tool rake and flank surface. Thus, it is necessary to consider a three-dimensional interference check process, while fully considering the tool morphology. In addition to U-LTP, this paper presents a universal tool interference check (U-TIC) method that can verify the interference risk of the rake face, flank face, and cutting edge simultaneously. The U-TIC method was performed before the machining experiments, as described in subsequent sections.
The rest of this paper is organized as follows. Section 2 describes the proposed U-LTP and U-TIC methods. Section 3 introduces the specific tool interference check and tool path generation cases. The feasibility of both methods is verified via STS experiments described in section 4. Finally, major conclusions drawn from this are presented in section 5.
2. Method
2.1 U-LTP generation method
An ultra-precision machining lathe is usually composed of three axes as shown in Fig. 1. Workpiece rotation is controlled by the C-axis spindle. The X- and Z-axes drive the workpiece and the tool to have linear movements, respectively, realizing material removal. As a result, the cutting tool follows a spiral trajectory, relative to the workpiece. According to the motion features, the lathe coordinate system OL-XLYLZL can be established using the origin point OL, which is always assumed to be coincident with the workpiece center OW. For the sake of formal unity, the diamond tool front tip is set as the CLP in this study. The diamond tool coordinate system OT-XTYTZT can be obtained when the OT is located at the diamond tip and the XTYTZT axis is always parallel to XLYLZL. The α, β, and r of the tool in Fig. 1 correspond to the nominal rake angle, clearance angle, and nose radius of the diamond tool, respectively. Furthermore, the workpiece coordinate system OW-XWYWZW can also be established with the initial position coincident with OL-XLYLZL. The workpiece coordinate system is fixed on the freeform surface and will rotate following the lathe spindle.
The tool path generation strategy of U-LTP method is illustrated in Fig. 2. Firstly, the parameters of diamond tool and machined freeform optic should be determined, so as to obtain their expressions in the corresponding coordinate system. In order to facilitate numerical calculation, the expressions should be converted into the same lathe coordinate system. Initially, the diamond tool is lifted into the safe plane, which is located above the freeform surface center with zd distance. Since the diamond tool follows an isometric spiral trajectory relative to the workpiece, the planar spiral tool path can be generated according to the optic aperture and machining feedrate. And the planar spiral can be discretized to obtain the points P as the initial position of CLPs according to the rule of constant-angle, constant-arc, mixed constant-angle-arcs [18], or the constant error method [31]. With the obtained mathematical expression of the cutting edge and workpiece at the ith discretized point Pi, the machining tool path of an arbitrary rake angle tool can be calculated through projecting the cutting edge along the –ZW direction, as illustrated in Fig. 3. The final position of the CLP can then be determined by the terminal condition, i.e., when the diamond tool contacts the workpiece surface at the CCP, as shown in subfigure II of Fig. 3. The CLP and CCP of the diamond tool are separated, which can be seen in the rake face cross section, as plotted in subfigure I of Fig. 3.
With the parameters of freeform optics and the diamond tool, the CLP coordinates can be computed successfully using the U-LTP method. In the following formula, the symbol subscript represents the coordinate system, and the superscript represents the object. Specifically, the freeform optic expression is established in the workpiece coordinate system as:
The cutting edge of the cylindrical diamond tool becomes:2.2 U-TIC method
An interference check is indispensable for preempting tool cracks and surface damages. It is necessary to verify whether there is an interference risk with the existing tools during the optical machining process, which occurs typically at tool curvature, rake face, and flank face, represented by the red volume in Fig. 4(a), 4(b), and 4(c), respectively. This paper presents a U-TIC method that utilizes the minimum distance between the tool and workpiece surfaces as the verification criterion.
Fig. 4. Tool interference at the (a) curvature, (b) rake face, and (c) flank face. (d) Diamond tool geometry depicting the rake, flank, and optional secondary flank faces.
Assuming the coordinates of the ith CLP to be (XCLi, YCLi, ZCLi), the corresponding rake face can be expressed as:
Nowadays, diamond tools with secondary flank surfaces are widely applied in minute concave surface machining, as shown in Fig. 4(d). The secondary flank surface design can effectively avoid interference while simultaneously maintaining tool stiffness. For a diamond tool with a secondary flank surface, the U-TIC method is still applicable. The secondary flank surface expression is as follows:
3. Analyses
Considering the variation features of the surface normal vector, the typical cosinusoidal freeform surface was selected as the workpiece with a diameter of 19 mm:
where Ax = Ay = 0.015 mm, λx = λy = 4 mm.3.1 Interference check
For the diamond tool with a nominal 1 mm nose radius, –25 ° rake angle, 1 ° clearance angle, 0.15 mm rake face height, and 0.5 mm flank face height, we observed flank surface interference, depicted by the red points in Fig. 5. When the clearance angle was set to 10 °, no interference was observed. This implies that the tool parameters were feasible for the aforementioned cosinusoidal surface machining. Hence, the same tool parameters were considered during experiments performed in this study.
3.2 Tool path analyses
To compare the characteristics of the conventional CTP method and the novel U-LTP method proposed herein, we analyzed the tool paths generated by the two methods and subsequently used them to machine freeform optics. The mixed constant-angle-arc sample method was applied with an arc length of 0.07 mm and a maximum angle of 0.7 °. After the interference check, the nominal 1 mm nose radius cylindrical diamond turning tool, with a rake angle of –25 ° and a clearance angle of 10 °, was taken for the calculation.
The tool path was analyzed with time and frequency domain information. The X/Z coordinates of the CLPs varied with the rotation angle, as plotted in Fig. 6, where the red solid line represents X coordinates, and the blue dotted line represents the Z coordinates of the CLPs. The tool path generated by the CTP method contained X-direction oscillation, as shown in Fig. 6(a). Because of the larger X direction movement inertia of the lathe, there is always a worse dynamic response error. This drawback does not exist in the tool path generated by the U-LTP method, as shown in Fig. 6(b). It can be concluded that in the proposed method, X coordinates have a linear relationship with θ, which is indicated by the steady movement of the tool in the X direction, without any oscillation. Moreover, the Z oscillation did not markedly increase. This proves the stability of the machining process and machining accuracy when using the U-LTP method.
Periodogram power spectral density (PSD) analysis with 720 points/s reading speed was employed on the tool paths generated using CTP and U-LTP methods, as shown in Fig. 7. The Z component spectral characteristics of both tool paths showed little diversity. Further, the X component spectral characteristics of the U-LTP method contained only a single dominant frequency. However, there were multiple under-frequencies for the X component in the CTP method, indicated by the unsteady oscillation along the X-axis during the machining process.
Fig. 7. PSD analysis of (a) X component of the U-LTP method, (b) Z component of the U-LTP method, (c) X component of the CTP method, and (d) Z component of the CTP method.
4. Experiment
We conducted comparative experiments involving both the CTP and U-LTP methods to further investigate their actual performance. More specifically, the machining following error was analyzed to compare the dynamic features of the tool paths. The surface roughness and form error of the machined optics were measured to compare the machining quality.
The STS turning experiment was set up as shown in Fig. 8(a). A 19 mm diameter chalcogenide glass (As40Se60) workpiece was fixed on the aluminum substrate, which was vacuum-chucked onto the spindle of the ultra-precision lathe. Two cylindrical diamond tools were used specifically in the rough and finish machine processes. The nominal values of the nose radius, rake angle, and clearance angle were 1 mm, –25 ° rake angle, and 10 ° flank angle. To prove the superiority of the U-LTP method and eliminate the influence of tool wear, one newly finished diamond tool was used to machine #1 optic part using the CTP method firstly, and then, #2 optic part using the U-LTP method, as shown in Fig. 7(b) and 7(c), respectively. The surface can be expressed by Eq. (8) with a maximum sag of 60 µm and a period of 4 mm in both the X and Y directions. The finish turning with a feed rate of 2.0 µm/rev, a 2 µm deep of cut and a spindle speed of 70–75 rpm was adopted in the comparative experiment and maintained invariantly. For clarity, the experimental parameters are summarized in Table 1.
Fig. 8. (a) Schematic of STS experiment setup, (b) CTP machined #1 optic part, and (c) U-LTP machined #2 optic part.
Table 1. STS machining experiment parameters
4.1 Following error
We collected the following error of the first 360 seconds in the machining processes and plotted it, as shown in Fig. 9. It is observed that the X-axis following error incurred when employing the CTP method far exceeds that corresponding to the U-LTP method. The enveloping curve of the X-axis following error incurred using U-LTP remains within the -1.88 nm to 2.19 nm range, whereas that corresponding to the CTP method remains within the -74.65 nm to 68.54 nm range. In the meantime, the Z-axis following error for both tool paths remains nearly identical, i.e., within the -110 nm to 110 nm range. These following error results further prove the superior machining dynamics of the U-LTP method, which greatly facilitates stable freeform machining.
4.2 Surface finish
The CTP and U-LTP methods were used to machine two optic parts, as shown in Fig. 8(b) and 8(c), respectively. Both workpieces were machined in the ductile regime and the surface roughness was measured using the white light interferometer (Contour GT-X). To obtain a moderate scanning field measuring of 0.220 × 0.165 mm2, 20× objective and 2× zoom lenses were used under in the vertical scanning interferometry measurement mode. Twenty-one areas on the machined optics were selected to measure the surface roughness, as shown in Fig. 10(a). Owing to the tool position and alignment error, both workpiece center areas left a machining tumor, which further influenced the central machining quality. As observed, when the central (0, 0) measurement result is ignored, the average surface roughness of the #2 optic part equals 9.271 nm, whereas that for the #1 optic part equals 13.426 nm. Compared to the conventional CTP method, the freeform surface finish is improved by 44.81% when using the novel U-LTP method. Figure 10(b) and 10(c) denote the morphologies of the #1 and #2 optic areas, respectively, located at (7, 0). As can be observed, there exist more apparent vibration traces on the #1 optic area compared to #2. This further indicates the instability of the traditional CTP generation method. No additional vibration traces exist in the #2 optic area.
Fig. 10. Results of surface roughness measurements performed on (a) multiple areas of two optics. (b) #1 optic area at (7, 0) and (c) #2 optic area at (7,0).
It is difficult to analyze the surface quality with amplitude-based parameters. Spectral distribution analysis was used to better represent randomly distributed peaks and valleys on the machined surface. The PSD of the surface roughness among the various surface frequencies was employed to analyze the quality of the surfaces.
The raw data were extracted from the same (7,0) area measurement results using white light interferometry. The measured information was decomposed by Fourier transformation into its component spatial frequencies to obtain the averaged PSD analysis results along the workpiece radial and turning direction, as shown in Fig. 11. It can be concluded that the averaged PSD signal intensity of the CTP machined optic part is larger than that of the U-LTP machined part. There are more spatial low-frequency components in Fig. 11(a), which refers to the more periodical components during the machining process. This can be attributed to the X-direction oscillation and would further affect the surface roughness. This conclusion is consistent with the dynamic characteristics of the CTP tool path, as shown in Fig. 7. Conversely, although the Z-axis following errors of CTP and U-LTP are only slightly different in terms of spatial amplitude, the X-oscillation of the CTP method affected the Z-axis dynamic response, leading to excessive peak strength of the cutting direction in the PSD analysis results, as shown in Fig. 11(b).
Fig. 11. Surface roughness PSD analysis along the (a) radial direction and (b) tangential direction.
4.2 Form error
The form error of the machined optics was measured using a freeform measurement instrument (UA3P-300). A diamond probe with a radius of 2 µm was used to scan the optics at a scanning speed of 0.8 mm/s. Finally, around 47800 measurement points of 16 mm diameter area were collected to analyze the form error. The #1 optic part machined by the CTP method showed a form error of 0.701 µm, whereas the U-LTP machined #2 optic part showed a form error of 0.476 µm, as shown in Fig. 12. Owing to the tool position error and build-in waviness error, the form error of the two optic parts presents the same pattern. The larger PV value of the #1 optic part can be attributed to the system error of the traditional CTP generation method. The tool geometry compensation of the CTP method is under the approximate assumption of an ideal circle cutting edge, which is, indeed, a varied curvature ellipse curve. By contrast, the U-LTP method generates a tool path using the accurate ellipse expression and leads to a more ideal form error.
5. Conclusion
The proposed study presents a tool path generation method along with a corresponding tool interference check method. The superior efficiency and machining quality of the proposed method were verified via systematic analysis and ultra-precision machining experiments performed on a typical cosinusoidal surface. The major conclusions drawn from this study are as listed below.
- • The U-LTP method is feasible for arbitrary rake angle cylindrical/conical tools, thereby supporting a wide range of applicability.
- • The tool path generated by the U-LTP method exhibited a steady, linear relationship between the X direction movement of the lathe and the rotation angle, θ, without any X direction oscillation, validating the machining stability and accuracy.
- • The U-TIC method successfully verified the rake, flank, and curvature interference for the cutting tool with one or more flank faces.
- • Experimentally, the surface finish was improved by 44.81% in case of the U-LTP method, while reporting lesser following error compared to the traditional CTP method.
- • A surface roughness of 9.3 nm and a form accuracy of 0.476 µm (in PV) were achieved on the chalcogenide glass with the U-LTP method, which is much better than the corresponding surface roughness of 13.4 nm and form accuracy of 0.701 µm (in PV) obtained using the traditional CTP method, further proving the feasibility of the proposed method.
Funding
National Key Research and Development Program of China (2016YFB1102203); National Natural Science Foundation of China (61635008); “111” project of the State Administration of Foreign Experts Affairs and the Ministry of Education of China (B07014); Postdoctoral Innovative Talent Support Program of China (BX20190230).
Acknowledgments
The authors would like to express their sincere thanks to Yongxu Xiang, Jiaming Dong and Zhen Li their assistance in the preparation of experiments performed in this study.
Disclosures
The authors declare no conflicts of interest.
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