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Freeform optics are defined as nonrotational symmetric optical surfaces in the manufacturing industry. Freeform optics are extensively applied to many areas in order to improve system performance. Fast tool servo (FTS) assisting single-point diamond turning technology has high application prospects in freeform optics machining. This paper discusses the interpolation algorithm for tool path generation of FTS through the application of a radial basis function (RBF) algorithm. For this purpose, a positive definite RBF with compact support was employed as the interpolant. The existence is mathematically proven. Numerical simulations were performed to compare the performances of the RBF algorithm and commonly used algorithms for satisfying the requirements of existence, smoothness, and accuracy. Machining experiments were also conducted to validate the applicability of the algorithm. The simulation results showed that the RBF interpolation algorithm outperformed other algorithms in terms of smoothness. The RBF algorithm also provided the highest interpolation accuracy. Furthermore, the RBF interpolation algorithm exhibited the highest accuracy for error distribution, with large errors distributed mainly in transition areas. The machining results were also in general agreement with the simulation results although obvious practical errors were observed. Overall, RBF interpolation can provide higher accuracy and better smoothness in the tool path generation of FTS.

Freeform optics are extensively applied in various areas, such as aerospace, energy, and life science, to improve system performance and reduce overall system complexity and system weight [

Freeform optics can be fabricated using various techniques such as micromilling, raster-cutting or fly cutting, profile and form grinding, and fast tool servo (FTS) [

A typical machine configuration for FTS machining is shown in Figure

Typical configuration of fast tool servo machining.

With the maturity of FTS equipment, many kinds of commercial FTS with various strokes and bandwidths have already been designed to fabricate different kinds of freeform optics [

The procedure of tool path generation of FTS can be described as follows [

Tool compensation: cutting tool compensation should be calculated because surfaces have mutative curvature and different cutting points exist between the tool and workpiece [

Regeneration of freeform surfaces: the scattered data should be interpolated into gridded data to guarantee the online interpolation speed and avoid frequent reciprocation of the

Surface decomposition: the original freeform surfaces can be decomposed into two parts—a rotational symmetric surface (generated by the

Online interpolation of cutting points: the cutting points are calculated online based on the interpolation algorithm. Then, the movements of FTS can be obtained based on the current position of the

Based on the procedure of FTS machining, it can be concluded that the interpolation algorithm is very important. An algorithm with high accuracy and calculation speed is required. However, the existing algorithms are not sufficient in speed and accuracy. This study focused on the tool path generation technique of FTS. An interpolation algorithm with higher accuracy in the tool path generation technique was investigated to improve the machining profile accuracy of FTS.

The interpolation algorithm in the tool path generation of FTS affects the machining accuracy of freeform optics. The interpolation algorithm totally decides the accuracy of the described surface in the computer, controlling the motion of the four axes. The requirements of interpolation in tool path generation of FTS are as follows. Firstly, the interpolation algorithm should have the ability to obtain a determined solution at any point on the surface (existence). Secondly, the interpolation function should be second-order continuous to guarantee smoothness of surface (smoothness). Finally, interpolation errors should be below the sub-nanometer scale in the

Hu et al. used the four-point mean value interpolator for FTS diamond turning to improve the surface accuracy of the freeform surface and meet the real-time requirement compared with nonuniform rational B-splines (NURBS) [

In this study, radial basis function (RBF) interpolation was applied as the interpolation algorithm in the tool path generation of FTS to process freeform optics. RBF interpolation was adopted because it can satisfy the three requirements of FTS. Firstly, it has the advantage of high accuracy, improving machining accuracy. Secondly, it has the advantage of fast convergence, which facilitates the acquisition of a determinate solution at any point on the surface. Thirdly, most RBFs are second-order differentials, ensuring high smoothness of the surface approximated by RBF interpolation. The better smoothness further ensures better surface roughness.

The interpolation algorithm for the tool path generation of FTS should be classified in the multivariate scattered data interpolation problem. The problem of interpolation for FTS can be described as follows: a set of 3D data {_{j}, _{j}, _{j}} _{j} _{=} _{1, 2, …, n} can be obtained based on equations of the freeform surface to be processed. Here, we define {_{j}, _{j}]} _{j} _{=} _{1, 2, …, n} as sampling points and {_{j}} _{j} _{=} _{1,2, …, n} as sagittal height. A function _{j}}. Thus,

Furthermore, the problem can be described into a general mathematical problem, which can be described as follows: ^{2}); for the given points

In this equation,

The RBF

An RBF ^{+} to _{j}, _{j}, _{j}} _{j} _{=} _{1,2, …, n} ∈ ^{2}

Equation (_{j}} _{j} _{=} _{1, 2, 3, …, n} are pending coefficients, which are calculated through {_{j}, _{j}, _{j}} _{j} _{=} _{1, 2, 3, …, n}. Then, satisfying interpolation results will be obtained after a proper RBF

RBFs are widely studied. Gaussian interpolant, Kriging interpolant, multiquadric interpolant, and inverse multiquadric are favorable RBFs. A positive definite RBF with compact support is constructed using convolution to satisfy the requirements of FTS:_{i, j} = |_{i} − _{j}| + |_{i} − _{j}|. The real-time requirement should be fulfilled by employing a controller with hardware multiplier considering the multiplication in equation (

The cutter location points (CLPs) or the machining points in the machine controller are all scattered data, which cannot be described by gridded data. In the FTS interpolation, a localized interpolation is adopted considering the realization of the interpolation algorithm in the controller. For the current machining point

In this paper, RBF interpolation to

Then, an interpolation function for the current machining point can be obtained as follows:

Simulation experiments were carried out to verify that the RBF interpolation algorithm has the advantage of high accuracy and good smoothness. Three interpolation algorithms commonly used in the industrial area (Hermite interpolation algorithm, four-point mean interpolation algorithm, and Shepard interpolation algorithm) were employed to compare the performance of different algorithms.

Simulation experiments are performed in MATLAB software. The simulated workpiece is a lens array, which is a typical workpiece for evaluating the performance of FTS (Figure

Freeform optics simulated in experiments.

Parameters of simulated workpiece.

Diameter of workpiece | 10 mm |
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Aperture of small lens | 1 mm |

Pitch between two lenses | 1 mm |

Radius of small lenses | 2.84 mm |

Sagittal height of lenses | −44.4 |

In the experiments of smoothness, the original data quantity is 4800. The original data describe freeform surfaces to be processed. The interval between two points on the surface is 0.125 mm. Although the intensity of data in this experiment is rarely used in actual fabrication, it can clearly show the smoothness of the four surfaces generated by different interpolation algorithms. Then, analysis can be performed based on the simulation results. The four surfaces obtained using the four different algorithms are shown in Figure

Simulation experiment results of four interpolation algorithms.

Curves of integral smoothness variation versus data amount are shown in Figure

Integral smoothness variation curve versus data amount.

In order to compare the actual interpolation accuracy of the four algorithms, original scattered data with different densities were simulated. In the simulation experiments, the mean errors of the interpolated surface were adopted to evaluate profile accuracy, which is described as follows:

Accuracy value of four interpolation algorithms with different intervals.

Accuracy (mm) | Interval (amount) | ||||
---|---|---|---|---|---|

0.125 mm (9600) | 0.08 mm (25350) | 0.04 mm (93750) | 0.02 mm (375000) | 0.01 mm (1500000) | |

RBF | 8.6338 | 3.5278 | 1.0250 | 2.5318 | 7.2162 |

Hermite | 0.0011 | 6.0755 | 3.1390 | 1.5349 | 7.9757 |

Four-point mean | 0.0059 | 0.0045 | 0.0034 | 0.0063 | 0.0044 |

Shepard | 0.0015 | 8.5607 | 4.3173 | 2.0430 | 1.0157 |

Profile accuracy variation curve versus data amount.

The simulation experiment showed the following. (1) The interpolation accuracy decreased with increasing data amount for the Shepard interpolation algorithm, Hermite interpolation algorithm, and RBF interpolation algorithm. However, the interpolation accuracy of four-point mean algorithm does not obey this law, and its accuracy is lowest among four interpolation algorithms. (2) The RBF interpolation showed the highest interpolation accuracy. Furthermore, with increasing data amount, errors of the RBF interpolation decreased at a faster rate than the other two algorithms (Hermite interpolation algorithm and four-point mean interpolation algorithm). (3) The interpolation accuracy was achieved at the nano level, fulfilling the accuracy requirement of ultraprecision machining. Considering machine accuracy, intervals between two points ranging from 0.01 mm to 0.02 mm are recommended for the RBF interpolation algorithm. Intervals between two points are recommended to be smaller than 0.01 mm for the Shepard algorithm.

Freeform optics processed by FTS should achieve a specific function. Thus, error distribution is an important factor to judge whether a specific interpolation algorithm is suitable. The designed function will be severely affected if interpolation errors are mainly distributed at functional areas of freeform optics. Error distributions of the four algorithms were simulated in our experiment. Intervals between two neighbor points were 0.04 mm in the original scattered data, which is recommended as a preferable interval by engineers and researchers [

Error distributions of four interpolation algorithms.

The error distribution simulation results show that errors are distributed on the whole surface for the four algorithms. Large errors are mainly distributed at the edges of each small lens in the Shepard and especially the RBF interpolation algorithms. Large errors of the four-point mean and Hermite interpolation algorithms are mainly distributed in each small lens. Errors of these two algorithms have the same changing trend with the sagittal height, i.e., the higher the sagittal height, the larger the errors. Generally, transition areas of freeform optics (edges in our experiments) need not be processed for high accuracy because most mechanical processing techniques, especially FTS technique, will smoothen transition areas to avoid resonance in the linear axis. After the smoothening, the accuracy of the original surface is reduced. Based on the simulation experiment of error distribution, the RBF interpolation algorithm exhibited the highest accuracy, with large errors distributed mainly in transition areas. Thus, according to simulation results and practical processing requirements, RBF interpolation will provide better performance.

According to the simulation experiments, it can be concluded that RBF performs the best in satisfying FTS requirements of existence, smoothness, and accuracy.

Machining experiments were carried out on a commercial ultraprecision lathe (UPL250), configured with an FTS (NFTS6000) as the Z′ axis. In order to compare the accuracy of the four interpolation algorithms, the same parameters of the workpiece were adopted, as shown in Table

Machining parameters.

Spindle speed | 800 rpm |
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Depth of cut | 3 |

Tool radius | 0.5 mm |

Finish machining cycle | 18 min |

Considering that accurate measurement of freeform optics is still an unresolved issue, each small lens on the processed workpiece was measured with the Taylor Hobson profiler in our experiment. In the machining experiment, the edges of the lenselets will be preprocessed by a new smoothing algorithm proposed by us, and thus the edges of the lenselet will not affect the machining and measuring results. Average measurement results of all small lenses on each workpiece are shown in Table

Measurement results of profile accuracy of four interpolation algorithms.

Algorithm | FPM (nm) | Hermite (nm) | Shepard (nm) | RBF (nm) |
---|---|---|---|---|

Profile accuracy (mean) | 682.2 | 225.6 | 209.8 | 103.2 |

Roughness | 5.84 | 5.58 | 5 | 4.8 |

Profile error of four algorithms. (a) Profile error of RBF interpolation algorithm. (b) Profile error of four-point mean interpolation algorithm. (c) Profile error of Shepard interpolation algorithm. (d) Profile error of Hermite interpolation algorithm.

Based on actual processing experiment, it can be concluded that (1) the profile accuracy of the freeform optics processed by FTS can be improved by improving interpolation accuracy; different profile accuracies will be obtained based on various interpolation algorithms; (2) the RBF interpolation algorithm can provide the highest accuracy; (3) the Shepard and Hermite interpolation algorithms provide good profile accuracy, but the error value increases with increasing scallop height; and (4) the four-point mean interpolation algorithm has the lowest profile accuracy. Nevertheless, considering practical processing applications, it can be used in applications with low accuracy requirement for its simplicity.

The machining experiment results agree with simulation results. Considering other systematic errors (errors of fabrication of the three axis, errors of displacements, errors of heat, and so on), it is not surprising that the actual machining errors are larger than the simulation value.

This paper discusses interpolation algorithms in the tool path generation of FTS assisting diamond turning for freeform optics. Mathematical equations were built for interpolation of original data, and a positive definite RBF with compact support was employed to interpolate scattered data. Simulation and machining experiments were carried out to validate the performance of the proposed algorithm. The algorithm was found to excel in accuracy, smoothness, and error distribution in simulation experiments. Moreover, the machining results showed that the RBF interpolation algorithm can provide higher accuracy than the other three algorithms. Intervals between two cutting points should be under 0.02 mm in order to achieve nanolevel accuracy of interpolation. Overall, the RBF interpolation algorithm can provide higher accuracy, better smoothness, and more rational error distribution. Therefore, RBF should be employed as the interpolator for the regeneration of the surface in the tool path generation of FTS.

Readers can access the data underlying the findings of the study by accessing the following link:

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was supported by the Tianjin Application Foundation and Advanced Technology Research Program (the Youth Fund Project) under grant no. 19JCQNJC04200, the Tianjin Higher Education Science and Technology Development Fund Project under grant no. 2017KJ102, the National Natural Science Foundation of China (NSFC) under grant no. 51905378, the Tianjin Science and Technology Planning Project under grant nos. 20JCQNJC00360 and 20YDTPJC00450, the Scientific Research Foundation of Tianjin University of Technology and Education under grant no. KYQD1901, and the Open Fund of Tianjin Aerospace Intelligent Equipment Technology Key Laboratory under grant no. TJYHZN2019KT001.