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Abstract
Chromatic confocal sensor-based on-machine measurement is effective for identifying and compensating for form errors of the ultra-precisely machined components. In this study, an on-machine measurement system was developed for an ultra-precision diamond turning machine to generate microstructured optical surfaces, for which the sensor probe adopts a uniform spiral scanning motion. To avoid the tedious spiral center alignment, a self-alignment method was proposed without additional equipment or artefact, which identified the deviation of the optical axis to the spindle axis by matching the measured surface points and the designed surface. The feasibility of the proposed method was demonstrated by numerical simulation with full consideration of noises and system dynamics. Practically, taking a typical microstructured surface as an example, the on-machine measured points were reconstructed after calibrating the alignment deviation, which was then verified by off-machine white light interferometry measurement. Avoiding tedious operations and special artefacts may significantly simplify the on-machine measurement process, thereby greatly improving the efficiency and flexibility for the measurement.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nowadays, microstructured optical surfaces have been widely used in aerospace, electronic, and biomedical fields owing to their outperforming optical performances induced by the surface structures at the microscale [1–3]. Due to the particularity of the topological structure of the microstructured surface, it is challenging to improve the form accuracy by conventional post-processing methods, i.e., polishing. Therefore, an effective technique is to iteratively correct form errors by on-machine measurement in turning the microstructured surfaces [4,5].
Recently, various on-machine measurement technologies have been developed, which can be mainly classified into full-field optical and probe scanning measurements [6]. Among those methods, the probe scanning measurement system based on chromatic confocal sensors (CCS) is widely recognized as a promising protocol in terms of its unique capability of capturing rough and specular surfaces with significant slopes [7–10]. Taking advantage of the compact size of CCS probes, a variety of CCS-based on-machine measurement systems have been developed for complex microstructured surfaces. For instance, Tong et al. developed a CCS-based measurement system for fast tool servo diamond turning to compensate for the form errors of micro-grooves [11]. Yu et al. employed two CCS probes for the on-machine measurement system, effectively reducing the peak-to-valley form error of the off-axis paraboloid mirror from 2.894 µm to 0.345 µm [7]. Wang et al. developed a CCS-based measurement system on a four-axis ultra-precision machine tool for measuring high-gradient freeform surfaces by adopting proper scanning trajectories with optimized parameters [8]. Similarly, Xi et al. [12], Chen et al [13]. and Zou et al [10]. integrated the CCS probe with the ultra-precision machine tool to realize the on-machine measurement of complex free-form surfaces.
For the aforementioned on-machine measurement systems, the spiral scan is formed by the workpiece rotation and the side-feeding of the CCS probe. The alignment accuracy between the optical axis and spindle axis is crucial for the measured form accuracy [14,15]. To realize accurate alignment, various probe center alignment methods were developed based on specially designed artefacts, for example, the oblique plane [16], reference sphere [10,12], and particular surface structure [13,14], to mention a few. For those methods, the center alignment is highly dependent on the particularly designed artefacts, and the probe must be fixed tightly during the iterative cutting, resulting in low flexibility and high system complexity.
In this paper, we developed a similar on-machine measurement system for a slow tool servo (STS)-based ultra-precision machine tool by integrating a CCS probe. To avoid using special artefacts and enhance the measurement flexibility, a center self-alignment method was proposed by matching the remote surface shape between the on-machine measured and design surfaces, which incorporated the center deviations to be identified. Since the alignment was conducted after the practical cutting without additional equipment and artefacts, re-installation of the probe can be flexibly conducted without affecting the surface reconstruction process. Furthermore, the self-alignment freed from tedious operations may significantly improve the efficiency of the on-machine measurement, as well as the iterative form corrections.
The remainder of the paper is organized as follows. Section 2 introduces the configuration and the rough alignment for the on-machine measurement system. Section 3 presents the detailed alignment process, including the influence, working principle, and numerical simulations, which are then demonstrated by experiments in Section 4. Finally, the main conclusion is drawn in Section 5.
2. Construction of the on-machine measurement system
2.1 System configuration
The on-machine measurement system is developed on an ultra-precision lathe, as shown in Fig. 1. The lathe is equipped with two rotational axes (B- and C-axis) and two linear axes (X- and Z-axis). The CCS probe is mounted on a height adjustment platform with a fixture, which is installed on the platform of the B-axis to provide height adjustment along the Y direction. Here, the height adjusting mechanism is an elaborately designed mechanism with both coarse and fine adjustments. The coarse mechanism can adjust the height in a few millimeters through the dip angle of Wedge 1. Meanwhile, through the small dip angle of Wedge 2, the fine mechanism can adjust the height within several hundred microns with a resolution at the submicron scale, which is guided by a parallelogram flexure mechanism. Owing to the flexure mechanism, it can achieve a height adjustment resolution at the sub-micron scale. The workpiece is held on the vacuum chuck of the spindle (C-axis), which is then carried by the X-axis slide to have a side-feeding motion. The relative position between the probe and the surface to be measured is adjusted by the Z-axis motion of the lathe to be within the measurement range. During the iteratively turning, the B-axis may rotate to switch the tool and probe for the operation.
Since the probe has an identical configuration to the diamond tool, a spatial spiral scanning trajectory in the Cylindrical coordinate system (ρ,θ,z) was implemented for the surface scanning. Assume the workpiece has a radius of ρ0, the spiral scanning trajectory can be expressed without considering the center misalignment as [17,18]
2.2 Rough positioning of the probe
Before measurement, the position of the optical axis of the probe related to the spindle axis needs to be calibrated. The parallelism between two axes will be guaranteed by the mating accuracy when assembling the probe on the height adjustment platform. To roughly align the probe along the X and Y axis, a two-step method was employed to position the optical axis of the probe as close to the spindle axis as possible.
In this study, the two-step rough alignment operates at the flat cutting process after the tool setting. When a flat surface is reclamped on the vacuum chuck, a slight inclination will happen. By rotating the inclined flat surface, the even feeding motion of the probe along the X-axis may have a periodic measurement signal for the CCS, which is caused by the surface inclination. Obviously, the position with the smallest amplitude may correspond to the center along the X-axis, which will be set as the X-axis zero point for the probe. After positioning the X-axis position, a rough adjustment along the Y-axis will be implemented with the rotated workpiece. Similarly, the vertical motion of the probe carried out by the height adjustment platform causes a periodic measurement signal for the CCS. The position with the smallest amplitude for the CCS may correspond to the zero point for the Y-axis. The two-step rough adjustment can adjust the relative position between the optical and spindle axes within 100 µm.
It is noteworthy that the two-step rough alignment is just required at the very beginning of the measurement. By recording the positions, direct measurements can be conducted even after removing and re-installation the probe, enabling the on-machine measurement to be implemented flexibly. The repeated installation accuracy for the probe re-installation is guaranteed by the fixture accuracy.
3. Ultra-precision alignment and compensation
The rough alignment accuracy is far from the requirement for surface measurement. To achieve an ultra-fine alignment, a self-alignment identification method is further proposed, which will be implemented after generating the desired surfaces.
3.1 Analysis of the alignment deviation
As schematically shown in Fig. 2, considering the alignment deviation δx and δy, the actual polar axis in the spiral scanning is
The Y-axis component of the alignment deviation will lead to the position-dependent polar angle change, which can be expressed as
Concerning the alignment deviations, the measured Z-axis coordinate ${z_a}$ corresponding to the actual CCS probe positions may satisfy the surface equation $S({x_a},{y_a},{z_a}) = 0$. The coordinate (xa,ya) is
Without considering the alignment deviation, the plane coordinates might be estimated as ${x_w} = \rho \cos \theta $, ${y_w} = \rho \sin \theta $, which may construct the measured point $({x_w},{y_w},{z_a})$ with relatively large form errors. Taking a typical harmonic grid surface as an example
With ${A_x} = {A_y} = 1$ µm, ${f_x} = {f_y} = 2.5$ mm-1, the ideal surface is shown in Fig. 3(a). Meanwhile, the re-constructed surfaces containing different alignment deviations are shown in Fig. 3(b) and 3(c). As shown in Fig. 3, the center alignment deviations will lead to position shifts and structure distortions for the surface patterns.
Fig. 3. Surface distribution characteristics corresponding to different alignment deviations. (a) $({\delta x,\delta y} )= ({0,0} )\mathrm{\ \mu m}$, (b) $({\delta x,\delta y} )= ({23,65} )\mathrm{\ \mu m}$, (c)$({\delta x,\delta y} )= ({65,23} )\mathrm{\ \mu m}$.
3.2 Self-alignment principle
Generally, the structure intensity is low at the center of the machined surface. At the central region, the low cutting velocity and low oscillation frequency for the tool servo system may lead to ultra-high form accuracy for the turned surface, suggesting that the resultant surface might be nearly identical to the designed one. Therefore, by incorporating the alignment errors into the surface reconstruction model $S({x_a},{y_a},{z_a}) = 0$ (Eqs. (2) to 4), the alignment errors can be identified by minimizing the difference between the measured points and the designed surface.
Assume the measured data from the CCS is ${z}_{p}({t} )$ in the time domain. From Eq. (4), the XY coordinates having an alignment error $({\delta _x},{\delta _y})$ are estimated as ${x_a}(t)$ and ${y_a}(t)$. By substituting the XY coordinates into the surface equation $S({x_a},{y_a},{z_a}) = 0$, the estimated Z-axis coordinate ${z_a}(t)$ of the designed surface can be estimated from. Therefore, the alignment identification can be converted into the following minimization problem
3.3 Numerical verification
To verify the effectiveness of the self-alignment method, a set of numerical simulations for three cases were carried out. In the three cases, the measured surfaces were simulated as ideal surfaces, idea surfaces with noises, and distorted surfaces caused by cutting dynamics. For all the simulations, the sampling points were set as N = 360 for each revolution of the spindle, and the feedrate for the probe was set as fr = 20 µm/rev. All the parameters for the DE solvers were set as the same in the minimization, and the search spaces for the two alignment deviations were identically set as ±0.1 mm. Without loss of generality, only the case with the CCS probe at the fourth quadrant of the coordinate system was discussed here.
In this case, only the measured points with alignment deviations were employed for the identification without considering any other disturbances. The randomly chosen alignment deviations for the simulation were summarized in Table 1, and the simulated measured points za were obtained by substituting the resultant xa and ya into Eq. (5). The radius of the selected central region was set as ${\rho _\varepsilon } = 0.1$ mm. After the minimization iterations, the identified alignment deviations and the identification errors were derived as presented in Table 1.
Table 1. Simulation results for ideal surfaces
As shown in Table 1, under the ideal conditions, the proposed alignment identification method can effectively find the alignment deviations with nearly no identification errors (less than 10 pm), suggesting a theoretically perfect method for the alignment identification.
The noise of measured data from the CCS was a digital signal. Therefore, random integers with uniform distribution were generated to simulate the noise. To simulate the noise, the practical CCS noise was measured in the on-machine environment with a sampling frequency of 1 kHz, which demonstrated a peak-to-valley value of 40 nm. Therefore, based on the actual results, the generated noise in the simulation is set to be within ±20 nm with an interval of 3 nanometres as shown in Fig. 4. To have a fair comparison, the alignment deviations used in the simulation are the same as those used in the ideal case. The resultant results are then summarized in Table 2.
Table 2. Simulation results for surfaces with noises
As shown in Table 2, under the influences of the noises, the identification error increases significantly from the picometer level accuracy under the ideal condition to tens of nanometres. However, the maximum identification error for all the simulated groups is about 68 nm, demonstrating its effectiveness for achieving sub-100 nm identification accuracy.
Form error induced by the tracking errors of the STS is the main error for a machined surface, especially for the small region at the center. To simulate the influence of form errors on the identification, the Z-axis slow slide was modelled as a first-order low-pass filter, which is defined as
where $\tau = {({2\pi {f_c}} )^{ - 1}}$ is the time constant for the slide. fc(Hz) is the cut-off frequency for the machine tool, which is set as 50 Hz according to the practical testing reported in Ref. [17].To simplify the surface generation process, the surface points passing through this model were directly adopted to reconstruct the surface for mimicking the machined surface with form errors. Note that the surface points were converted to the time domain by substituting the spindle speed and sampling frequency into Eq. (1), where the continuous measurement time was discretized into a number of time points according to the sampling frequency. With spindle speeds of 15 rpm, 30 rpm, and 90 rpm, the projected microstructured surfaces in the XY plane are illustrated in Fig. 5. As shown in Fig. 5, the phase lag caused by the slide dynamics may lead to a certain degree of distortion. In common, a higher spindle speed may lead to a larger phase lag, resulting in a more severe distortion of the machined surface.
Fig. 5. Machining surface corresponding to different spindle speeds, (a) spindle speed n = 15 rpm, (b) spindle speed n = 30 rpm, (c) spindle speed n = 90 rpm.
To investigate the influence of surface form errors on the identification, the identification results for spindle speeds of 15 rpm, 30 rpm, 60 rpm, and 90 rpm were calculated with ${\rho }_{\varepsilon }$=0.1 mm. In this case, the pre-set alignment deviations were fixed as ${\delta }_{x}$=42 µm and ${\delta }_{y}$=53 µm. The identification results were shown in Table 3. As shown in Table 3, the identification error may increase gradually as the spindle speed increases. Moreover, the identification error in the Y direction is more significant than that in the X direction. This is due to the fact that larger rotational speeds may lead to larger phase lags, which may cause larger torsional errors on the surfaces. In general, the torsional error might be coupled with the structural distortion caused by the Y-axial alignment deviation, as pointed out in Eq. (3).
Table 3. Simulation results of alignment deviations under different spindle speed (${{\rho }_{\varepsilon }}$=0.1 mm)
To further investigate the influences of the evaluation region size on the identification, the simulated data set within ${\rho }_{ \varepsilon}$=0.2 mm and ${\rho }_{ \varepsilon}$=0.05 mm was employed for the simulation. Using various spindle speeds, the identification results for ${\rho }_{ \varepsilon}$=0.2 mm and ${\rho }_{ \varepsilon}$=0.05 mm are summarized in Table 4 and Table 5, respectively. For the larger area evaluation (${\rho }_{ \varepsilon}$=0.2 mm), the identification errors are larger than those obtained within a smaller area. Meanwhile, larger spindle speeds may also lead to larger identification errors, due to the larger phase lags coupled with the structure distortion caused by the Y-axis alignment deviation. For the smallest evaluation area with ${\rho }_{ \varepsilon}$=0.05 mm, the maximum identification error occurred for a spindle speed of 90 rpm, which was about 0.943 µm. Therefore, sufficient small evaluation areas and spindle speeds were required to guarantee sub-micron identification accuracy.
Table 4. Simulation results of alignment deviations under different spindle speed (${{\rho }_{\varepsilon }}$=0.2 mm)
Table 5. Simulation results of alignment deviations under different spindle speed (${{\rho }_{\varepsilon }}$=0.05 mm)
4. Experiment results and discussion
4.1 Experiment setup
The developed on-machine measurement system is photographically shown in Fig. 6, which was established based on a four-axis ultra-precision machine tool. The measurement resolution of the CCS (Chrcodile 2S, Precitec, Germany) is 3 nm, and the spot diameter is 5 µm. The measuring range and the maximum slope for the CCS are 282 µm and 30°, respectively. As shown in Fig. 6, the CCS is first installed on the height adjustment mechanism with an elaborately designed fixture and then installed on the B-axis platform. A diamond tool (Contour fine tooling, UK) with a nose radius of 0.1 mm was fixed on a tool holder, which was installed perpendicular to the CCS probe. Therefore, a rotation of around 90 deg was implemented by the B-axis to enable the switch between the diamond turning and on-machine measurement. The basic performance of the developed on-machine measurement system is summarized in Table 6.
Fig. 6. Photograph of the developed on-machine system, where 1: diamond turning tool holder, 2: the height adjustment mechanism, 3: fixture, 4: workpiece, 5: workpiece holder, 6: vacuum chuck.
Table 6. Basic performance of the developed on-machine measurement system
To validate the proposed compensation method, the typical harmonic grid microstructure surface, as described in Eq. (5), was fabricated based on the STS function of the ultra-precision machine tool. The photograph of the machining process is shown in Fig. 7(a). In the machining, the spindle speed was set as 30 rpm, and the feedrate along the X axis was 2 µm/rev. The aperture of the machined surface was 5 mm. After machining, a white light interferometer-based optical surface profiler (Zygo, Newview 8200, USA) was employed for the measurement of surface topography, for which a 5×objective lens was adopted. The captured surface topography is then presented in Fig. 7(b).
Fig. 7. Results of a diamond turned micro-grid surface, (a) photograph of the turning process, and (b) the obtained surface measured by the optical surface profiler.
4.2 Surface measurement and reconstruction
For the measurement, the roughly aligned zero point for the CCS probe was defined as the zero point of the machine tool. The X-axis of the machine tool fed towards the defined zero point with a feedrate of fr = 20 µm/rev, while the Z-axis was kept stationary. The spindle speed was set as n = 100 rpm in the STS mode. The sampling frequency for the CCS was set as 1 kHz. Practically, the evaluation area was set as ${\rho }_{ \varepsilon}$=0.1 mm, and the search spaces for the X and Y axial deviation were set as ±0.03 mm and ±0.08 mm, respectively. Note that a slightly larger search space was set for the Y-axial deviation due to the low height adjustment accuracy of the adjustment platform.
After the iterative optimization, the alignment deviations were estimated as ${\delta }_{x}$=16.301 µm and ${\delta }_{y}$=57.890 µm. The reconstructed measured surface with consideration of the alignment deviations was illustrated in Fig. 8. Compared with the design points, a good agreement between the measured and design points was obtained with a deviation of about 43.4 nm (rms), demonstrating the effectiveness of the proposed alignment method.
After substituting the identified alignment deviations into Eqs. (2) to (4), the obtained XY coordinates $({x}_{a},{{y}_{a}} )$ and ${z}_{a}$ values from the CCS were employed to construct the on-machine measured surface. By recording related motions of the ultra-precision lathe, the on-machine measured surface points after the center alignment deviation compensation are illustrated in Fig. 9(a). Concerning the central area without scanning, the measured data was fused with the design surface to construct the complete surface through the regional edge intensity algorithm [19]. The resultant 3-D surface is illustrated in Fig. 9(b). The highly uniform structure well demonstrated the effectiveness of the proposed alignment deviation identification method, as well as the surface reconstruction method.
Fig. 9. The reconstructed surface from the on-machine measurement, (a) the on-machine measured surface points, and (b) the reconstructed 3-D surface.
To verify the accuracy of the on-machine measurement results, the reconstructed surface was compared with the surface form obtained by the optical surface profiler (WLI), which is shown in Fig. 7(b). In the comparison, the two measurement results were matched through the hybrid two-stage registration algorithm developed in Ref. [19]. After the registration, the two measurement results are presented in the same coordinate system, as shown in Fig. 10(a), showing a high coincidence with each other. Accordingly, the deviation between the two results is shown in Fig. 10(b), which was about 142 nm (rms). The comparison suggests that the proposed self-alignment method is effective for the spiral scanning-based on-machine measurement.
Fig. 10. Results of the on-machine measured surface, (a) the registration result, and (b) the form deviation between the on-machine measurement and optical surface profiler measurement.
5. Conclusion
A chromatic confocal sensor (CCS) based on-machine measurement system was developed on an ultra-precision machine for measuring microstructured optical surfaces. In the cylindrical coordinate system, a spiral scanning path was adopted for the CCS probe to measure the surface form. A self-alignment method was then proposed for identifying the deviations between the optical axis and spindle axis, which was based on the matching between the measured and design surface within a specified region at the center. Since the alignment was operated after the machining, it essentially avoided special artefacts and tedious operations. The effectiveness of the self-alignment method was well demonstrated by numerical simulations including cases with ideal surfaces, surfaces with noises, and surfaces with distortions. Theoretically, the proposed method can identify the alignment deviations perfectly.
To practically demonstrate the proposed method, a harmonic micro-grid surface was fabricated and characterized by the on-machine measurement system. For the alignment, the deviation between the matched measurement points and the designed surface was about 43.4 nm (rms), and the deviations were accordingly identified as ${\delta _x}$=16.301 µm and ${\delta _y}$=57.890 µm. Taking the surface measured by the white light interferometry as the benchmark, the deviation for the on-machine measured surface was about 142 nm (rms), demonstrating the feasibility of the proposed on-machine measurement system.
Funding
National Natural Science Foundation of China (51705254, U2013211); Outstanding Youth Foundation of Jiangsu Province of China (BK20211572); Shenzhen Natural Science Foundation University Stability Support Project (20200821110721002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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