1. Introduction

With the development of space technology and astronomy, the demand for high-precision and high-efficiency manufacture of optical components is increasing. Due to high figuring precision, high convergence rate, and low reliance on manual experience, the advanced deterministic optical finishing technologies have been developed over past decades, including computer controlled optical surfacing (CCOS) [1], magnetorheological finishing (MRF) [2,3], ion beam figuring (IBF) [4,5], bonnet polishing (BP) [6,7], and jet polishing (JP) [8,9]. The material removal of optics surface is realized based on the convolution principle of tool influence function (TIF) and dwell time distribution. Hence the dwell time algorithm is critical to the precision and efficiency of optics fabrication.

Recently the researchers have proposed a variety of methods to compute the dwell time. Nevertheless, there exist still some of the following shortcomings when those methods were employed in the actual production. First, the calculation efficiency is not high enough and incompetent to accurately calculate the dwell time of large-aperture optics. Secondly, the dynamic performance of the machine tool is insufficiently considered in the optimization solution of dwell time. The dynamic constraints of machine tools are unneglectable factors in controlling low and middle frequency errors of optical surfaces due to the influence on the realization accuracy of dwell time. Thirdly, the low continuity of dwell time profile leads to feedrate and acceleration fluctuation during the process [10], declining the convergence efficiency. Finally, the existing dwell time algorithms rely on the commercial CNC interpolator where the acceleration/deceleration tactics is not open to the users, hindering the high-precision realization of scheduled dwell time. According to differences in computational schemes, the existing dwell time algorithms are categorized into iterative method, deconvolution method, and matrix equation method.

The iterative methods employ certain iteration rules to compute the dwell time according to residual error and TIF, and the iteration procedure is terminated when the residual error lies within the allowable range. Van Cittert’s method [11] was introduced into the CCOS field to solve the dwell time by Jones [12]. The original version of Van Cittert’s method converges to an exact solution without noise [13]. Nevertheless, there exists the noise for the actual TIF and surface figure, and thus the stable convergence cannot be guaranteed. The improved iterative methods [14,15] has higher efficiency and are less susceptible to noise interference, but those methods are only applicable to rotationally-symmetric TIF. For the non-rotationally-symmetric TIFs of polishing technologies such as MRF, the pulse iteration method causes serious surface residual error in the practical process, and thus is not suitable for MRF. Additionally, the dynamic constraints of machine tool were not respected in the works [11–15]. Zhu et al. [16] utilized Zernike polynomials to construct the optimum dwell time, generating the smooth dwell time map and reducing the feedrate fluctuation. Nevertheless, the dynamic constraints during the dwell time solution were not rigidly confined, and the Zernike approximation and evolution optimization involved yielded to low computational efficiency. Han et al. [17] provided the region adaptive scheduling under confined dynamic constraints for deterministic polishing, whereas the feedrate model depends on the continuity of the surface shape, and the possible failure in the iteration procedure influences the robustness and computational efficiency. Li et al. [18] proposed a B-spline surface approximation method for obtaining the optimum and smooth dwell time in deterministic polishing, helping to reduce the demands on the dynamic characteristics of the machine tools. Nevertheless, the dynamic constraints were not rigidly confined during the dwell time solution, and the numerical integration and non-negative least square problem involved during the iteration procedure also increased the computational complexity.

The deconvolution methods calculate dwell time based on the deconvolution restoration principle with a high calculation efficiency. Nonetheless, the existing deconvolution methods generally cannot ensure that the dynamic constraints of machine tools are rigidly respected. The existing deconvolution methods mainly include Fourier transformation method, Bayesian algorithm (e.g., Lucy-Richardson (LR) method), Wiener filtering method, and singular value decomposition method. Wilson [19] transformed the dwell time solution into an inverse Fourier transform of the dwell time function in the frequency domain, and established the Fourier transform method. Nevertheless, the parameters of Fourier transform method need to be set empirically. Taniguchi et al. [20] employed the fast Fourier transform (FFT) method to compute the dwell time for focused ion beam figuring, and the computational efficiency was further improved. Wang et al. [21] provided an iterative FFT to compute dwell time for ion beam figuring, and then developed FFT algorithms based on RISE [22], where the parameters of Fourier transform method were automatically adjusted. For the negative value generated by FFT, Tanebe et al. [23] developed an iterative blind deconvolution method to adjust dwell time. Jiao et al. [24] proposed a Bayesian-based approach to calculate the dwell time by iteratively maximizing the posterior probability using LR algorithm, achieving a smooth dwell time distribution with a high computational efficiency. Nonetheless, since it is difficult to incorporate the dynamic constraints into the dwell time solution, in the works [19–24] the dynamic constraints of machine tool were not rigidly respected during the optimization.

The matrix equation methods construct the solution model that expresses a convolution equation as a system of linear equations through dwell time discretization. The goal of that model is to solve the quadratic programming problem with constraints. Generally, the linear equation is ill-conditioned, and the existing methods are difficult to achieve highly-efficient optimization and solution under the dynamic constraints. For ill-conditioned characteristics of the linear equations [25], Deng et al [26] and Wu et al. [27] employed Tikhonov regularization method to balance the residual error and computational efficiency by adding a dwell time item to the objective function. That operation successfully overcame the ill-conditioned nature of the solution and accelerated the solution process. Li et al. [28] combined the non-negative least squares method and adaptive selection of Tikhonov regularization factors to improve the accuracy and efficiency of the dwell time calculation. Nevertheless, in the methods [26–28] did not include the dynamic constraints of machine tools in the dwell time solution. Furthermore, Li et al. [29] considered the dynamic constraints of machine tool, providing a dwell time algorithm with minimum equal extra material removal. Nevertheless, the minimum equal extra material removal needs to be empirically set, and there still exists the improving space of computational efficiency. Wang et al. [30] proposed a constrained least squares method to solve the linear equations, whereas the computational efficiency was not significantly improved. Huang et al. [31] proposed a method using the monotone projection gradient method to search the optimal dwell time based on the constrained least squares method, avoiding the inversion of large-size matrix. Nonetheless, the solution and storage of the gradient matrix also needed a large amount of memory and CPU occupation. Li et al. [32] proposed a novel time gradient-based constraint based on the least squares method with upper and lower bounds of dwell time, and verified the model through experiments, whereas the computational efficiency is still low. In the works [30–32] the acceleration constraint was not considered. Song et al. [33] introduced the dwell time gradient to the dwell time solution model for reducing the possibility of acceleration exceeding the limitation, and employed the interior point method to solve the non-linear optimization problem. Nevertheless, the method cannot guarantee the dwell time generated by the optimization algorithm rigidly satisfies the acceleration constraint, and the computational efficiency is low especially for large-aperture optical elements. Zhang et al. [34] transformed the feedrate and acceleration constraints into the bilateral constraints in the dwell time, and proposed bounded constrained least squares (BCLS) algorithm based on the sparse matrix and Newton iteration method. Nonetheless, the calculation efficiency is still low for large-aperture workpiece, and this method has certain limitations on the setting of line and step spacings of tool path, hindering the suppression of low and middle frequency errors.

The comparison among the existing dwell time algorithms is shown in Table 1. From the perspective of the development of dwell time algorithms, the existing methods failed to achieve highly-efficient dwell time solution under dynamic constraints. The dwell time computation is essentially the solution of large-scale variables under the complicated implicit constraints, and the conventional methods are difficult to high-efficiently find the optimal solution. The insufficient consideration of the dynamic constraints leads to the serious deviation from the ideal dwell time, deteriorating the surfacing quality. Additionally, in the recent methods, the model of depicting the dwell time along the path lacks reasonability, and the generation of feedrate profiles relies on the commercial NC system where the acceleration/deceleration strategy in the interpolator is not open to the users, hampering the realization precision of scheduled dwell time. Furthermore, the latent insufficient continuity of feedrate profile causes the feedrate and acceleration fluctuation, reducing the surfacing smoothness. To overcome the shortcomings of the existing methods, this work proposes a highly-efficient dwell time scheduling method based on a dwell time density (DTD) repairing scheme using trigonometric-spline model, where a self-adaptive offset algorithm under the dynamic constraints of machine tool is developed. Additionally, the real-time interpolator based on trigonometric DTD profile is also developed. The results show that this algorithm generates DTD profile rigidly respecting the dynamic constraints of the machine tool, and achieves higher computational efficiency and convergence rate simultaneously.

Table 1. Comparison among the existing dwell time algorithms

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The remainder of this paper is arranged as follows. Section 2 provides the generation method of the initial DTD sequence. Section 3 proposed the DTD scheduling scheme under the dynamic constraints of machine tool. Section 4 introduces the design of real-time interpolator based on trigonometric-spline DTD profile. Section 5 provides the simulative validation and comparison. Experiment and comparative analysis are conducted in Section 6. The conclusion is presented in Section 7.

2. Generation of the initial dwell time density sequence

According to Preston equation [35], the material removal amount during the process is linearly related to dwell time. When the polishing tool sweeps across the workpiece surface, the material removal process could be described as a two-dimensional convolution of tool influence function and the dwell time, expressed as follows.

Reconstruct the TIF matrix such that the resolution of TIF matrix is the same resolution as that of the removal amount matrix. The dwell time distribution $T({x,y} )$ without considering dynamic constraints can be high-efficiently solved by LR algorithm. The improved LR method [24] generates the smoother dwell time distribution using the non-negative smoothing factor. The improved iteration formula is presented as follows

The dwell time density (DTD) is defined as the duration per unit distance along the tool path. To characterize DTD more conveniently, a fixed-spacing raster tool path is employed. Assume that both the step and line spacings are constant, denoted as D. Denote the tool path sequence as ${\boldsymbol{P}_i} = ({x({{s_i}} ),y({{s_i}} )} )$, where ${s_i}$ is the accumulative length of the path at ${\boldsymbol{P}_i}$.The dwell time ${T_i}$ at ${\boldsymbol{P}_i}$ is evaluated by the average dwell time of the grid points around ${\boldsymbol{P}_i}$. The DTD at ${\boldsymbol{P}_i}$ is denoted by ${\tau _i} = {T_i}/D$.

The cubic Hermit interpolation [36–38] for the discrete DTDs can generates the C¹-continuous feedrate profile. Nonetheless, the simple manner fails to guarantee that the dynamic constraints of machine tool are rigidly respected and possibly causes the practical dwell time to significantly deviate from the scheduled one. Wang et al. [39] employed the PVT interpolation to generate the smooth feedrate profile under confined feedrate and acceleration for DTD sequence. Nonetheless, the optimization model proposed failed to reflect the weight difference of dwell time due to the nonuniformity of surface error. Additionally, that method needed to solve a quadratic programming problem with $6N$ variables and $3N - 1$ constraints, yielding to a rather time-consuming procedure.

3. DTD scheduling under dynamic constraints

3.1 Trigonometric-spline DTD model

The non-negative smoothing factor $\mu $ in Eq. (2) improves the smoothness of the dwell time profile. Nonetheless, the dynamic variables of machine tools cannot be directly controlled by the non-negative smoothing factor $\mu $, and therefore the algorithm needs to detect and correct DTD sequence to satisfy the dynamic constraints of machine tools. Additionally, the DTD profile should keep C¹ continuity to prevent the drastic change of acceleration.

For a fixed-spacing tool path sequence $\{{{\boldsymbol{P}_i}} \}_{i = 1}^N$, denoted $\boldsymbol{\xi } = \{{{\boldsymbol{\xi }_i}|{\boldsymbol{\xi }_i} = ({{s_i},{\tau_i}} ),i = 1 \ldots N} \}$. Denote DTD profile as $\tau (s )$. The relationship between $\tau (s )$ and feedrate $v(s )$ is

The line/step spacing of tool path is $D = {s_{i + 1}} - {s_i}$. The DTD ${\tau _i}$.at ${\boldsymbol{P}_i}$ is computed by ${\tau _i} = {T_i}/D$. Assume that two adjacent points in the sequence are connected by a half-period trigonometric-spline function, as illustrated in Fig. 1. For $s \in [{{s_i},{s_{i + 1}}} ]$, $\tau (s )$ is expressed as

 

Fig. 1. Trigonometric-spline DTD profile between adjacent dwell positions.

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Differentiate Eq. (3) in the time parameter t, and then the acceleration is

In the practical machining, the dynamic axis of machine tool always remains in motion. The existing methods calculated the duration between adjacent points according to the scheduled dwell time. Nevertheless, the dwell time at each dwell position should be deemed as the duration near the dwell position. As illustrated in Fig. 2, the duration between the adjacent dwell positions ${\boldsymbol{P}_i},{\boldsymbol{P}_{i + 1}}$ is

 

Fig. 2. The equal distribution of dwell time ${T_i}$ at ${\boldsymbol{P}_i}$ to adjacent path segments.

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3.2 Feasible DTD profile under dynamic constraints

The dynamic constraints of machine tools are significant factors in controlling low and medium frequency errors due to the influence on the realization accuracy of dwell time distribution. The dynamic performance parameters include the feedrate and acceleration constraints. Let ${V_{max}}$ be the maximum feedrate of machine tools. The feedrate constraint is written as

Due to the monotonicity of $\tau (s )$, $\tau ({{s_i}} )\ge {\tau _{min}}$ and $\tau ({{s_{i + 1}}} )\ge {\tau _{min}}$ can guarantee that the constraint (7) holds. Then the feedrate constraint is translated into

According to Eq. (5), the acceleration constraint is

For the given ${M_1}$, determine the range of ${\tau _i}$ and ${\tau _{i + 1}}$ by the following rules.

Case I. If ${\tau _{i + 1}} \ge {\tau _i}$, the polishing tool decelerates or keep constant-feedrate within the interval. The acceleration satisfies

Hence if $g({{\tau_i}} )\le {A_{max}}$, then $|{a(s )} |\le {A_{max}}$ can be deducted. Let

Then

Combining with the upper bound of feedrate, ${\tau _i}$ and ${\tau _{i + 1}}$ should satisfy

Case II. If ${\tau _{i + 1}} < {\tau _i}$, the polishing tool accelerates within this interval.

Therefore, if $F({{\tau_{i + 1}}} )\le {A_{max}}$, then $|{a(s )} |\le {A_{max}}$ can be deducted. Let

Then

Combining with the upper bound of feedrate, ${\tau _{i + 1}}$ and ${\tau _i}$ should satisfy

When dynamic constraints are disobeyed, the whole DTD sequence needs to be corrected. The condition (7) is a criterion of judging whether the feedrate constraint is satisfied. According to the acceleration/deceleration case, condition (13) or (17) is a criterion of judging whether the acceleration constraint is satisfied. From the Eq. (10) and (14), given the ${M_1}$, when ${\tau _{i + 1}}$ or ${\tau _i}$ is sufficiently large, the acceleration constraint between adjacent dwell positions is satisfied. The conditions (13) and (17) guarantee that the whole DTD profile satisfies the dynamic constraints.

3.3 DTD repairing scheme based on self-adaptive offset algorithm

Without considering the dynamic performance of machine tool, the initial DTD sequence generated by Section 2 can achieve the deterministic removal. Nevertheless, the realization of dwell time is limited to the dynamic constraints, and thus the dwell time algorithm under dynamic constraints should be studied. In the method [34], the setting of line and step spacings of tool path is limited to dynamic constraints, hindering high-precision calculation of dwell time. Additionally, due to the optimization with large-size variables, the method is still time-consuming for large-aperture workpieces, and the trapezoidal ACC/DEC feedrate profile employed in that work only has C⁰ continuity. To overcome those shortcomings, this work proposes a highly-efficient DTD repairing scheme based on a self-adaptive offset algorithm using trigonometric DTD model under the dynamic constraints of machine tool.

The DTD gradient distribution yielded by Section 2 can achieve the desirable removal distribution. From the analysis in Section 3.1, when DTD gradient distribution is fixed, as the DTDs become sufficiently large, the feedrate and acceleration constraints are satisfied. Hence for the fixed DTD gradient distribution, it is feasible and convenient to provide an offset to the initial DTDs, preserving the ideal dwell time gradient under the dynamic performance.

First, find the minimum DTD in the initial DTD sequence. Then if the minimum is less than lower bound of DTD, lift all the DTDs until the minimum one is just upward to lower bound of DTD, as illustrated in Fig. 3.

 

Fig. 3. DTDs are located above the lower bound after “lifting” the DTD sequence.

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Secondly, scan from the first dwell position and examine whether the DTD profile between adjacent dwell positions satisfy the acceleration constraint. Let the previous DTD be ${\tau _i}$. According to Eq. (13) or (17), check whether ${\tau _i}$ or ${\tau _{i + 1}}$ needs to be adjusted. As illustrated in Fig. 4(a), for the deceleration segment, if ${\tau _i} < \textrm{max}\{{{G_i},{\tau_{min}}} \}$, then lift all the DTDs by $\delta = \textrm{max}\{{{G_i},{\tau_{min}}} \}- {\tau _i}$. For the acceleration segment, if ${\tau _{i + 1}} < \textrm{max}\{{{F_{i + 1}},{\tau_{min}}} \}$, then lift all the DTDs by $\delta = \textrm{max}\{{{F_{i + 1}},{\tau_{min}}} \}- {\tau _{i + 1}}$, as illustrated in Fig. 4(b). The time complexity of scanning procedure is $\mathrm{{\cal O}}(N )$. After completing the scanning of all the DTDs, DTD profile between adjacent dwell positions is generated by trigonometric-spline function. Since the derivatives of DTD profile in the displacement at the dwell positions are zero, DTD profile generated is C¹-continuious, yielding the acceleration-continuous feedrate. The algorithm of computing DTD profile is elaborated in Algorithm 1 DTDREP. By algorithm DTDREP, the DTD profile after the repair satisfies the condition (13) or (17). According to the theoretical derivation in Section 3.2, the dynamic constraints of machine tool during the process are rigidly satisfied, and the feedrate profile has C¹ continuity.

 

Fig. 4. Diagram of DTD repairing algorithm: (a) deceleration segment; (b) acceleration segment.

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Algorithm 1. DTDREP

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When the dwell time without considering dynamic constrains is directly used for the practical polishing, the commercial CNC system automatically modifies the corresponding feedrate of positions where the dynamic constraints are violated. Nevertheless, the automatic feedrate modification in the commercial CNC system is only related to the machine tool performance, and does not preserve the ideal dwell time gradient. That operation inevitably leads to the over-polishing or under-polishing, reducing the convergence rate of surface error. The proposed algorithm traverses the DTD sequence and employes a self-adaptive offset tactics to adjust the DTD sequence such that the ideal relative distribution of dwell time is preserved. From Ref. [17], the DTD offset $\delta $ introduces an extra waviness profile ${W_\delta }(x )$ on the residual figure

4. Real-time interpolator based on trigonometric-spline DTD profile under dynamic constraint

The existing dwell time algorithms rely on the commercial CNC interpolator where the acceleration/deceleration strategy is not open to the users, hindering the high-precision modelling of practical dwell time. This work designs a self-developed interpolator based on the trigonometric-spline DTD profile. The versatility of machine tool (such as applicability to five-axis polishing) can be extended based on the same principle and model proposed.

According to the dwell position spacing, generate the uniform-spacing tool path. And the dwell time solution without dynamic constraints is computed using LR deconvolution scheme, and the initial DTD sequence is generated. Then compute the trigonometric-spline DTD profile $\tau (s )$ using the algorithm DTDREP. The feed rate profile is

The path and feedrate between adjacent dwell positions are expressed by seven elements. Hence the information including the position and feedrate between adjacent path points is stored into a NC code file with the following format.

N00001 $X,Y,Z,{\tau _1},{\tau _2},{s_1},{s_2}$

N00002 $X,Y,Z,{\tau _2},{\tau _3},{s_2},{s_3}$

N00003 $X,Y,Z,{\tau _3},{\tau _4},{s_3},{s_4}$

. . . . . .

After the path and feedrate profiles are generated offline, the NC file is loaded into NC system. The feedbacks of X, Y, Z-axis positions are sampled through encoders to measure the motion accuracy of worktables for the tracking tasks. PID controller is adopted for closed-loop position control. Real Time Extensions (RTX) software is embedded into operation system (Windows XP) to generate both the look-ahead and interpolation timer.

The procedure of real-time interpolation is elaborated as follow. The information after the code of look-ahead block is interpreted and stored into memory buffer. In the stage of calculating referenced interpolation position, the preliminary information is read from the memory buffer. Then determine the referenced interpolation position according to the interpolation period ${T_s}$ and the feedrate profile $v({s({k{T_s}} )} )$ $({k = 1,2,3, \ldots } )$ by the method in Ref. [40]. The information of referenced interpolation position is transferred into feed driver system. PID controller is adopted for closed-loop position control. The system arranges 1 ms for performing the fine interpolation and 0.1 ms for position control. Fig. 5 illustrates the whole flow chart of trigonometric-spline DTD profile generation and real-time interpolation.

 

Fig. 5. The whole flow chart of DTD profile generation and real-time interpolation.

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5. Simulative validation and comparison

5.1 Validation of proposed method

In the simulation, as a typical deterministic polishing technology, MRF is employed to validated the proposed method. Implement the proposed algorithm on a certain surface for example. The initial surface error used is acquired by measuring a circular fused-silica workpiece with a 100 mm diameter, using a laser interferometer. The initial surface error and the residual error map simulated by proposed method are shown in Fig. 6.

 

Fig. 6. Surface errors and DTD maps: (a) initial surface error; (b) ideal DTD map; (c) DTD map after repairing; (d) residual surface error by the proposed method.

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The step and line spacing of raster path is $D = 0.2\textrm{mm}$. On the assumption that the machine tool has infinite dynamic performance, the ideal DTD map is generated, as illustrated in Fig. 6(b). The ideal dwell time matrix without considering dynamic constraints has minimum value 7.8482e-07s, and thus the average feedrate near the dwell position reaches 254.835 m/s, far exceeding the feedrate constraint of the machine tool. Set ${V_{max}} = 4000\textrm{mm}/\textrm{min}$ and ${A_{max}} = 2000\textrm{mm}/{s^2}$. First, lift all the DTDs by initial offset to satisfy the feedrate constraint. Then perform the scanning to examine the acceleration, and an extra offset is yielded. Finally, the total offset is added to each DTD after repairing. The offset operation preserves the ideal dwell time distribution gradient, helping to realize the deterministic removal of material. The DTD map computed by proposed method is Fig. 6(c). The residual surface error by the proposed method is illustrated in Fig. 6(d). From that, the deterministic material removal without counting edge effect is achieved.

The DTD, the feedrate and the acceleration under different dynamic parameters are illustrated in Fig. 7. From Fig. 7(c)-(f), the proposed algorithm generates the DTD profile rigorously satisfying the dynamic constraint by increasing the offset removal amount, and the feedrate profile has C¹ continuity. For the first group of dynamic parameters, the DTD offsets caused by the acceleration and feedrate constraints are 0.0003s/mm and 0.015s/mm, as illustrated in Fig. 7(a). For the second group of dynamic parameters, the DTD offsets caused by the acceleration and feedrate constraints are 0.00045s/mm and 0.005s/mm, as illustrated in Fig. 7(b). From that, the feedrate constraint has far stronger influence on the dwell time than acceleration constraint.

 

Fig. 7. The DTD, feedrate and acceleration profiles under different dynamic parameters: (a),(c),(e) are respectively DTD, feedrate and acceleration under ${V_{max}} = 4000\textrm{mm}/\textrm{min},{A_{max}} = 2000\textrm{mm}/{{\textrm{s}}^2}$; (b),(d),(f) are respectively DTD, feedrate and acceleration under ${V_{max}} = 12000\textrm{mm}/\textrm{min},{A_{max}} = 200\textrm{mm}/{{\textrm{s}}^2}$.

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5.2 Comparison of different methods

In this section, LR method [24], BCLS method [34] and the proposed are compared in the computational efficiency, polishing time and residual map. LR method [24] computes the ideal dwell time distribution, and then performs the feedrate adjustment such that the dynamic constraints are respected. BCLS method [34] transforms feedrate and acceleration constraints into bilateral constraint of dwell time, and employ the Newton iteration to compute the matrix equation.

Set ${V_{max}} = 4000\textrm{mm}/\textrm{min}$ and ${A_{max}} = 2000\textrm{mm}/{{\textrm{s}}^2}$. The ideal residual error map and the residual error maps simulated by different methods are illustrated in Fig. 8. For LR method, the feedrate adjustment cannot preserve the ideal dwell time distribution, inevitably deteriorating the residual error, as illustrated in Fig. 8(b) and the detail. BCLS method has limitations on the lower bound of line and step spacings of tool path, and too small spacing of tool path declines the allowable maximal feedrate and drastically increases heavy computational load, and the feedrate profile generated based on trapezoidal ACC/DEC model has only C⁰ continuity.

 

Fig. 8. Ideal residual error and residual error maps simulated by different methods: (a) ideal residual error; (b) LR method; (c) BCLS method; (d) proposed.

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Considering the prevention of under-polishing and the computational capability, the modulation ratio (the ratio of the minimal-to-maximal feedrate) in BCLS method is set as $\eta = 0$ and the characteristic feedrate (the allowable maximal feedrate) is set as ${V_{max}}$, and thus the spacing of tool path under the same dynamic constraints should satisfy $d \ge V_{max}^2/({2{A_{max}}} )= 1.11\textrm{mm}$. As illustrated in Fig. 8(c) and the detail, the residual error map simulated by BCLS method has obvious ripples, causing the deterioration of middle frequency error. The largest difference of figure error between Fig. 8(a) and Fig. 8(d) is only 0.067 nm, revealing that the residual surface error simulated by the proposed method is nearly consistent with the ideal one. Due to a smaller feasible spacing, the proposed method performs better in suppressing surface ripple than BCLS method. Compared with LR and BCLS methods, the proposed method effectively minimizes residual error under confined feedrate and acceleration, and simultaneously the feedrate profile has C¹ continuity.

The results for different methods are provided in Table 2. From that, the proposed method takes about 2.35 minutes more than LR method to finish the machining in this simulation, achieving much higher convergence than LR method. Since the length of tool path for BCLS method is only about 1/6 of those for LR method and the proposed, BCLS method generates the least offset time, achieving the least polishing time. Nevertheless, BCLS method performs the worst in computational efficiency, and yields the larger residual error than the proposed. Both LR and the proposed methods have good performance in computational efficiency. The proposed method performs the best in restraining the residual error.

Table 2. Simulation results for different methods

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Remark. The line and step spacings of tool path for BCLS method are highly dependent on the dynamic constraints. Assume that ${A_{max}} = 2000\textrm{mm}/{\textrm{s}^2}$ and modulation ratio $\eta = 0$. If the line and step spacing for BCLS method is set as $d = 0.2\textrm{mm}$, the characteristic feedrate is ${V^\mathrm{\ast }} = \sqrt {2d{A_{max}}/({1 - {\eta^2}} )} = 28.3\textrm{mm}/\textrm{s}$. Then DTD offset of BCLS method will exceeds 0.0354s/mm, which is 28.3 times and 11.8 times of that the proposed method generates for ${V_{max}} = 12000\textrm{mm}/\textrm{min}$ and ${V_{max}} = 4000\textrm{mm}/\textrm{min}$ respectively, yielding to low polishing efficiency. Although a larger modulation ratio $\eta $ yields to a larger characteristic feedrate, minimal feedrate constraint ${V_{min}}$ is inevitably imposed on the solution procedure, leading to the potential under-polishing on the region with high removal amount. Additionally, too small spacing for BCLS method increases computational load, and the algorithm does not work without extremely high computing resource.

6. Experiment and comparative analysis

6.1 Experimental design and condition

The validation of algorithm proposed is implemented through MRF of the actual workpiece. The layout of the system architecture for the method proposed is illustrated in Fig. 9.

 

Fig. 9. The layout of the system architecture for the method proposed.

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The configuration of CNC system is 2.5-GHz CPU and 4-G memory. RTX software is embedded into the operation system to generate both the look-ahead and interpolation timer, guaranteeing real-time performance of the interpolation process. The proposed algorithms are coded with VC++2012. The MRF machine tool has the following performance parameters: maximal power ${P_{max}} = 4.5\textrm{KW}$, maximal feedrate ${F_{max}} = 10\,\textrm{m}/\textrm{min}$, maximal acceleration ${A_{max}} = 3500\textrm{mm}/{\textrm{s}^2}$, wheel diameter ${D_w} = 400\textrm{mm}$, rotation speed of wheel $\omega = 75\textrm{r}/\textrm{min}$.

Perform LR method [24], BCLS method [34] and the proposed in the same machine tool respectively. LR method and BCLS method are conducted based on the interpolator of the commercial CNC system (Siemens 840D). The proposed method is performed based on the self-developed NC interpolator. Considering the dynamic stability of MRF ribbon, set maximal feedrate ${V_{max}} = 4000\textrm{mm}/\textrm{min}$, and maximal acceleration ${A_{max}} = 3000\textrm{mm}/{\textrm{s}^2}$.

Except for the initial surface error, the other experimental conditions for three methods are the same. The raster tool path is employed, and the step and line spacing of tool path for LR method and the proposed is $D = 0.2\textrm{mm}$. In BCLS method [34], the step and line spacing is limited to the dynamic constraints of machine tool. Considering the dynamic constraints of machine tool and the computational capability of PC, the step and line spacing for BCLS method is set as 0.8 mm. The experimental conditions of MRF are shown in Table 3 and Fig. 10. The duration of spot-taking is 3s. Three Φ100 mm fused-silica workpieces are polished for comparison. The figures of three workpieces before and after polishing are measured by a Φ150mm-aperture laser interferometer, and the roughness is measured by a white light interferometer with the vertical resolution better than 0.1 nm.

Table 3. Experimental condition parameters of MRF

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Fig. 10. Experimental conditions: (a) polishing spot used; (b) MRF machine tool.

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6.2 Experimental results and analysis

All of three methods take one round of processing respectively. To exclude the influence of the edge effect, the figures of edge area with 2 mm width for three workpieces is cut off. The figures before and after polishing by different methods are illustrated in Fig. 11, where the figures before polishing are filtered to facilitate the calculation of smooth dwell time, and the figures after polishing are unfiltered. The figure error, the roughness average (Ra) and the polishing time are shown in Table 4. From that, the proposed method performs the best in convergence rate and roughness reduction. For the proposed method, PV converges from 992.03 nm to 41.31 nm, achieving 95.84% convergence rate, and the RMS converges from 248.23 nm to 6.02 nm, achieving 97.57% convergence rate. Besides, the Ra is reduced from 0.99 nm to 0.61 nm, achieving 38.38% reduction rate. Although the initial surface errors of LR and BCLS methods are larger than that of proposed, the convergence rate of LR and BCLS methods are lower than that of proposed. For the small targeted removal amount, as illustrated in Fig. 11(a) and (b), the dwell time offset dominates in the polishing time, and the dwell time offset is closely related to the length of tool path. When the targeted removal amount is large, as illustrated in Fig. 11(c), the targeted removal time dominates in the polishing time.

 

Fig. 11. The figures before and after polishing: (a) LR method; (b) BCLS method; (c) proposed method.

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Table 4. The PV, RMS, Ra and polishing time for different methods

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From Fig. 11, the surface error convergence level of BCLS method is much higher than LR method, since the under-polishing positions exist due to the exceeding to the dynamic constraints of the machine tool. And the proposed method performs the best in restraining the error in each frequential range. The proposed algorithm generates C¹-contiuous DTD profile fully satisfying the dynamic constraints of the machine tool compared with LR method, and the impact during the process is significantly suppressed. For BCLS method [34], the lower boundary of the step and line spacing is required, decreasing the discretization level of the dwell time algorithm and hindering the convergence level of the middle and low frequency errors.

Fig. 12 illustrates the PSD profiles in the directions parallel and perpendicular to feed. LR method could not sufficiently consider dynamic constraints in the dwell time solution, and the automatic feedrate adjustment does not consider the preservation of the ideal dwell time gradient, leading to the deterioration of low and middle frequency error, as illustrated in Fig. 12(a) and (b). The line and step spacings of tool path of BCLS method are limited to dynamic constraints. The large spacings leads to the ripple on the surface, hampering the control of the middle frequency error. As shown in Fig. 12(d), the significant peak happens at the frequency around 1.25mm^-1. Additionally, LR and BCLS method generate C⁰ feedrate profile and yields to the drastic feedrate and acceleration fluctuation [41], decreasing the stability of MRF ribbon and deteriorating middle and high frequency errors. The proposed method performs the best in restraining the multi-frequency errors among all methods, as illustrated in Fig. 12(e) and (f).

 

Fig. 12. PSD profiles before and after polishing in the directions parallel and perpendicular to feed for different methods: (a) and (b) for LR method; (c) and (d) for BCLS method; (e) and (f) for proposed method.

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6.3 Comparison of computational efficiency

The offline calculation of dwell time is performed on the computer with 2.30-GHz CPU and 8-G RAM. From Table 4 and Table 5, the proposed method only takes 2.47% more calculation time (0.054s) than LR method, whereas PV and RMS convergence rates of the proposed method are reduced by 45.08% and 37.22% respectively, compared with LR method. The proposed method handles the workpiece with the initial surface error matrix size 1641 × 1641, and totally takes 2.237 seconds to compute the DTD profile, including the initial DTD sequence generation and the DTD scheduling under dynamic constraints of machine tool. BCLS method [34] is performed on the same platform and takes 60.153 seconds to solve the dwell time. BCLS method [34] also considered the dynamic constraints of the machine tool in the computation of dwell time, but the lower boundary of the step and line spacing of tool path is required, and the coefficient matrix of BCLS algorithm is much smaller than that of the proposed. Nonetheless, the computational efficiency of the propose method is raised by 96.28% compared with BCLS method.

Table 5. The calculation time of three methods

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7. Conclusion

Dwell time scheduling is a critical stage of deterministic polishing for ultra-precision manufacturing of optics. The existing dwell time algorithms have some of the following deficiencies, including low computational efficiency, large memory consumption, insufficient consideration of dynamic constraints, and poor smoothness of the feedrate profile, reliance on non-open CNC interpolator. To address those problems, this work proposes a highly-efficient algorithm of computing the dwell time that satisfies the dynamic constraints of machine tools. The method generates the initial DTD sequence by a non-blind deconvolution algorithm, and provides the feasible set of DTD profiles based on trigonometric-spline model, and then develops a DTD repairing tactics using a self-adaptive offset scheme under confined feedrate and acceleration, achieving the time complexity $\mathrm{{\cal O}}(N )$. Finally, C¹-continuous DTD profile satisfying dynamic constraints is generated adopting the trigonometric-spline function. A real-time NC interpolator based on trigonometric-spline DTD profile is developed.

Compared with the previous dwell time algorithms, the proposed method has the following advantages. (1) The method proposes a trigonometric-spline DTD model more reasonably depicting the dwell time distribution near the dwell position. (2) The method preserves ideal dwell time gradient distribution under the sufficiently consideration of dynamic constraints, and simultaneously achieves a high computational efficiency under a small spacing of tool path, helping to restrain the ripple of residual figure. (3) The proposed method generates trigonometric-spline DTD profile with C¹ continuity, and the smooth feedrate profile can restrain the impact caused by acceleration fluctuation, helping to reduce the middle-high frequency error. (4) A real-time NC interpolator based on DTD profile is developed, facilitating the accurate realization of scheduled dwell time. And the principle and method proposed can be also extended to five-axis polishing.

The simulation results show that the proposed method generates C¹-continuous feedrate profile rigidly respecting dynamic constraints, and achieves more ideal residual error with high computational efficiency compared with the LR and BCLS methods. The comparative polishing experiment are conducted by different methods, demonstrating that the proposed method performs better in suppressing the low frequency error and the middle-high frequency error, and achieves high computational efficiency. The PV and RMS convergence rates after actual process by the proposed method reach 95.84% and 97.57% respectively. The proposed method totally takes only 2.237 seconds, reducing 96.28% computational time compared with BCLS method and increasing only 2.47% computational time compared with LR method. The algorithm is applicable to highly-precise and highly-efficient manufacturing of large-aperture optical components.

Appendix

Funding

National Natural Science Foundation of China (61605182, 12162008); Equipment Development Department of the Central Military Commission (211ZW22001); National Key Research and Development Program of China (2022YFB3403402); Innovation Development Foundation of China Academy of Engineering Physics (K1267-2022-TCF).

Acknowledgments

The authors gratefully acknowledge Jun Liu and Chao Cheng, from Institute of Machinery Manufacturing Technology of China Academy Engineering Physics, for providing the assistance for the experimental validation. The authors appreciate the constructive and valuable suggestions from the reviewers.

Disclosures

The authors declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with this work.

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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