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Abstract
The magnetorheological finishing (MRF) of surfaces often results in tool mark errors. A prediction model can effectively guide subsequent processing, necessitating thorough research. To address this issue, this paper introduces an enhanced continuous tool influence function method. This method involves sub dwell time convolution with varying tool influence functions, enabling tool mark prediction. Numerical simulations demonstrate the proposed method’s effectiveness, while the data size is estimated to confirm its economic properties. Subsequently, a MRF experiment was conducted, affirming the practicability through power spectral density evaluation. A fast algorithm is given to guide tool mark predictions on large-aperture mirrors fabrication engineering subjected to sub-aperture polishing.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the advancement of astronomy, there is an increasing demand for higher resolution astronomical telescopes. According to the Rayleigh criterion, angular resolution is directly proportional to the aperture, posing greater challenges for the fabrication of large-aperture mirrors. Currently, the manufacturing of such mirrors predominantly employs sub-aperture polishing technologies, including bonnet polishing, magnetorheological finishing (MRF), and ion beam figuring. A common characteristic of these methods is that the size of the polishing tool is significantly smaller than the mirror's dimensions. However, due to the periodic feeding path, these techniques may result in tool mark errors on the mirror surface [1]. Surface quality plays a crucial role in the performance of a machined part throughout its life cycle [2]. Errors in surface quality can impact the functional characteristics of the part [3], such as Optical Transfer Function deterioration in imaging systems [4,5]. Additionally, surface characteristics affect the energy absorption of laser systems owing to the coating's impact [6]. Predicting tool mark becomes essential to assess whether the surface will achieve the expected function before manufacturing, such that it can also participate in the optical system quality evaluation [7]. Furthermore, an accurate tool mark prediction is also helpful for tolerance analysis and tool mark desensitization design [8].
There are four primary methods for predicting tool mark on surfaces in material removal processes: analytical, numerical, experimental, and artificial intelligence (AI)-based methods [9]. In the realm of analytical models, a Gaussian mixture model method has been proposed to fit the tool influence function [10]. The analytic tool mark model is derived by convolving a primitive Gauss tool influence function with the dwell time. This method considers the interaction of cutting edge topography, material deformation, and cutting edge path error to achieve a reliable cutting tool mark prediction on ball end milling surfaces [11]. Furthermore, a theoretical model based on computational fluid dynamics was established to predict and characterize Free Jet Polishing material removal characteristics and surface formation, elucidating the tool mark formation mechanism under different polishing conditions [12]. Cha et al [13] considered the behavior of the ejected/redeposited melt and the non-linear interaction between successive pulses when a laser beam is fed along a given path. They presented a novel formulation to establish an analytical expression for groove tool mark. Additionally, a theoretical tool mark prediction model was developed based on the relationship between the valve cover cutter feed speed and material removal [14]. Analytical methods are suitable for lower-order derivable tool influence functions and surfaces, and can have good approximate tool mark predictions. In the domain of numerical methods, the predicted residual error on the surface is primarily obtained by subtracting the convolution of the tool influence function and path dwell time. To validate the PVT model for machine tool movement, the prediction method via up-sampled tool path has been utilized to simulate tool mark on different paths [15]. In exploring the suppression of mid-spatial error, several sets of simulations are conducted, establishing a simulation model for tool mark generated on the surface [16]. Numerical methods need to consider computational feasibility both in time and space. They are commonly used in small- and medium-aperture prediction tasks. In the realm of experimental methods, the study on the suppression effect of multi-pitch path polishing on tool mark involves obtaining multi-period tool mark through machining experiments [17]. To verify the dwell time algorithm's dynamic limitation in a MRF machine tool, a machining experiment is carried out, resulting in tool mark with a significant period [18]. Experimental methods incur in extensive material consumption, and the empirical rules can be summarized for reference. Regarding AI-based methods, an approach for implementing an ultrafast laser ablation simulator using a deep neural network was established. This simulator can calculate the 3D structure of tool mark created by multiple laser pulses irradiated at any position and with any pulse energy [19]. Additionally, an artificial neural network model for milling has been developed to analyze and predict tool mark based on the relationship between cutting conditions and the corresponding fractal parameters of the machined surface [20]. AI-based methods obtain the corresponding surface through extensive process practice, and attempt to establish the mapping relationship between the two with the aid of complex mathematical models.
For engineering tasks involving large-aperture mirrors without analytic tool influence functions, such as MRF, and lacking multiple experiments or large datasets for neural network training [21], numerical simulation methods are preferable. However, for large-aperture mirror manufacturing tasks, the existing dense path numerical methods require a massive amount of data, and the time to calculate the tool mark via the dense path method far exceeds the time allocated for the calculation by the engineering task. The main purpose of tool mark prediction is to change the parameters repeatedly to observe whether the tool mark reach the target. Therefore, such time consumption is not suitable for mass simulation experiments. The existing methods are not conducive to engineering use, emphasizing the need for a new fast tool mark prediction method.
In this paper, an improved continuous tool influence function numerical method, considering the dynamic performance of machine tools, is proposed to predict tool mark on the surface of large-aperture mirrors undergoing sub-aperture polishing. The proposed numerical method allows for quick predictions without data explosion or the need for extensive big data support. Both simulation and experimental results validate the feasibility of the proposed method.
The remainder of this paper is organized as follows. Section 2 briefly introduces the technical background. Section 3 establishes the continuous tool influence function method, considering the dynamic performance of the machine tool. Section 4 provides the calculation formula for predicting tool mark using the proposed method. Section 5 explains the effectiveness and economic properties of the proposed method compared to other numerical methods via simulations. Section 6 demonstrates the practicability of the method through experimental verification. Section 7 discusses the time cost of the proposed method. Finally, Section 8 concludes the paper.
2. Technical background
In deterministic polishing, the relationship between surface error, the tool influence function (TIF), and dwell time is given by Eq. (1) as follows:
where $E({x,y} )$ represents the surface error matrix, $R({\omega ,\upsilon } )$ denotes the TIF matrix, ${\ast}{\ast} $ represents 2D convolution and $T({\varepsilon ,\eta } )$ denotes the dwell time matrix.The dwell time is solved via deconvolution. To solve the dwell time, the linear algebra method is widely used. The discrete matrix of Eq. (1) can be expressed as a linear equation, given by:
Equation (2) can be simplified to Eq. (3):
Because most of the linear problems from deconvolution in Eq. (3) are ill-posed, and the discrete convolution kernel matrix $R^{\prime}$ in the calculation process of large-aperture mirrors is mostly stored by a large sparse matrix, the method presented in Ref. [22] is used to calculate the dwell time.
Therefore, the calculated error $\Delta E$ is the difference between the actual fabrication surface residuals $E^{\prime\prime}$ minus the calculated surface residuals $({{\mathbf {E}^{\prime}} - R^{\prime}{\mathbf {T}^{\prime}}} )$, which is given by Eq. (4), as follows:
3. Continuous tool influence function model
The continuous tool influence function (CTIF) serves as continuation of the TIF to incorporate the removal process, with the foundational CTIF model outlined in Ref. [23]. They utilized the basic CTIF model to compute more precise dwell times. However, for predicting tool mark as ${\mathbf {\Delta}_{\textrm{ToolMark}}}$ in Eq. (4), this basic CTIF model falls short in accounting for the CTIF calculation, considering the dynamic performance of the machine tool involves three primary steps. The first step entails inserting ${P_n}$ sub dwell points that include endpoints. The second step is to calculate the sub dwell time of the sub dwell points; this step is divided into two sub-steps, namely, compute the sub dwell point node and allocate the sub dwell time. The third step is to integrate the CTIF. Assuming that ${P_n}$ sub dwell points have been inserted equidistant. This section focuses on the CTIF model, emphasizing the limitations imposed by the dynamic performance of the machine tool, particularly in the MRF context.
3.1 Sub dwell point computation node
The machine in this study operates in position loop control mode. The machine tool acceleration model, as referenced in this study [24], is divided into two segments based on the operation case: the segments with acceleration are termed acceleration segments, while those without acceleration are referred to as uniform velocity segments. The computational node ${t_s}$ mark the moments that separate these operation cases from time. The equation to determine the computational node is presented in this section.
Following the dwell time distribution calculated as discussed in Section 2, two adjacent points on the feeding path are identified and labeled as points A and B. Taking the grating path as an example, as illustrated in Fig. 1. The feeding and scanning directions of the grating path, as well as the feeding ${f_g}$ and scanning ${s_g}$ gaps are also defined in Fig. 1. The dwell time of point A is ${t_1} = {\mathbf {T}^{\prime}}({{x_A},{y_A}} )$, while that of point B is ${t_2} = {\mathbf {T}^{\prime}}({{x_B},{y_B}} )$; then, the theoretical feeding time of right-left equipartition from point A to point B is approximately $t \approx 0.5 \times ({{t_1} + {t_2}} )$.
In the process of moving from point A to point B, the operation time of the acceleration segments is $t^{\prime}$, while that of the uniform velocity segments is $t^{\prime\prime}$, as shown in Fig. 2(a) and Fig. 2(b). The distance (or feed amount) vector from point A to point B is $\mathbf {s} = \overrightarrow {AB}$.
Fig. 2. (a) Acceleration and (b) Deceleration in Case 1; (c) Acceleration and (d) Deceleration in Case 2.
Case 1: Position loop control mode allows the tool to gently reach the desired position as shown in Fig. 2(a) and Fig. 2(b). The theoretical dwell time t and feed amount $\mathbf {s}$ are satisfied as:
The time node solution from scalar Eq. (6) under Case 1 is:
Therefore, the dwell time node is ${t_s} = \min (t^{\prime})$; taking the minimum value to make ${t_s} < t$.
Case 2: Position loop control mode forces the tool to reach the position to proceed to the next segments, and the tool maintains its acceleration during the feed amount $\mathbf {s}$, as shown in Fig. 2(c) and Fig. 2(d). Therefore, the actual operational time ${t_r}$ satisfies scalar Eq. (8):
The time node solution from scalar Eq. (8) under Case 2 is:
Therefore, the dwell time node is ${t_s} = \min ({t_r})$. Taking the minimum value because $\min ({t_r})$ is closer to t than $\max ({t_r})$.
The operation case of the machine tool can be judged according to the criterion diagrams shown in Fig. 3.
Fig. 3. Operation case criteria. (a)$s > 0,{v_A} > 0$, (b)$s < 0,{v_A} < 0$, (c)$s > 0,{v_A} < 0$ and (d)$s < 0,{v_A} > 0$.
3.2 Sub dwell time allocation
Since the sub dwell points are equidistant in space and not equidistant in time, it should solve the moment at each sub dwell point according to the relationship between displacement and time. The scalar displacement for Cases 1 and 2 are solved with moment ${t_l}$, $0 \le l \le {P_n} - 1$ for each moment using Eq. (10) and Eq. (11), respectively:
For Case 1:
For Case 2:
It takes the difference ${\mathbf {d}_{\boldsymbol {t}}} = diff ({{t_l}} )$ as shown in Fig. 4 and the right-left equipartition approximation on sub dwell time. The sub dwell time allocation vector is Eq. (12):
Fig. 4. Difference schematics for the relation between feed amount and time referring to (a) Fig. 2(a) and (b) Fig. 2(c).
3.3 Continuous tool influence function computation
The forward convolution operation is performed on the sub dwell points from points A to B. Taking the TIF of MRF as an example in Fig. 5(a), the TIF is expanded according to the continue domain as:
The motion reflected in the TIF is the translation of the support set, while change functions affect the expansion and contraction of support set. The TIF is translated for distance $l \times d,0 \le l \le {P_n} - 1$ and operated on the zero-continuation of $CD$. zero-continuation is achieved by interpolation or Kronecker product. The slice of the CTIF can be obtained from Eq. (14), as shown in Fig. 6.
The amplitude of removal rate of the TIF can also vary, given as Eq. (15):
Further, the CTIF considering the dynamic performance of the machine tool is as follows Eq. (16):
4. Proposed tool mark prediction method
The CTIF set ${\{{C{R_m}} \}_{\textrm{set}}}$, $1 \le m \le N - 1$, are obtained for all adjacent points on the path. The CTIF set are operated on the zero-continuation according to the expansion domain as Eq. (17):
where $ED$ is expansion domain for continuation and ${D_e}$ is the mirror diameter.The slice of the full-aperture removal can be obtained from Eq. (18), as shown in Fig. 6.
The set of expansion slice set ${\{{E{R_m}} \}_{\textrm{set}}}$ is obtained by ${\{{C{R_m}} \}_{\textrm{set}}}$. To add the expansion slices together, the full-aperture removal is obtained by Eq. (19), as follows in Fig. 6:
where $FAR({x,y} )$ is the full-aperture removal.Finally, the predicted residuals $\mathbf{Res}$ are equal to the surface error $E({x,y} )$ minus the full-aperture removal $FAR({x,y} )$ by Eq. (20), an example of which is shown in Fig. 7(c).
The CTIF interpolation algorithm (IA) flowchart is illustrated in Fig. 7. Where $\textrm{criteria} ({{s_\varepsilon },{s_\eta },{v_A},a} )$ denotes a criteria function written according to Fig. 3, whose return value is the operation case number of the machine tool; $\textrm{interp} ({{R_l},C{D_m}} )$ represents the interpolation of the surface of the form ${R_l}({{\omega_l},{\upsilon_l}} )$ at the specified query point in $C{D_m}$ and returns the inserted value. It is worth noting that the CTIF Kronecker product algorithm (KPA) can also achieve good prediction results and requires less time; however, it requires more sampling constraints.
5. Simulations
5.1 Tool mark prediction comparison between different numerical methods
To determine the maximum sampling for which the dense path method can predict tool mark at a certain scanning gap ${s_g}$, several simulation experiments were considered.
Primarily, the simulation assumes an original surface with diameter ${D_e} = 100\textrm{ }mm$ and a sampling gap along x and y of ${d_{\textrm{xg}}} = 1\textrm{ }mm$ and ${d_{\textrm{yg}}} = 1\textrm{ }mm$, respectively, as shown in Fig. 8(a). The grating path with a scanning gap of ${s_g} = 1\textrm{ }mm$ is shown in Fig. 8(c). The path feeding gap ${f_g}$ is always consistent with the sampling gap along the feeding direction of the surface data. These parameters are called the original surface and path (OSP). The TIF of MRF with a sampling gap of ${T_s} = 0.1\textrm{ }mm$ along both x and y is shown in Fig. 8(b). To mitigate edge effects, only tool mark within a clear aperture (CA) of $CA = 60mm$ were considered.
Fig. 8. Prediction residuals simulation diagrams. (a) Surface with ${D_e} = 100\textrm{ }mm$; (b) TIF of MRF in a certain process, (c) Grating path with ${f_g} = unfixed$ and ${s_g} = 1\textrm{ }mm$; (d) Dwell time under the following parameters: ${d^{\prime}_{\textrm{xg}}} = 0.5\textrm{ }mm$, ${d^{\prime}_{\textrm{yg}}} = 0.5\textrm{ }mm$, ${f^{\prime}_g} = 0.5\textrm{ }mm$, and ${s_g} = 1\textrm{ }mm$; (e) Dense path method prediction from (d); (f) CTIF method prediction from (d); (g) Dwell time under the following parameters: ${d_{\textrm{xg}}} = 1\textrm{ }mm$, ${d_{\textrm{yg}}} = 1\textrm{ }mm$, ${f_g} = 1\textrm{ }mm$, and ${s_g} = 1\textrm{ }mm$, (h) Dense path method prediction from (g); and (i) CTIF method prediction from (g).
Subsequently, the dense path method involves an increasing density in the surface sampling and feeding gaps, but not in the scanning gap. The surface in this case assumes an increasing sampling density with ${d^{\prime}_{\textrm{xg}}} = 0.5\textrm{ }mm$ and ${d^{\prime}_{\textrm{yg}}} = 0.5\textrm{ }mm$ and an increasing sampling density path with ${f^{\prime}_g} = 0.5\textrm{ }mm$. These parameters are called the dense surface and path (DSP).
The CTIF sampling gaps are ${T_{\textrm{xg}}} = {f^{\prime}_g}{[{path = dense\textrm{ }path} ]_{\textrm{Ib}}} + {f_g}{[path = origin\textrm{ }path]_{\textrm{Ib}}}$ and ${T_{\textrm{yg}}} = ({{1 / p}} )\times {s_g},p \in {Z^ + }\backslash {\{1 \}_{\textrm{set}}}$, here $p = 2$. This is to satisfy the Nyquist sampling theorem of tool mark without numerical aliasing. One thing to consider is that only the grating path feeding parameters along x are chosen parameters in this manner. If another path is used, such as a spiral path, the sampling gap of the CTIF in both directions must satisfy the $({{1 / p}} )\times {s_g},p \in {Z^ + }\backslash {\{1 \}_{\textrm{set}}}$ relation. The other parameters are selected as ${P_n} = 3$, $a ={\pm} 5m/{s^2}$, ${v_{\textrm{temp}}} = 0$.
Figures 8(d)-(f) show the DSP tool mark prediction. Figures 8(g)-(i) show the OSP tool mark prediction. Peak to valley (PV), root mean square (RMS) and the convergence rate of surface $\iota = ({RM{S_{\textrm{err}}} - RM{S_{\textrm{res}}}} )/RM{S_{\textrm{err}}} \in [{0,1} ]$ are for the full-aperture not CA. The ring structure on the predicted residuals resulted from the unavoidable ringing effect induced by the deconvolution algorithm.
The dense path method prediction with the DSP is shown in Fig. 8(e), which is almost identical to that by the CTIF method with DSP as shown in Fig. 8(f). This underscores the inherent relationship between the dense path and CTIF methods, both representing different forms of adding sub dwell points.
Finally, the dense path method with the OSP fails to predict the tool mark as shown in Fig. 8(h). It means that tool mark appears only when the surface and path reach a certain density. A comparison between Fig. 8(e) and Fig. 8(h) highlights the fact that ${f^{\prime}_g} = 0.5\textrm{ }mm$ can be used for tool mark prediction residuals in the dense path method at ${s_g} = 1\textrm{ }mm$. Meanwhile, the CTIF method with the OSP can predict tool mark, as shown in Fig. 8(i). This demonstrates that the CTIF method holds significant advantages on sparse surface and path, thereby simultaneously validating the effectiveness and data size advantages for tool mark prediction.
5.2 Computational complexity estimation
To demonstrate that the CTIF method saves memory, the data size for three methods, namely, the dense path method with the OSP (hereinafter referred to as DMOSP), dense path method with the DSP (hereinafter referred to as DMDSP), and CTIF method with the OSP (hereinafter referred to as CTIFM), were estimated. The computational complexity is denoted as $O({f(n )} )$. ${D_p}$ is the diameter of the path domain, as shown in Fig. 9(b).
Fig. 9. Efficient element estimation diagram. (a) The red dots are the surface data vector elements, (b) The green dots are the path vector elements and (c) The yellow dots are the non-zero valid elements in the sparse matrix $R^{\prime}$.
The main memory of the DMOSP with sparse matrix $R^{\prime}$ in Eq. (3) is estimated by Eq. (21) as follows:
Fig. 10. Estimated data size for ${d_{\textrm{xg}}} = 5\textrm{ }mm,{d_{\textrm{yg}}} = 5\textrm{ }mm$. (a) Sparsity of matrix $R^{\prime}$ with a varying feeding gap at ${s_g} = 1\textrm{ }mm$, (b) Proportion of data points of recorded matrix with the aperture increases. (c) Sparsity changes of matrix $R^{\prime}$ with ${D_e}$.
Therefore, the computational complexity of the DMOSP is estimated as:
The main memory of the DMDSP with sparse matrix $R^{\prime}$ in Eq. (3) is estimated by Eq. (23), as follows:
Finally, the main memory of the CTIFM with $C{R_m}$ matrix elements is estimated by Eq. (24) for the TIF referring to Fig. 8(b):
From the computational complexity expressions, it is not clear which method has a larger data size; therefore, a numerical simulation was carried out. The OSP with different apertures ${D_e}$ adopts as ${d_{\textrm{xg}}} = 5\textrm{ }mm$, ${d_{\textrm{yg}}} = 5\textrm{ }mm$, ${f_g} = 5\textrm{ }mm,{s_g} = 1\textrm{ }mm$. The TIF of MRF is as shown in Fig. 8(b). The CTIF sampling gaps are ${T_{\textrm{xg}}} = 5\textrm{ }mm$ and ${T_{\textrm{yg}}} = 0.5\textrm{ }mm$ (minimum parameter to display tool mark) and ${T_{\textrm{yg}}} = 0.1\textrm{ }mm$ (parameter to display more details). According to the discussion in Section 5.1, the parameters ${d^{\prime}_{\textrm{xg}}} = 0.5\textrm{ }mm$, ${d^{\prime}_{\textrm{yg}}} = 0.5\textrm{ }mm$ and ${f^{\prime}_g} = 0.5\textrm{ }mm$ are taken for estimation of ${S^{\prime}_n}$. It is worth noting that When parameter ${f^{\prime}_g}$ is equal to parameter ${T_{\textrm{yg}}}$, the predicted tool mark is the same. The denser the DSP, the more accurate predictions are; therefore, ${d^{\prime}_{\textrm{xg}}} = 0.1\textrm{ }mm$, ${d^{\prime}_{\textrm{yg}}} = 0.1\textrm{ }mm$ and ${f^{\prime}_g} = 0.1\textrm{ }mm$ are added to observe larger data sizes by the DMDSP.
A simulation was carried out with the above mentioned parameters, and the results are shown in Fig. 11 which is displayed using the semiology method. The five curves in Fig. 11 all show an increasing trend. Curve ${S_n}$ of ${D_e}$ represents the data size for calculated surface residuals $({{\mathbf {E}^{\prime}} - R^{\prime}{\mathbf {T}^{\prime}}} )$. The large gap among curves ${S_n}$, ${C_n}$, and ${S^{\prime}_n}$ indicates that the tool mark ${\mathbf {\Delta}_{\textrm{ToolMark}}}$ simulations incur in necessary data size costs.
The difference between curve ${S^{\prime}_n}$ according to ${D_e}$ for ${f^{\prime}_g} = 0.1\textrm{ }mm$ and curve ${C_n}$ according to ${D_e}$ for ${T_{\textrm{yg}}} = 0.1\textrm{ }mm$ is within three orders of magnitude. But in most large-aperture mirror manufacturing engineering tasks, extremely accurate tool mark predictions are not required; therefore, it is reasonable to choose a fast prediction with low accuracy. Curve ${S^{\prime}_n}$ according to ${D_e}$ for ${f^{\prime}_g} = 0.5\textrm{ }mm$ represents the minimum data size for tool mark prediction by DMDSP. The CTIFM with ${T_{\textrm{yg}}} = 0.5\textrm{ }mm$ exhibits an advantage, and its data size is less than that of the DMDSP by approximately two orders of magnitude at the same prediction effect. Comparing Eq. (23) and Eq. (24), we observed that the smaller ${R_{\textrm{length}}}$ and ${R_{\textrm{width}}}$ are, the more feasible it is for the CTIFM to predict tool mark fast. In addition, according to Section 5.1, the DSP parameters of CTIFM are not limited to equal to DMDSP, which can also be enlarged nearly to OSP. Further, ${C_n}$ will be reduced but always higher than ${S_n}$. The save of data size are beneficial to large-aperture mirrors by sub-aperture polishing.
6. Experiments
To validate the practicality of the CTIFM, experiments were conducted on the MRF machine, whose processing range can reach ${D_e} = 4000\textrm{ }mm$, as shown in Fig. 12(a). The MRF wheel's diameter used in this experiment was 160 mm. The polishing liquid employed was silicon liquid, and the slurry consisted of a mixture of pure water and cerium oxide with a particle size of 1.5 um, a concentration of 20 g/L, and a polishing gap of 2 mm. The diameter of the mirror was ${D_e} = 100\textrm{ }mm$ (fused quartz glass mirror), as shown in Fig. 12(b). To prevent edge effects, $CA = 60\textrm{ }mm$ was set for the observation. The mirror surface was measured before and after polishing via Zygo HDX interferometry.
Fig. 12. Experimental conditions. (a) MRF machine and (b) fused quartz glass mirror with ${D_e} = 100\textrm{ }mm$.
The initial error of the experimental surface is shown in Fig. 13(a). With ${d_{\textrm{xg}}} = 1\textrm{ }mm,{d_{\textrm{yg}}} = 1\textrm{ }mm$, the inner red dashed circle is the $CA$ region, the outer blue dashed circle is the ${D_e}$ region, and the data outside ${D_e}$ is analytically extended to suppress the ringing effect. The TIF under the above MRF parameters is shown in Fig. 13(b) with ${T_s} = 0.1mm$. The grating path is selected with ${f_g} = 1\textrm{ }mm$ and ${s_g} = 0.5\textrm{ }mm$, as shown in Fig. 13(c). The ${s_g} = 0.5\textrm{ }mm$ was chosen in order to make the calculation more accurate and not time-consuming. The dwell time is calculated via the method presented in Section 2, as shown in Fig. 13(d). To match the Zygo HDX sampling rate, the parameters of the CTIF method were set to ${T_{\textrm{xg}}} = 1\textrm{ }mm$ and ${T_{yg}} = 0.1\textrm{ }mm$. The CTIFM prediction residuals are shown in Fig. 13(e). A partially zoomed image of the area within the purple box in Fig. 13(e) is shown in Fig. 13(f). A profile along the vertical blue line in Fig. 13(e) is shown as the blue curve in Fig. 14(a). The power spectral density (PSD) analysis results are shown as the blue curve in Fig. 14(b).
Fig. 13. Tool mark prediction results. (a) Initial surface error, (b) TIF, (c) Grating path, (d) Dwell time, (e) Residual of the CTIF method in CA and (f) Zoomed-in view of the purple box in (e).
Based on the dwell time depicted in Fig. 13(d), a CNC code was generated, and actual machining was executed. The actual residuals by the interferometer after processing are illustrated in Fig. 15(a). Due to noise and fringe effect, they are very high about the PV and RMS values in the interferometer measurement software. While the residuals without noise within the CA are presented in Fig. 15(b). It is much denser sampling gap than Fig. 13(e) due to data from the interferometer, so the image is much smoother. A partially zoomed image of the area within the purple box in Fig. 15(b) is shown in Fig. 15(c). A profile along the vertical red line in Fig. 15(b) is shown as the red curve in Fig. 14(a). The PSD analysis results are shown as the red curve in Fig. 14(b).
Fig. 15. Tool mark prediction experimental results. (a) Interferometer detection residuals, (b) Residuals of the experiment in CA and (c) Zoomed-in view of the purple box in (b).
When calculating the PSD curve, cut line data is processed into the same sampling density as $0.1\textrm{ }mm$. The PSD curves in Fig. 14(b) are presented using the semiology method. The period of the tool mark on the surface should align with the scanning gap ${s_g} = 0.5\textrm{ }mm$. A notable characteristic peak is evident at $SF = {1 / {{s_g}}} = 2({1/mm} )$ for both the actual residuals and CTIFM predict residuals. This observation underscores the efficacy of the CTIFM for tool mark prediction.
However, the difference is that there is also a characteristic peak at $SF = 4({1/mm} )$ in the CTIFM, which we believe that it is the frequency doubling effect in the numerical calculation, and has no direct impact on the tool mark prediction for large-aperture mirrors in engineering practice.
Although the experimental verification was conducted on small-aperture mirrors, the CTIFM can be generalized to large-aperture mirrors as the splicing of a limited amount of small sub-aperture mirrors.
7. Discussion
Let us now discuss the time consumption of the proposed method. Note that the interpolation is in two-level loop in Fig. 7, which means that it is the main time consuming item of the IA. In fact, the computation time of the IA is almost the same as that of the DMDSP. As mentioned earlier, the KPA can reduce the computational burden, but the sampling grid size cannot be controlled arbitrarily. Despite reducing the computational time, it introduces other more significant problems. It seems that the CTIFM does not improve in terms of time consumption, only in terms of memory utilization and prediction accuracy.
However, we made a very critical approximation that can improve the time consumption. When entering the MRF process, this assumption ensures the dynamic performance of the machine tool for most of the grating path processing. Furthermore, the errors of the mirror are already very small everywhere, and we believe that the gradient of the errors will not be exceedingly large for each feeding segment on the mirror. Therefore, within the segment, the sub dwell time ${t_l}$ is almost equal. Further, the $C{R_m}$ for each segment are different only at the removal peak that depends on the dwell time of each segment, but the distribution is the same. Therefore, it calculates only one $CR$ distribution and the dwell time is used to control the removal peak near the dwell point to achieve the tool mark prediction residuals. The IA-Simplification (IAS) algorithm flowchart is shown in Fig. 16. The IAS performs only one interpolation in each loop, which sacrifices some precision compared with the DMDSP and IA; however, it exhibits an extremely high computational efficiency. The KPA-Simplification (KPAS) algorithm can achieve nearly the same effect, with faster computation and more limited sampling. These two simplification algorithms have been used in different large-aperture mirror fabrication engineering tasks.
8. Conclusion
In this paper, a continuous tool influence function (CTIF) method for predicting tool mark on the surface of large-aperture mirrors via magnetorheological finishing (MRF) is introduced. This innovative approach dynamically allocates like-convolution on dwell time and a variable tool influence function (TIF). The precise residuals with tool mark for the full-aperture are determined by the CTIF. The proposed method exhibits significant potential in predicting the tool mark of large-aperture mirrors via MRF. Furthermore, compared with other numerical methods, our method requires a smaller data size to achieve the same effect, making it an effective and economical tool mark prediction method. Through an MRF experiment, the actual tool mark on a fused quartz glass mirror with a diameter of 100 mm closely matched the predicted tool mark, particularly at the spectral characteristic peak under power spectral density (PSD) analysis. This verification underscores the practicability of the proposed CTIF method. A reasonable approximation is proposed and a fast prediction algorithm for tool mark is presented to faster computation in different large-aperture mirrors fabrication engineering task.
It is worth noting that the proposed method is not limited to the MRF process; it extends to other deterministic sub-aperture polishing processes such as ion beam computing. The CTIF method outlined in this paper provides valuable technical guidance for predicting tool mark on the surface of large-aperture mirrors by sub-aperture polishing.
Despite its many advantages, the proposed method suffers from certain limitations. For example, we must admit that random paths are not particularly friendly to our method as it has a lot of variable feeding directions. However, our method can be applied to lower-order differentiable paths, such as spiral paths. In future work, we will aim further to provide a faster tool mark prediction algorithm and explore the causes of frequency doubling peaks in more detail for the fabrication of large-aperture mirrors.
Nomenclature
N1. Matrix [element] (coordinate)
$E({x,y} )$ | Surface error |
$T({\varepsilon ,\eta } )$ | Dwell time |
$R({\omega ,\upsilon } )$ | Tool influence function |
$R^{\prime}{[{{r_{ij}}}]}$ | Discrete convolution kernel |
${R_\omega }$ | $\omega $-coordinate matrix of $R({\omega ,\upsilon } )$ |
${R_\upsilon }$ | $\upsilon $-coordinate matrix of $R({\omega ,\upsilon } )$ |
$S{R_l}({\varsigma ,\xi } )$ | Slice of the CTIF at the $({l + 1} )- th$ sub dwell point |
${R_l}$ | TIF at the $({l + 1} )- th$ sub dwell point |
$CR({\varsigma ,\xi } )$ | CTIF from point $A$ to point $B$ |
$FAR({x,y} )$ | Full-aperture removal |
$ER({x,y} )$ | Slice of $FAR({x,y} )$ at the $m - th$ segment |
$\delta $ | Higher order $R({\omega ,\upsilon } )$ error |
N2. Vector [element] (coordinate) <scalarization>
${\mathbf {E}^{\prime}}{[{{e_i}}]}({{x_i},{y_i}} )$ | Vectorized $E({x,y} )$ |
${\mathbf {T}^{\prime}}{[{{t_j}}]}({{\varepsilon_j},{\eta_j}} )$ | Vectorized $T({\varepsilon ,\eta } )$ |
$\mathbf{\Delta}\mathbf{E}$ | Calculated error |
${\mathbf {E}^{\prime\prime}}$ | Actual fabrication surface residual |
${\mathbf {\Delta}_{\textrm{ToolMark}}}$ | Tool mark error |
${\mathbf {\Delta}_{\textrm{Other}}}$ | Other errors |
$\mathbf{Res}$ | Predict residuals with tool mark error |
$\mathbf {s}\left\langle s \right\rangle $ | ‘distance vector from point $A$ to $B$’ OR ‘feeding amount’ |
$\mathbf {a}\left\langle a \right\rangle $ | Acceleration vector of the tool |
${\mathbf {v}_{\boldsymbol{A}}}\left\langle {{v_A}} \right\rangle $ | Velocity vector of point $A$ |
$\mathbf {d}\left\langle d \right\rangle $ | Sub feeding amount |
${\mathbf {d}_{\boldsymbol {t}}}$ | Sub dwell time difference |
$\mathbf{DT}$ | Sub dwell time allocation vector |
N3. Scalar
${D_p}$ | Diameter of the path domain |
${D_e}$ | Diameter of the mirror domain |
$N$ | Vectorized path points number |
$M$ | Vectorized surface error points number |
${P_n}$ | Sub dwell points number |
${t_1}$ | Dwell time of point $A$ |
${t_2}$ | Dwell time of point $B$ |
$t$ | Dwell time from $A$ to $B$ |
$t^{\prime}$ | Operation time of the acceleration segment |
$t^{\prime\prime}$ | Operation time of the uniform velocity segment |
${t_s}$ | Time nodes separating the accelerated and uniform velocity |
${t_r}$ | Actual operation time |
${t_l}$ | Sub dwell time to reach the $({l + 1} )- th$ sub dwell point |
${s_\varepsilon }$ | Projection of $\mathbf {s}$ along $\varepsilon $ |
${s_\eta }$ | Projection of $\mathbf {s}$ along $\eta $ |
${T_{\textrm{xg}}}$ | CTIF sampling gaps along $x$ |
${T_{\textrm{yg}}}$ | CTIF sampling gaps along $y$ |
${d_\varepsilon }$ | Projection of $\mathbf {d}$ along $\varepsilon $ |
${d_\eta }$ | Projection of $\mathbf {d}$ along $\eta $ |
${R_{\mathrm{\omega}\textrm{min}}}$ | Minimum coordinate of ${R_\omega }$ |
$RM{S_{\textrm{err}}}$ | RMS of surface errors |
$RM{S_{\textrm{res}}}$ | RMS of surface residuals |
${R_{\mathrm{\omega}\textrm{max}}}$ | Maximum coordinate of ${R_\omega }$ |
${R_{\mathrm{\upsilon}\textrm{min}}}$ | Minimum coordinate of ${R_\upsilon }$ |
${R_{\mathrm{\upsilon}\textrm{max}}}$ | Maximum coordinate of ${R_\upsilon }$ |
${R_{\textrm{length}}}$ | Valid area length of $R({\omega ,\upsilon } )$ |
${R_{\textrm{width}}}$ | Valid area width of $R({\omega ,\upsilon } )$ |
${\kappa _\gamma }$ | Perturbed frequency of variate $\gamma $ |
${d_{\textrm{xg}}}$ | Surface original sampling gap along $x$ |
${d_{\textrm{yg}}}$ | Surface original sampling gap along $y$ |
${d^{\prime}_{\textrm{xg}}}$ | Surface dense sampling gap along $x$ |
${d^{\prime}_{\textrm{yg}}}$ | Surface dense sampling gap along $y$ |
${f_g}$ | Feeding gap |
${s_g}$ | Scanning gap |
${f^{\prime}_g}$ | Feeding dense gap |
${T_s}$ | TIF sampling gap both along x and $y$ |
$CA$ | Observation clear aperture |
$SF$ | Spatial frequency |
${S_n}$ | Main memory elements of DMOSP |
${S^{\prime}_n}$ | Main memory elements of DMDSP |
${C_n}$ | Main memory elements of CTIFM |
${R_{{P_n}}}$ | Proportion of effective data points in the data recording matrix |
${S_r}$ | Sparsity of $R^{\prime}$. |
$\iota $ | Convergence rate of surface for full-aperture |
${v_{\textrm{temp}}}$ | Velocity temporary argument in algorithm |
N4. Operator / Function (variable)
${\ast}{\ast} $ | 2D convolution |
${\ast} $ | Function argument |
${[{\ast} ]_{\textrm{Ib}}}$ | Iverson bracket |
${\otimes} $ | Cartesian product |
${\{{\ast} \}_{\textrm{set}}}$ | Argument set |
$\varOmega \backslash \sigma $ | Difference set of set $\varOmega $ and set $\sigma $ |
$\textrm{min}({\ast} )$ | Obtained minimum value of argument |
$\textrm{atan}({\ast} )$ | Obtained arc tangent of argument |
$\textrm{diff}({\ast} )$ | Obtain the difference of argument |
$\textrm{cat} ({a,b} )$ | Concatenated array a and $b$ |
$\textrm{vec} ({\ast} )$ | Vectorized the argument matrix |
$\textrm{size} ({\ast} )$ | Obtained data size of the argument matrix |
${\textrm{f} _\varepsilon }({{\varepsilon_l}} )$ | Change of TIF along $\varepsilon $ |
${\textrm{f} _\eta }({{\eta_l}} )$ | Change of TIF along $\eta $ |
$\textrm{g} ({{t_l}} )$ | Change of TIF with time lapses |
$\textrm{supp} (^{\ast} )$ | Obtained support set of argument |
$\textrm{criteria} ({{s_\varepsilon },{s_\eta },{v_A},a} )$ | Criteria function for operation case |
$\textrm{interp} ({{R_l},C{D_m}} )$ | Interpolates the surface of the form ${R_l}({{\omega_l},{\upsilon_l}} )$ at the specified query point in $C{D_m}$ and returns the inserted value |
N5. Domain / Set
$CD$ | Continuous domain for continuation |
${Z^ + }$ | Set of positive integers |
$ED$ | Expansion domain for continuation |
N6. Symbol / Abbreviation
MRF | Magnetorheological finishing |
TIF | Tool influence function |
CTIF | Continuous tool influence function |
$A$ | Original point of $\mathbf {s}$ |
$B$ | Terminus point of $\mathbf {s}$ |
$n$ | Data scale |
$O(n )$ | Big $O$ notation |
IA | CTIF-Interpolation algorithm |
KPA | Kronecker products algorithm |
OSP | Original surface and path |
DSP | Dense surface and path |
DMOSP | Dense path method by OSP |
DMDSP | Dense path method by DSP |
CTIFM | CTIF method by OSP |
PSD | Power spectral density |
IAS | IA-Simplification |
KPAS | KPAS- Simplification |
PV | Peak to valley value |
RMS | Root mean square value |
Funding
Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021215); National Natural Science Foundation of China (12003034, 12203048, 62075218, 62275246); National Key Research and Development Program of China (2022YFB3403405).
Acknowledgment
The authors would like to thank the anonymous reviewers for their valuable suggestions and comments that have helped us greatly improve the paper.
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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