Region-adaptive path planning for precision optical polishing with industrial robots

Songlin Wan; Xiangchao Zhang; Min Xu; Wei Wang; Xiangqian Jiang

doi:10.1364/OE.26.023782

1. Introduction

Ultra-precision optical components are increasingly employed in the fields of astronomical observation and opto-electronic industries [1-3]. Polishing is an essential step to obtain an ultra-precision surface. The computer controlled optical surfacing (CCOS) [4] first proposed by Itek Inc. is a commonly used technology of optical polishing. In CCOS, the material removal behavior is described as the convolution between the tool influence function (TIF) and the dwell time. Controlling the dwell time at each sampling point can make the form error of the workpiece converge rapidly [5,6].

Various polishing tools have been developed for different applications, such as the dual-rotation tool [7], magnetorheological tool [8] and ion-beam tool [9]. But the cost of the commercial polishing machines is relatively high, consequently limiting their popularization. The robot-based sub-aperture polishing method has the potential to reduce the cost by 20 times compared to existing commercial solutions [10]. But the low control accuracy and low stiffness of the industrial robots worsen the stability of the removal amount. This issue is especially severe when the material removal amount is small, in which case the running speed is high and the material removal behavior is difficult to be controlled. Henceforth the resulting form error is still in the micron level, which cannot meet the requirements of the ultra-precision optical components.

In the traditional computer controlled polishing, the paths are usually planned along a regular route to cover the whole surface, e.g. in equispaced concentric circles or squared grids. It is common to amend the material removal depth all over the whole surface, and the surface quality cannot improve further by implementing more polishing cycles, making the robotic polishing undeterministic and inefficient. Some new path planning methods have been developed in recent years [11–13], but these methods mainly focus on improving the mid-frequency quality by increasing the randomness of the paths. The paths have too many sharp turning angles, thus they suffer from the polishing instability. As a result, an effective path planning method is desired to address the effects of the controlling inaccuracy and running instability of the robotic polishing, so as to improve the surface quality and fabricating efficiency of large optical mirrors.

This paper is organized as follows. A region-adaptive path planning method is proposed in Section 2. Then a space-variant deconvolution algorithm is developed for calculating the dwell time in Section 3. Section 4 illustrates the experiment setup and results of this new polishing method. Finally the paper is summarized in Section 5. See Table 1 for nomenclature.

Table 1. Nomenclature.

View Table | View all tables in this article

2. Region-adaptive path planning

To improve the convergence rate of robotic polishing, the paths are arranged in such a way that they only go through the regions with relatively large form errors, henceforth the fabricated area and material removal depth can both be significantly reduced, and in turn the form quality will not be deteriorated. And we named it “region-adaptive” method. The idea of this method is similar to the conventional manual polishing. A schematic of this method is shown in Fig. 1.

Fig. 1 Schematic of region-adaptive method.

Download Full Size | PDF

However, the resulting moving speed of the polisher can be much higher than the traditional regular paths, because the overall material removal amount is decreased and the required dwell time is shortened accordingly. Therefore the polishing path should be sufficiently smooth to ensure that the polisher can run stably; and this is especially critical for the robot polishers because of their low stiffness and low control accuracy. As a result, a new region-adaptive path planning method is proposed.

2.1 Determine the processing regions

The processing region Reg is defined as the area with its form error err_m greater than a threshold t₊

\begin{array}{l} R e g = {(x, y) | e r r_{m} (x, y) > t_{+}} \\ with e r r_{R} = {e r r (x, y) | 0 < x^{2} + y^{2} < m^{2} \cdot R^{2}} \\ e r r_{m} = e r r_{R} + 3 \cdot r m s (e r r_{R}) - t_{-}, \end{array}

where err is the measured residual error map, with the piston and tilt terms removed. err_R is the error map within the reachable area of polishing tool. The parameter m specifying the reachable region is set between 0 and 1 for small-tools and it can be greater than 1 for magnetohelogical-tools and ion-beam tools. err_m is a non-negative error map obtained by lifting err_R to a suitable height. In order to avoid the effect of outliers, the lifting amount is determined by 3 rms of the residual adjusted by a tolerance t_-, rather than the lowest point.

Obviously the choice of two parameters t₊ and t_- are critical for the selection of the processing region. For the sake of simplicity we set the default values of both parameters as λ/40, because the desired peak-to-valley (PV) form error of most optical components is not higher than λ/20. Users can select appropriate values according to their specific situations.

We exemplify the processing regions of some surfaces, where the diameters of the workpieces are set to be 200 mm and the parameter m is set to be 0.9, as shown in Fig. 2.

Fig. 2 Processing regions of different surface shapes.

Download Full Size | PDF

Evidently when the residual error is large enough, the processing regions can almost cover all the reachable area of the polishing tools. While the area of the processing regions will become smaller when the residual error decreases. To make the results more intuitive, the surface shown in Fig. 2(c) is taken as an example in the following sections.

2.2 Hexagonal meshing of the processing regions

The processing regions are first converted into hexagonal grids, mainly because the routes connecting neighboring cells have higher flexibility and smaller turning angles in the hexagonal grids, which makes the generated paths smoother and more diverse, and this is critical for improving the polishing accuracy and efficiency. On the contrary, in the squared grid each cell has only four neighboring cells, and the generated paths can be jagged and unfriendly to the processing regions of various complex shapes.

The hexagonal cells are numbered in order, and the conversion formula between the numbering indices and the Cartesian coordinates can be expressed as:

{\begin{cases} x = \frac{3}{4} \cdot p \cdot d_{h e x} \\ y = - \frac{\sqrt{3}}{2} \cdot (q + \frac{1}{2} \cdot p) \cdot d_{h e x} \end{cases},

where x and y are the Cartesian coordinates of a cell, and p and q are the corresponding sequential numbers. d_hex is the diagonal length of the of the hexagonal cells. A schematic is shown in Fig. 3.

Fig. 3 Schematic of hexagonal grids numbering system.

Download Full Size | PDF

In the planned polishing path, the tool can only move to an adjacent cell from the current cell. The minimum step number from cell A to cell B can be expressed as:

\begin{array}{l} D (A, B) = \max (| Δ p |, | Δ q |, | Δ p + Δ q |) \\ with Δ p = p_{A} - p_{B}, \end{array}

where (p_A, q_A) and (p_B, q_B) are the sequential numbers associated with two hexagonal cells A and B, respectively.

According to Eq. (2), the processing regions can be easily meshed by traversing the hexagonal cells in the reachable region. For each cell, if its center is within the processing region, we treat this cell as a passable cell; otherwise this cell is treated as impassable. The passable cells are plotted as the hexagonalized processing region in Fig. 4(b).

Fig. 4 The adjustment of processing regions after hexagonal meshing.

Download Full Size | PDF

However, it is found that there exist some small isolated passable or impassable cells, in which case the path planning can be hindered to some extent. We change these isolated passable cells into impassable ones and change the isolated impassable cells into passable ones to improve the continuity of the polishing paths. This manipulation can improve the connectivity of the hexagonal grid and it is helpful to ease the following path planning. The passable hexagonal grids after adjustment are shown in Fig. 4(c). Evidently the boundaries of the processing regions are made much smother.

2.3 Path generation

Here we propose to generate spiral-like paths based on the hexagonal meshing. The generation of a spiral path can be divided into three steps, generating primary paths, merging into complete paths, and finally smoothing.

A primary spiral-like path is generated based on the spiral filling rule. The polishing tool keeps moving along the boundary of the “obstacles” counterclockwise inwards to the interior region, and during the movement the impassable cells and the passed cells are regarded as “obstacles”. Each time an obstacle is encountered which obstructs the path going further, the path turns 60° counterclockwise to a neighboring cell. However, it is usually infeasible to traverse the whole processing regions solely in one path with the spiral filling rule due to the complexity of the regions. Instead we continue to generate another path from a passable cell closest to the terminating node each time the planned path terminates prematurely by chance. Then new primary traversal paths are generated by repeating this spiral filling step until all the passable cells are traversed. The generated traversal paths are shown in Fig. 5(a).

Fig. 5 Primary traversal paths diagram.

Download Full Size | PDF

However, the number of paths can be very large when the shape of the processing region is complicated, leading to low efficiency and poor continuity in optical polishing. Thus some primary paths need to be merged or deleted. Two parameters L₁ and L₂ are defined for this purpose. If the starting point of the new path is adjacent to any cell of the last L₁ nodes of the previous path, these two paths can be merged and the fragment in the previous path, i.e., the nodes after the adjacent point should be ignored. Then those paths containing less than L₂ nodes are deleted as well. The processing procedure is presented below.

    FOR (FROM the first path TO the last path)
    IF (any of its last L₁ nodes is adjacent to the 1st node of the new path)
    delete the fragment of L₁ nodes;
              merge these two paths;
      END
      IF (adjacent node not found)
           save the integrated path;
           define another integrated path;
      END
    END
    FOR (FROM the first integrated path TO the last one)
      IF (the node number of this path < L₂)
           delete the path;
      END
    END

The greater the L₁ is, the more paths will be merged, with more nodes ignored as fragments. And the greater the L₂ is, the more paths will be deleted. For the sake of simplicity we set the default values of both two parameters to be D_TIF/d_hex, where D_TIF is the diameter of TIF. After that, the number of the paths can be greatly reduced, thereby improving the applicability of the paths.

In order to improve the moving stability, the paths are smoothed using a Gaussian filter, which can be expressed as a convolution operation:

\begin{array}{l} X'^{(n)} (t) = X^{(n)} (t) * G (t; σ) \\ Y'^{(n)} (t) = Y^{(n)} (t) * G (t; σ) \\ with G (t; σ) = \frac{1}{\sqrt{2 π} σ} \exp (- \frac{t^{2}}{2 σ^{2}}) - 3 σ < t < 3 σ, \end{array}

where the symbol * denotes the convolution operation. X⁽ⁿ⁾ and Y⁽ⁿ⁾ represent the coordinates of the n-th path, while X’⁽ⁿ⁾ and Y’⁽ⁿ⁾ are the corresponding coordinates after smoothing. The standard deviation σ of the Gaussian window can be adjusted, usually set to be 1.

In Fig. 5, the two parameters L₁ and L₂ are both set to be 10, and σ is set to be 1. The paths in Fig. 5(c) correspond to the surface shown in Fig. 2(c).

3. Calculation of dwell time of region-adaptive path

During the polishing process, the material removal amount can be calculated as the convolution between the TIF and the dwell time:

z (x, y) = T I F (x, y) * T (x, y),

where * indicates the convolution operation, z(x,y) is the material removal amount, T(x,y) is the dwell time matrix and TIF(x,y) is the tool influence function.

Conventionally, the convolution in Eq. (5) is represented in a matrix form, and the TIF is considered as space-invariant. But the sampling points in the generated paths are non-uniformly distributed, thus the dwell time should also be calculated at non-regular discrete points. When polishing a complex curved surface, the polishing pressure is affected by the form deviation between the polishing pad and the workpiece [14]. In addition, when the polishing tool overhangs the workpiece, the pressure increases suddenly at the edge region [15]. According to the pressure distribution and the relative velocity between the pad and the workpiece, a reliable space-variant TIF model can be obtained with the Preston equation [15].

To describe the material removal behavior more precisely by considering non-uniform sampling points and the space-variant TIFs, the convolution operation is replaced by a linear transformation:

\hat{z} = \hat{H} \cdot \hat{T},

Here $\hat{z}$ is a column vector obtained by stacking either the rows or columns of z(x,y), and $\hat{T}$ is a column vector representing the dwell time. Suppose z(x,y) is a M × N matrix, then $\hat{z}$ is a MN × 1 vector. Suppose $\hat{T}$ is a J × 1 vector, where J is the total number of the sampling points, then $\hat{H}$ is a MN × J matrix composed of vectorized TIFs.

To work out the dwell time, we adopt the infinite norm of the residual error as the objective function to minimize the peak-to-valley (PV) of the error map, which is of major concern in the ultra-precision polishing area. For a workpiece with an effective optical aperture k time of the workpiece diameter, the optimization function is expressed as:

\begin{array}{l} \min E (\hat{T}) = \underset{i}{\min \max} {| {\overset{⌢}{δ}}_{i} | | 0 < x_{i}^{2} + y_{i}^{2} < k^{2} \cdot R^{2}} \\ s .t \overset{⌢}{δ} = {\hat{e}}_{p} - \hat{H} \cdot \hat{T} \\ e_{p} = e r r + 3 \cdot r m s (e r r_{R}) - t_{-} \\ T_{\min} < \hat{T} < T_{\max}, \end{array}

where δ represents the predicted error map after polishing, err and err_R are the matrices mentioned in Eq. (1), and T_min and T_max are the allowed minimum and maximum values of the dwell time, respectively.

It should be noticed that the mini-max optimization problem in Eq. (7) is sensitive to measurement noise and outliers. Therefore an additional denoising process is implemented before optimization to eliminate defects and outliers. The gradient of the measurement data can be used as a criterion to distinguish the outliers, and a median filter is adopted for denoising. Moreover, the mini-max optimization problem is numerically unstable and non-differentiable, thus it is difficult to be solved directly. Here an exponential penalty function is used to convert Eq. (7) into a continuously differentiable problem [16]:

\min E_{p} (\hat{T}) = \frac{1}{p} \log \sum_{i} \exp (p \cdot d_{i}) with d_{i} = {\hat{δ}}_{i}^{2.}

Here $E_{p} (\hat{T})$ is the exponential penalty function associated with $E (\hat{T})$ . According to the numerical approximation theories, the optimal solution of Eq. (8) becomes closer to that of Eq. (7) with p increasing. As a consequence derivative-based methods, e.g. the Newton’s method or the conjugate gradient method can be applied to solve Eq. (8) [16]. The gradient vector is calculated as:

\begin{array}{l} \nabla E_{p} = \frac{\partial E_{p}}{\partial \hat{T}} = {\sum_{i} {\hat{ζ}}_{i} \nabla d_{i} | 0 < x_{i}^{2} + y_{i}^{2} < k^{2} \cdot R^{2}} \\ with \nabla d = 2 \cdot {\hat{H}}^{T} \cdot d i a g (\hat{ζ}) \cdot ({\hat{e}}_{p} - \hat{H} \cdot \hat{T}) \\ d i a g (\hat{ζ}) = (\begin{matrix} {\hat{ζ}}_{1} & 0 \\ ⋱ \\ 0 & {\hat{ζ}}_{M N} \end{matrix}) \\ {\hat{ζ}}_{i} = \exp (p \cdot d_{i}) / \sum_{i} \exp (p \cdot d_{i}) . \end{array}

In practice, the calculation concerning $\hat{H}$ and ${\hat{H}}^{T}$ in Eq. (9) is difficult to be implemented due to their huge matrix sizes. But $\hat{H}$ and ${\hat{H}}^{T}$ are sparse matrices and they can be further simplified. Dong et al simplified the calculation using the sparse properties [17], but the TIF therein is considered to be space-invariant, and interpolation is required to calculate the dwell time, which can reduce the computational efficiency and polishing accuracy. Here generalized convolution and correlation operations are developed to simplify the calculation of large matrices into that of small matrices, which directly employs non-uniform sampling points and space-variant TIFs. The generalized operations can be expressed as:

\begin{array}{l} convolution: {\hat{B}}_{(M N \times 1)} = {\hat{H}}_{(M N \times J)} \cdot {\hat{A}}_{(J \times 1)} \Leftrightarrow B (x, y) = \sum_{j = 1}^{J} T I F^{(μ_{j}, η_{j})} (x - μ_{j}, y - η_{j}) \cdot \hat{A} (j) \\ correlation: {\hat{A}}_{(J \times 1)} = {\hat{H}}^{T}_{(J \times M N)} \cdot {\hat{B}}_{(M N \times 1)} \Leftrightarrow \hat{A} (j) = \sum_{x} \sum_{y} T I F^{(μ_{j}, η_{j})} (x - μ_{j}, y - η_{j}) \cdot B (x, y), \end{array}

where (μ_j, η_j) are the coordinates of the j-th sampling point, and TIF⁽^μj^, ^ηj⁾ is the tool influence function at this point.

\hat{A}

,

\hat{B}

and B are matrices with the dimensions J × 1, MN × 1 and M × N, respectively.

In Eq. (10), the calculation associated with $\hat{H}$ and ${\hat{H}}^{T}$ is replaced by the superposition of the TIFs at different positions, which is similar to the convolution and correlation operations. The size of the matrix TIF is generally small, and the TIFs of all the sampling points can be stored in a hash table, thereby improving the efficiency of deconvolution and reducing the space occupied. This improvement makes the deconvolution process realizable using a personal computer. The detailed deconvolution algorithm is presented below.

Obtain the coordinates of all the sampling points (μ_j, η_j) in the region-adaptive path;
Calculate the TIFs at each sampling point

T I F^{(μ_{j}, η_{j})} (x, y)

;
    delete the fragment of L₁ nodes;
    t = 0;
    Denoise err(x,y);
    Calculate e_p(x,y) by Eq. (7);
    Calculate the initial dwell time matrix T₀(x,y):

T_{0} (x, y) = e_{p} (x, y) \cdot J / \sum_{j = 1}^{J} \iint_{x, y} T I F^{(μ_{j}, η_{j})} (x, y) d x d y

;
    _{${\hat{T}}_{0}$} integrated by the T₀(x,y) based on the Voronoi diagram [18];
    WHILE (t<N_iter)
              Compute the residual error:

δ (x, y) = {e_{p} (x, y) - \sum_{j = 1}^{J} T I F^{(μ_{j}, η_{j})} (x - μ_{j}, y - η_{j}) \cdot \hat{T} (j) | 0 < x^{2} + y^{2} < k^{2} \cdot R^{2}}

              IF (PV of δ(x,y)< PV_thres)
              break;
              END
              Compute the gradient:

\nabla E_{p} (j) = {\sum_{x} \sum_{y} T I F^{(μ_{j}, η_{j})} (x - μ_{j}, y - η_{j}) \cdot ζ (x, y) \cdot δ (x, y) | 0 < x^{2} + y^{2} < k^{2} \cdot R^{2}}

Update the dwell time at each sampling point:

{\hat{T}}_{t + 1} = {\hat{T}}_{t} + γ \cdot \nabla E_{p}

Correct T out of range limits:c

\begin{array}{l} {\hat{T}}_{t + 1} ({\hat{T}}_{t + 1} < T_{\min}) = T_{\min}; \\ {\hat{T}}_{t + 1} ({\hat{T}}_{t + 1} > T_{\max}) = T_{\max}; \end{array}

t=t+1;
END

Take the shape shown in Fig. 2(c) as an example, the dwell time and the resulting error map after polishing are shown in Fig. 6

Fig. 6 The results calculated by proposed deconvolution algorithm (parameter setting: N_iter = 20; γ = 0.1; T_min = 0.05 s; T_max = 5 s;).

Download Full Size | PDF

The total polishing time is only 2.7 minutes due to the small removal amount. And the form error associated with the 90% effective aperture is shown in Fig. 6(b). The residual errors at the top and bottom of the workpiece become relatively large, mainly because the paths do not pass through these areas. But these errors can converge in the following polishing cycles.

4. Experimental demonstration

4.1 Experimental setup

A polishing facility is built based on an ASEA Brown Boveris (ABB) IRB 6620 robot, with a polishing tool mounted on the robot flange. The polishing tool is a dual-rotation structure with a circular pad. The industrial robot controls the position and posture of the polishing pad; and the polishing tool changes the shape of the TIF and the material removal rate by adjusting the velocity ratio, eccentricity and the pressure applied on the pad. The spin and orbital motions of the polishing pad are controlled by the motors on the polishing equipment via RAPID commands. An adjustable pressure cylinder inside the polishing tool is used to control the pressure. The robotic polisher is shown in Fig. 7.

Fig. 7 ABB robotic polisher and the polishing pad.

Download Full Size | PDF

The specifications of the robotic polisher are shown in Table 2.

Table 2. Specifications of the robotic polisher

View Table | View all tables in this article

To demonstrate the validity of the region-adaptive path, we conducted a polishing experiment on a concave mirror whose diameter is 200 mm and the radius of curvature is 1500 mm. As long as the path covers the entire surface, the material removal amount needs to be large enough, then severe polishing error will be caused due to the instability of robotic polishers. Thus the PV of the resulting form error polished by such paths will be similar regardless of the setting of the path trajectory. For the purpose of comparison, in the experiment covering the entire surface is applied recursively until the form error cannot improve further. Then the surface is polished by using the new path proposed in this paper to observe whether the surface can converge continuously; and the working conditions are kept identical in the whole polishing process. The schematic diagram of the experiment plan is illustrated in Fig. 8. The detailed working conditions of the experiments are listed in Table 3.

Fig. 8 The schematic diagram of the experiment plan.

Download Full Size | PDF

Table 3. Detailed working conditions

View Table | View all tables in this article

To ensure the TIFs are Gaussian-like, the curvature of the polishing pad should be greater than the workpiece. The profile of the polishing pad is measured by a scanning profiler Form Taylsurf PGI BLU. The polishing pad and the measurement results are shown in Fig. 9. The fitted radius of the curvature is 1.45 m and the shape of the TIF is Gaussian-like.

Fig. 9 The measured profile of polishing pad and the corresponding tool influence function.

Download Full Size | PDF

4.2 Experiment results

The experiments contain 14 polishing cycles, where the traditional spiral path planning method is used in the first 10 cycles and the proposed new planning method is used in the following 4 cycles. And their form error of 90% effective aperture and the corresponding polishing paths are shown in Fig. 10.

Fig. 10 Form error maps and polishing paths.

Download Full Size | PDF

To assess the surface quality and the polishing efficiency more reliably, we introduce a parameter PVr to assess the form error of the workpiece due to its better robustness than PV [19]; and the polishing time of each cycle is recorded to evaluate the polishing efficiency. The PVr and the polishing time of each cycle are plotted in Fig. 11. The coordinate axes in Fig. 11 are both set to be logarithmic to make the result intuitive.

Fig. 11 PVr of form error map and polishing time.

Download Full Size | PDF

In Fig. 10(a) and Fig. 11, it is straightforward to find that the form error converges quickly at the early stage. But when the form error reduces down to λ/10, the PVr fluctuates in the first 10 cycles, and it even increases in the 4th, 7th, 8th and 10th polishing cycles. The residual error cannot be improved further due to the controlling instability of the industrial robotic polisher, and the total removal amount associated with the regular Archimedes spiral path needs to be amended; In addition, the polishing time is sustained around 15 minutes because of the low convergence rate.

The planned region-adaptive paths and the corresponding error maps are shown in Fig. 10(b). It is worth noting that the path for each cycle is calculated based on the error map of the previous cycle. In the experiment, the form error continues to converge until the low frequency form error is almost eliminated, and the PVr of the form error finally decreases down to 0.061 λ, i.e 38.6 nm. And the resulting form error contains only mid-frequency components, which is difficult be eliminated by the sub-aperture polishing. This is the main reason why the form quality cannot improve further. More remarkably, the polishing time is reduced by 80% compared with the conventional routes, and the time of the final polishing cycle is less than 1 minute.

5. Conclusions

A new path planning method adaptive to the form errors is developed, and a deconvolution algorithm is proposed accordingly to calculate the dwell time. This path planning method is similar to the manual polishing process, where only the “convex area” of the form error is polished without affecting the “concave portion”, thereby greatly saving the polishing time. In addition, this method can reduce the polishing error caused by the low control accuracy of the robotic polisher, and it can be employed in other sub-aperture polishing machines to improve the polishing stability and accuracy.

The experiment results verify that the region-adaptive method can significantly improve the form quality of the robotic polishing. As far as the authors know, the PVr of the robot polishing achieves λ/15 for the first time, which makes this polishing technology competent with conventional polishing machines. In addition the total polishing time can be greatly reduced by more than 80%. As a result these improvements have a great significance to the development of robotic polishing and precision optical manufacturing.

Funding

Science Challenging Program (JCKY2016212A506-0106); National Natural Science Foundation of China (NSFC) (51875107); National Key Research and Development Program of China (2017YFB1104700); Horizon 2020 EMPIR project (15SIB01 FreeFORM).

References and links

1. D. D. Walker, A. T. H. Beaucamp, V. Doubrovski, C. Dunn, R. Freeman, G. McCavana, R. Morton, D. Riley, J. Simms, and X. Wei, “New results extending the precessions process to smoothing ground aspheres and producing freeform parts,” Proc. SPIE 5869, 79–87 (2005). [CrossRef]

2. M. Tricard, P. Dumas, and G. Forbes, “Subaperture approaches for asphere polishing and metrology,” Proc. SPIE 5638, 284–299 (2005). [CrossRef]

3. F. Klockea, C. Brecherb, R. Zunkea, and R. Tuecksb, “Corrective polishing of complex ceramics geometries,” Precis. Eng. 35(2), 258–261 (2011). [CrossRef]

4. R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16(1), 218–224 (1977). [CrossRef] [PubMed]

5. M. Schinhaerl, R. Rascher, R. Stamp, L. Smith, G. Smith, P. Sperber, and E. Pitschke, “Utilisation of time-variant influence functions in the computer controlled polishing,” Precis. Eng. 32(1), 47–54 (2008). [CrossRef]

6. H. Li, W. Zhang, and G. Yu, “Study of weighted space deconvolution algorithm in computer controlled optical surfacing formation,” Chin. Opt. Lett. 7(7), 627–631 (2009). [CrossRef]

7. R. A. Jones, “Computer controlled polisher demonstration,” Appl. Opt. 19(12), 2072–2076 (1980). [CrossRef] [PubMed]

8. W. I. Kordonski and S. D. Jacobs, “Magnetorheological finishing,” Int. J. Mod. Phys. B 10(23n24), 2837–2848 (1996). [CrossRef]

9. A. I. Stognij, N. N. Novitskii, and O. M. Stukalov, “Nanoscale ion beam polishing of optical materials,” Tech. Phys. Lett. 28(1), 17–20 (2002). [CrossRef]

10. X. Tonnellier, P. Comley, X. Q. Peng, and P. Shore, “Robot based sub-aperture polishing for the rapid production of metre-scale optics,” in Proceedings of LAMDAMAP 2013 - Laser Metrology and Performance X (2013).

11. C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16(23), 18942–18949 (2008). [CrossRef] [PubMed]

12. H. Y. Tam, H. Cheng, and Z. Dong, “Peano-like paths for subaperture polishing of optical aspherical surfaces,” Appl. Opt. 52(15), 3624–3636 (2013). [CrossRef] [PubMed]

13. T. Wang, H. Cheng, W. Zhang, H. Yang, and W. Wu, “Restraint of path effect on optical surface in magnetorheological jet polishing,” Appl. Opt. 55(4), 935–942 (2016). [CrossRef] [PubMed]

14. S. Wan, X. Zhang, H. Zhang, M. Xu, and X. Jiang, “Modeling and analysis of sub-aperture tool influence functions for polishing curved surfaces,” Precis. Eng. 51, 415–425 (2018). [CrossRef]

15. S. Wan, X. Zhang, X. He, and M. Xu, “Modeling of edge effect in subaperture tool influence functions of computer controlled optical surfacing,” Appl. Opt. 55(36), 10223–10228 (2016). [CrossRef] [PubMed]

16. X. Zhang, H. Zhang, X. He, M. Xu, and X. Jiang, “Chebyshev fitting of complex surfaces for precision metrology,” Measurement 46(9), 3720–3724 (2013). [CrossRef]

17. Z. Dong and H. Cheng, “Toward the complete practicability for the linear-equation dwell time model in subaperture polishing,” Appl. Opt. 54(30), 8884–8890 (2015). [CrossRef] [PubMed]

18. M. De Berg, M. Van Kreveld, M. Overmars, and O. C. Schwarzkopf, Computational Geometry (Springer, Berlin, Heidelberg, 2000), Chap. 7.

19. C. J. Evans, “PVr-a robust amplitude parameter for optical surface specification,” Opt. Eng. 48(4), 043605 (2009). [CrossRef]


Reg:	the selected processing regions
err_m:	the modified form error
err_R:	the form error in the reachable area
err:	the measured residual error
R:	the radius of curvature of the workpiece
m:	the ratio specifying the reachable area of polishing tool
t_-, t₊:	the thresholds used to determine the processing region
p, q:	the sequential numbers of hexagonal cells
d_hex:	the diagonal length of the hexagonal cells
D(A,B):	the minimum step number from cell A to cell B
L₁:	the threshold used for path merging
L₂:	the threshold used for path deletion
X’⁽ ⁿ ⁾, Y’⁽ ⁿ ⁾:	the node coordinates in the n-th smoothed path
X⁽ ⁿ ⁾, Y⁽ ⁿ ⁾:	the node coordinates in the n-th integrated path
G(t;σ):	the Gaussian kernel with standard deviation σ
z(x,y):	the material removal amount at point (x, y)
T(x,y):	the dwell time at point (x, y)
TIF:	the tool influence function
$\hat{z}$ :	the vectorized z(x,y)
$\hat{T}$ :	the vectorized T(x,y)
$\hat{H}$ :	the matrix consisting of vectorized TIFs
k:	the ratio between the effective aperture and workpiece diameter
e_p^:	the error map used in deconvolution
${\hat{e}}_{P}$ :	the vectorized e_p
PV_thres:	the threshold used for terminating the iteration
N_iter:	the maximum number of iterates
γ:	the stepping ratio

Industrial robot	model	ABB IRB6620
	repetitive positioning error	<0.1 mm
	maximum load	150 kg
Polishing tool	velocity range of spin motion	0~200 rpm
	velocity range of orbital motion	0~200 rpm
	pressure range	0~100 N
	optional diameters of polishing pad	10 mm, 15 mm, 20 mm, 25 mm, 40 mm
	Eccentricity range	0~40 mm

Tool diameter	25 mm
Tool material	Polyurethane
Spin velocity	200 rpm
Orbital velocity	−119 rpm
Eccentricity	5 mm
Force	10 N
Workpieces	concave mirror with 200 mm aperture
Workpiece material	BK7
Slurry	10% wt. CeO₂ (grain radius: 0.8 μm)
Parameter selection in the algorithm	m = 0.9, k = 0.9, t₊ = λ/40, t_- = λ/40; PV_thres = 0.05 λ; N_iter = 20; γ = 0.1; T_min = 0.05 s; T_max = 5 s;d_hex = 4 mm, L₁ = 10, L₂ = 10, σ = 1


Reg:	the selected processing regions
err_m:	the modified form error
err_R:	the form error in the reachable area
err:	the measured residual error
R:	the radius of curvature of the workpiece
m:	the ratio specifying the reachable area of polishing tool
t_-, t₊:	the thresholds used to determine the processing region
p, q:	the sequential numbers of hexagonal cells
d_hex:	the diagonal length of the hexagonal cells
D(A,B):	the minimum step number from cell A to cell B
L₁:	the threshold used for path merging
L₂:	the threshold used for path deletion
X’⁽ ⁿ ⁾, Y’⁽ ⁿ ⁾:	the node coordinates in the n-th smoothed path
X⁽ ⁿ ⁾, Y⁽ ⁿ ⁾:	the node coordinates in the n-th integrated path
G(t;σ):	the Gaussian kernel with standard deviation σ
z(x,y):	the material removal amount at point (x, y)
T(x,y):	the dwell time at point (x, y)
TIF:	the tool influence function
$\hat{z}$ :	the vectorized z(x,y)
$\hat{T}$ :	the vectorized T(x,y)
$\hat{H}$ :	the matrix consisting of vectorized TIFs
k:	the ratio between the effective aperture and workpiece diameter
e_p^:	the error map used in deconvolution
${\hat{e}}_{P}$ :	the vectorized e_p
PV_thres:	the threshold used for terminating the iteration
N_iter:	the maximum number of iterates
γ:	the stepping ratio

Industrial robot	model	ABB IRB6620
	repetitive positioning error	<0.1 mm
	maximum load	150 kg
Polishing tool	velocity range of spin motion	0~200 rpm
	velocity range of orbital motion	0~200 rpm
	pressure range	0~100 N
	optional diameters of polishing pad	10 mm, 15 mm, 20 mm, 25 mm, 40 mm
	Eccentricity range	0~40 mm

Region-adaptive path planning for precision optical polishing with industrial robots

Abstract

1. Introduction

2. Region-adaptive path planning

2.1 Determine the processing regions

2.2 Hexagonal meshing of the processing regions

2.3 Path generation

3. Calculation of dwell time of region-adaptive path

4. Experimental demonstration

4.1 Experimental setup

4.2 Experiment results

5. Conclusions

Funding

References and links

Cited By

Figures (11)

Tables (3)

Equations (10)

Optics Express