 Original Article
 Open Access
 Published:
Wheel Setting Error Modeling and Compensation for Arc Envelope Grinding of LargeAperture Aspherical Optics
Chinese Journal of Mechanical Engineering volume 35, Article number: 108 (2022)
Abstract
Precision grinding is a key process for realizing the use of largeaperture aspherical optical elements in laser nuclear fusion devices, largeaperture astronomical telescopes, and highresolution space cameras. In this study, the arc envelope grinding process of largeaperture aspherical optics is investigated using a CM1500 precision grinding machine with a maximum machinable diameter of Φ1500 mm. The form error of the aspherical workpiece induced by wheel setting errors is analytically modeled for both parallel and cross grinding. Results show that the form error is more sensitive to the wheel setting error along the feed direction than that along the lateral direction. It is a bilinear function of the feeddirection wheel setting error and the distance to the optical axis. Based on the error function above, a method to determine the wheel setting error is proposed. Subsequently, grinding tests are performed with the wheels aligned accurately. Using a newly proposed partial error compensation method with an appropriate compensation factor, a form error of 3.4 μm peaktovalley (PV) for a Φ400 mm elliptical K9 glass surface is achieved.
Introduction
Large ultraprecision aspheric optical elements are highly demanded for the development of laser nuclear fusion devices [1], largeaperture astronomical telescopes [2, 3], highresolution earth observation systems [4, 5], and lithography machines [5]. However, their diameter and accuracy restrict the performance of related equipment. For example, the primary mirrors of the Thirty Meter Telescope [6] and the 42m European Extremely Large Telescope (EELT) [2] consist of 492 and 798 quasihexagonal mirror segments, respectively, with a diagonal length of 1.45 m. The National Ignition Facility [1] in the USA and the Megajoule Laser [7] in France require 7360 and 4200 largeaperture optical elements, respectively. The typical manufacturing process of aspheric optics comprises grinding [8], polishing [9, 10], and focused ion beam figuring [11]. The material removal rates of the polishing and figuring processes are extremely low. Therefore, the form error and subsurface damage depth [12] induced by the grinding process should be reduced to minimize the required material removal depth in the subsequent processes.
Hence, considerable technological advancements pertaining to ultraprecision grinding of largeaperture mirrors have been achieved. The Steward Laboratory at the University of Arizona developed the large optical generator to grind and polish 8 m mirrors [13]. It was successfully used to machine the 6.5m primary mirror of the Magellan telescope [14] and 8.4m primary mirrors of the Large Binocular Telescope [14, 15]. The form error of these mirrors can be reduced to less than 10 μm RMS after grinding. Researchers at the Cranfield University developed a large ultraprecision grinding machine, OAGM2500 [16,17,18], for the manufacturing of large mirrors used in telescopes. Its maximum machinable aperture and achievable relative form error are Φ2500 mm and 1/10^{6} PV, respectively. Subsequently, another large ultraprecision grinding machine, Big OptiX [17, 18], was developed to achieve highefficiency and lowdamage grinding of hard and brittle materials. By employing a novel Rtheta grinding mode with an inclined toroidal shape diamond wheel, a high material removal rate of 187.5 mm^{3}/s was achieved when grinding the 1.45m Zerodur mirror [18] used for the EELT telescope. The form error and subsurface damage depth were reduced to less than 1 μm RMS and 8 μm, respectively, within a manufacturing cycle of < 20 h. The ULTRASONIC1005 machining center developed by DMG MORI Company Ltd. integrates ultrasonic vibration with traditional grinding to improve the machining efficiency of hard and brittle materials. Using this machine, the form error of a 700 mm × 700 mm SiC highorder offaxis aspheric mirror was efficiently reduced to 2.13 μm RMS [19]. Other companies, such as Blohm, Satisloh, Schneider, and Optotech, provide large aspheric grinding machines with a machinable diameter of Φ500‒2000 mm.
China has conducted comprehensive studies pertaining to the ultraprecision grinding of largeaperture optics in recent years. For example, researchers at Tsinghua University developed a sixaxis, largescale precision grinding machine. Grinding tests of a Ф770 mm K9 glass show that the form error was less than 10 μm, and the surface roughness can reach the submicron level. The AOCMT ultraprecision grinding machine [20] developed by the National University of Defense Technology can machine optical elements up to a diameter of 650 mm. It has been successfully used to grind a Ф116 mm parabolic SiC workpiece to a form error of 8.9 μm. The Changchun Institute of Optics and Fine Mechanics developed a series of fouraxis aspheric machine tools, namely, the 800mm FSGJ1 [21], 1.2m FSGJ2 [22], and 2m FSGJ3. These machines integrated the functions of rough grinding, fine grinding, polishing, and online measurement. Using the computercontrolled optical surfacing technique, a form precision of 12 nm RMS for a 1m mirrors was achieved. In recent years, Jiang et al. at Xi’an Jiaotong University developed two ultraprecision aspheric grinding machines with maximum machinable diameters of Ф900 mm [23, 24] and Ф1500 mm [25, 26]. The achievable form error for a 400mm mirror was smaller than 5 μm PV using the arc envelope grinding method.
The studies above indicate that it is challenging to efficiently achieve an extremely high form accuracy by grinding, e.g., 5 μm/m PV, for large aspheric optics. During the grinding of such optics, the tool setting error is one of the main error sources of the workpiece form error. For a specified geometry of the machined surface, the grinding path is determined by the geometry of the grinding wheel and the spatial locations of the wheel and workpiece. Therefore, the tool position in the X, Y, and Zdirections must be adjusted during the grinding process. The tool setting in the X and Ydirections ensures the lowest point of the grinding wheel coincides with the axis of the turntable. The tool setting in the Zdirection determines the height of the grinding wheel relative to the workpiece.
The outer cylindrical surface and upper plane of the workpiece can be used as reference surfaces during the rough setting of the wheel. When the wheel is set in the Xdirection, it approaches the outer cylindrical surface from both the positive and negative directions of the Xaxis. The Xcoordinates of the machine tool in the contact state are recorded as x_{1} and x_{2}. The middle point between x_{1} and x_{2}, i.e., x_{0} = (x_{1} + x_{2})/2, is the zero point of the workpiece in the Xdirection. However, the accuracy of the wheel setting method mentioned above is limited by the form error of the cylindrical surface. In addition, the contact state between the wheel and workpiece is difficult to be determined accurately. Therefore, other methods should be developed by clarifying the evolution of the workpiece form error with the wheel setting errors.
The tool setting problem has been intensively investigated [27, 28] to improve machining precision. Typical contacttype tool setting methods include onmachine touch probes [28] and force sensors [29, 30], whereas examples of noncontact methods include acoustic emission sensors [31], digital microscopes [32], and digital holography [33]. However, studies pertaining to wheel setting for the grinding of aspheric optics are rare. Chen et al. [34] discovered that inward and outward offsets generated Vshaped and Λshaped profiles, respectively. However, the relationship between wheel setting error and workpiece form error has not been analytically modeled. Kang et al. [35] modeled and analyzed the form error of aspheric surfaces subjected to grinding with a cup wheel. The key error sources were discovered to be the tool setting error and radius error of the grinding wheel. Nevertheless, elaborate numerical computations were required to obtain the form error. Wei et al. [26] and Xi et al. [24] developed analytical tool setting error models that encompassed both the radial and lateral directions. However, these models did not account for the variation in the grinding point during arc envelope grinding; as such, the prediction accuracy was limited, especially for steep aspheric surfaces.
To overcome these challenges, the relationship between the form error of aspheric optics and the setting error of an arc grinding wheel was modeled analytically and numerically by considering the variation in the grinding point on the wheel. A grinding example is presented herein to verify the form error model and a newly proposed compensation method.
Modeling of Aspherical Surface Generation
Geometric Representation of Aspherical Surface
An aspheric surface is a rotationally symmetrical surface that deviates from a spherical surface in shape. A spherical or aspherical surface with a vertex at the origin and an optical axis along the Zaxis can be expressed as
where \(c = {1 \mathord{\left/ {\vphantom {1 R}} \right. \kern\nulldelimiterspace} R}_{0}\) is the vertex curvature; R_{0} is the vertex curvature radius; h is the radial distance from the optical axis (Figure 1); k is the conic constant that determines the shape of the surface; \(A_{m} h^{2m}\) represents the highorder terms of the aspherical surface, where m is an integer.
Eq. (1) is reduced to a quadratic aspheric surface if higherorder terms are absent. It can be rewritten as
For a point P (x, y, z) on a quadratic aspherical surface Figure 2), its coordinates can be expressed as a parametric equation, i.e.,
where θ is the angle between the positive direction of the Xaxis and the projected vector of \(\overrightarrow {OP}\) on the XOY plane, as shown in Figure 2.
Geometric Representation of Arc Grinding Wheel
A parallel grinding wheel with a circular arc was used for the arc envelope grinding of aspheric optical elements. The coordinate system of the grinding wheel is illustrated in Figure 3. The Y′axis was along the wheel axis, and the origin O′ was located at the intersection of the Y′axis and the axial symmetrical surface of the wheel. The wheel radius in this symmetry plane, i.e., X′O′Z′ coordinate plane, was R. The wheel surface with abrasive grains was rotationally symmetrical along the Y′axis. Its generatrix is an arc with a radius QP′ = r. The distance from the arc center Q to the wheel axis is \(a = R  r\). The distance from point P′ on the wheel surface to the Y′axis is denoted as l. The angle between the line P′O′ and X′O′Y′ plane is θ′ (clockwise), which ranges from 0° to 360°. The angle between the radius QP′ and the X′O′Z′ plane is γ′.
In the O′X′Y′Z′ coordinate system, the wheel surface can be described by the following equations:
where the range of γ′ is \( \arcsin \left( {{W \mathord{\left/ {\vphantom {W {2r}}} \right. \kern\nulldelimiterspace} {2r}}} \right) \le \gamma^{\prime} \le \arcsin \left( {{W \mathord{\left/ {\vphantom {W {2r}}} \right. \kern\nulldelimiterspace} {2r}}} \right)\), where W is the width of the grinding wheel.
Modeling of Aspherical Surface Generation
During the modeling of the generation process of the aspherical surface, the X, Y, and Zaxes of the workpiece coordinate system are assumed to be parallel to the X′, Y′, and Z′axes of the wheel coordinate system, respectively, as shown in Figure 4. The workpiece surface was tangential to the wheel surface during grinding at the grinding point. The positions of the grinding point on both the wheel and workpiece changed during grinding. The coordinates in the wheel coordinate system can be transformed into the workpiece coordinate system via the following translation transformation:
where B = \(\overrightarrow {{OO^{\prime}}}\) is the translation vector, which corresponds to the coordinates of O′ in the workpiece coordinate system. O′ is the programming point and its trajectory is the grinding path.
In the initial grinding state, the X and Ycomponents of B_{0} represent the wheel setting errors along the tangential and axial directions of the wheel, respectively.
For parallel grinding, the wheel moves along the Y and Zaxes, and the velocity directions of the grinding wheel and workpiece are parallel at the grinding point, as shown in Figure 5a. Therefore, X_{s} in vector B remains constant and is the lateral wheel setting error. For cross grinding, the wheel moves along the X and Zaxes, and the velocity directions of the wheel and workpiece are perpendicular to each other at the grinding point, as shown in Figure 5b. Hence, Y_{s} remains constant during grinding and is the lateral wheel setting error.
The grinding point is the intersection point between the workpiece and wheel surfaces. Combining Eqs. (3)‒(5) yields
In addition, the workpiece surface is tangential to the wheel surface at the grinding point. Therefore, the normals of the workpiece and wheel surfaces are collinear at the grinding point. The normal vector at point P(h, θ) on the aspherical surface can be expressed as
where
Similarly, the normal vector at point P′(γ′, θ′) on the grinding wheel can be written as
The normals of the workpiece and wheel surfaces at the grinding point are collinear. Therefore, the following equations can be obtained:
The coordinates of the grinding point can be obtained by solving Eqs. (7) and (11). Subsequently, the corresponding grinding path Z_{s} = f(X_{s}) or Z_{s} = f(Y_{s}) can be obtained as follows:
If the grinding path is known, then the expression of \(z\left( h \right)\) can be obtained by solving Eqs. (11) and (12).
Effect of Wheel Setting Error on Workpiece Form Error
The effect of the wheel setting error on the workpiece geometry can be clarified using the generation model of aspherical surface established above. The aspheric surface is a long ellipsoid with c = 1/3600 and k = − 0.2. The wheel is an arcshaped parallel wheel with R = 175 mm and r = 30 mm. This r value renders the wheel suitable for grinding aspheric surfaces with vertex curvature radii ranging from hundreds to thousands of millimeters. It is assumed that the grinding wheel moves from the center of the workpiece to the outside during grinding.
Wheel Setting Error in Feed Direction
If the wheel setting error is in the outward feed direction (Figure 6a), then the resultant surface profile of the workpiece can be expressed by the following piecewise function:
where \(\Delta h\) is the wheel setting error along the feed direction. The value of h_{1} can be determined by the continuity of the piecewise function; \(h_{1} \approx \Delta h\) if \(\Delta h \ll {1 \mathord{\left/ {\vphantom {1 {c_{s} }}} \right. \kern\nulldelimiterspace} {c_{s} }}\). Figure 6a indicates that the surface profile is a circular arc in the region h ≤ h_{1}. The radius of this arc is equal to r and R for parallel and cross grinding, respectively. c_{s} in Eq. (13) refers to the vertex curvature. If parallel and cross grinding are performed using an arcshaped parallel wheel, then \(c_{s} = {1 \mathord{\left/ {\vphantom {1 r}} \right. \kern\nulldelimiterspace} r}\) and \({1 \mathord{\left/ {\vphantom {1 R}} \right. \kern\nulldelimiterspace} R}\), respectively. Eq. (13) indicates that the form error in the circular arc segment will be less sensitive to the wheel setting error if a larger radius of the wheel arc profile is used. Compared with parallel grinding, the form error for cross grinding is more sensitive to the wheel setting error in these segments because r < R.
Figure 6b compares the workpiece profiles with and without wheel setting error. When the setting error was along the outward feed direction, the workpiece profile was composed of a circular arc and an elliptic curve. As shown in Figure 7, the workpiece form error was negative and decreased linearly with h in the region h ≥ h_{1}. The form error at 400 mm induced by a wheel setting error of 0.4 mm was − 0.0446 mm during parallel grinding.
For the inward tool setting error (Figure 8a), the surface profile of the workpiece after parallel grinding can be expressed by the following piecewise function:
The values of c_{s}, c, and k are known for a specified wheel and aspheric surface. Therefore, the values of h_{2} and h_{3} can be determined using the continuity condition of the curve; \(h_{2} \approx \Delta h\) if \(\Delta h \ll {1 \mathord{\left/ {\vphantom {1 {c_{s} }}} \right. \kern\nulldelimiterspace} {c_{s} }}\). The profile of the workpiece is composed of two elliptic curves and one circular arc, as shown in Figure 8. Figure 9 shows that the form error increased linearly with h in the region h ≥ h_{3}. A tool setting error \(\Delta h\) of − 0.4 mm resulted in a form error of 0.0447 mm at a radius of 400 mm. Its absolute value was extremely close to that induced by \(\Delta h\) = 0.4 mm. Therefore, the direction and magnitude of the wheel setting error determined the shape and magnitude of the form error profile, respectively.
The wheel setting error determines the slope between the form error and h. Simulations (see Figure 10) show that the absolute value of the slope increased linearly with the wheel setting error in the feed direction. This relationship can be used to determine the value of the wheel setting error.
The relationship between the slope of the form error and the wheel setting error can be obtained via theoretical analysis. When \(\Delta h\) is small, the form error in the region \(h \ge h_{1}\) (or \(h \ge h_{3}\)) can be expressed via the Taylor expansion, as follows:
where z_{1} and z are the actual and ideal profiles, respectively. e_{z} for a quadratic aspheric surface is expressed as
\(h \ll R_{0}\) and \(\left( {k + 1} \right)c^{2} h^{2} \ll 1\) because h ≤ 400 mm and R_{0} = 3600 mm in this study. Eq. (16) can be reduced to \(e_{z} \approx  ch\Delta h\). Therefore, the form error is a bilinear function of h and \(\Delta h\). This is consistent with the simulation results shown in Figures 7, 8, 9, 10. The value of e_{z} can be calculated using Eqs. (13) or (14) when \(h \ll R_{0}\) is not satisfied.
Lateral Wheel Setting Error
The form error of the workpiece in the region h ≥ Δl caused by the lateral wheel setting error Δl can only be solved using numerical methods. X_{s} equals Δl for parallel grinding, whereas Y_{s} and Z_{s} follow the trajectory without the wheel setting error. For cross grinding, Y_{s} is equal to Δl, whereas X_{s} and Z_{s} follow the trajectory without the wheel setting error. Subsequently, the surface profile and form error of the workpiece in the region h ≥ Δl can be obtained by numerically solving Eqs. (11) and (12).
The surface profile of the workpiece in the region h < Δl is a circular arc expressed as follows:
For parallel and cross grinding using arc grinding wheels, c_{s} is equal to 1/R and 1/r, respectively.
The effects of the lateral wheel setting error on the surface profile and its form error are shown in Figure 11. The maximum form error was recorded at the center of the aspherical surface. It is smaller than 0.1 μm because c_{s} was extremely small in this study. The form error remained almost constant in the region h ≥ Δl, with a slight decrease as h increased.
Because the c_{s} value for cross grinding was much greater than that for parallel grinding, the maximum form error for cross grinding was much greater than that for parallel grinding, as indicated by Figures 11 and 12. The form error of the workpiece in the region h ≥ Δl was extremely small and almost constant, which is similar to parallel grinding. It is noteworthy that the abrupt change shown in Figure 12b might be induced by the error in the numerical calculation, since the form error in this region was as low as 2.24 × 10^{−5} mm.
In summary, the form error was more sensitive to the feeddirection wheel setting error than the lateral wheel setting error during the grinding of largeaperture aspherical surfaces. The maximum form error induced by the feeddirection wheel setting error was recorded at the rim of the workpiece. It exhibited a bilinear relationship with the wheel setting error and the distance to the optical axis. By contrast, the maximum form error of the workpiece caused by the lateral wheel setting error was recorded at the center of the workpiece. The form error in the central region was more sensitive to the wheel setting error in the axial direction of the grinding wheel than that in the tangential direction.
Error Compensation and Grinding Tests
Error Compensation
Because the form error was insensitive to the wheel setting error along the lateral direction, only the feeddirection wheel setting error was considered during the grinding of the largeaperture aspheric surfaces. After determining the slope between the form error and h via measurement, the value of the wheel setting error Δh was calculated using Eq. (16).
The dashed line in Figure 13 represents the measured error curve of the workpiece after the wheel changing process. It comprises two straight lines forming the V shape, which is consistent with the theoretical error curve induced by the wheel setting error Δh = 2.89 mm, i.e., the solid line in Figure 13. This indicates that inappropriate wheel changing process results in a significant wheel setting error. The wheel setting error determined by the above method was used for wheel adjustment during the subsequent grinding process.
In this study, offline measurement was adopted during the fine grinding stage. Hence, repetitive workpiece clamping was required within the compensation cycles. Figure 13 and Ref. [24] indicate that wheel setting errors of 2.89 mm and 0.866 mm occurred during the grinding of largeaperture aspherical optics, respectively. Figure 6 and Figure 8 indicate that a tool setting error of 0.4 mm resulted in a form error of 0.0447 mm at a radius of 400 mm. Therefore, it is essential to accurately adjust the wheel position during the grinding of largeaperture aspherical optics.
After adjusting the wheel to the correct position, the error compensation method shown in Figure 14 was used to further reduce the machining error. First, a coordinate measuring machine (CMM) (Leitz Reference HP, Hexagon, China) was used to measure the profile of the aspherical generatrix, and the error curve E was obtained by subtracting the theoretical curve from it. Subsequently, the lowpass filtering of E was performed to eliminate the effects of random machining errors, which resulted in a filtered error curve E_{f}. The theoretical compensation curve \(\Delta_{t}\) was the opposite number of E_{f}. A compensation factor f was introduced to obtain the actual compensation curve \(\Delta_{a}\). Finally, the current grinding curve C_{after} was obtained by summing up \(\Delta_{a}\) and the former grinding curve C_{before} (i.e., the envelope of the outer circumference of the grinding wheel).
The original grinding path was obtained using the method described in Section 2.3, and the errors of the machine tool, wheel wear, and wheel setting were not considered. During the error compensation stage, the trajectory of the wheel center, i.e., the grinding path Z_{s} = f(X_{s}) or Z_{s} = f(Y_{s}), was obtained by solving Eqs. (7) and (11) using C_{after} as the input. Subsequently, the grinding path was used to machine the workpiece. If the PV value of the form error measured by the CMM was out of the tolerance, then a compensation grinding process was performed based on the measured error curve until the form error satisfied the requirement.
The tilt of the error curve was adjusted before calculating the form error to eliminate the effect of the workpiece clamping error. The equation for the symmetry axis of the aspherical surface was assumed to be \(z = kx + b\). The slope k and intercept b were varied at certain intervals to determine their optimal values. For each k and b value, the measurement points P_{l}(h_{1}, z_{1}) on the left side of the symmetry axis were mirrored to the right side to obtain the mirror points \(P^{\prime}_{l} \left( {h^{\prime}_{1} ,z^{\prime}_{1} } \right)\). The measurement points P_{r}(h_{2}, z_{2}) on the right side of the symmetry axis were interpolated to obtain the interpolation points \(P^{\prime}_{r} \left( {h^{\prime}_{1} ,z^{\prime}_{2} } \right)\). Subsequently, the Zdirection difference \(e_{z} \left( {h^{\prime}_{1} } \right)\) between each mirror point and its corresponding interpolation point was calculated. The k and b values resulting in the smallest root mean square of \(e_{z} \left( {h^{\prime}_{1} } \right)\) defined the optimal symmetry axis. Finally, the error curve was rotated to achieve a vertical symmetry axis. The error curves before and after tilt correction are shown in Figure 15a and b, respectively.
The optimal compensation factor depends on the ratio of the actual material removal depth to the nominal grinding depth; it is governed by the machine stiffness, machine precision, and process parameters. Currently, it is difficult to theoretically model the quantitative relationship between the nominal grinding depth and actual material removal depth. Therefore, the compensation factor was determined via grinding tests in this study. An error compensation factor of 1:1 was appropriate for rough grinding. However, it was discovered that a compensation factor of 1:1 resulted in overcompensation in the fine grinding stage, as revealed by Figure 16. Consequently, the locations of the peaks before compensation corresponded well to the valleys after compensation. The form error of 6.9 μm PV after compensation was similar to that before compensation, i.e., 7.9 μm PV. Therefore, a compensation factor of 0.5 was adopted in the fine grinding stage. The form error was reduced significantly to 3.3 μm PV after grinding using the abovementioned factor.
Grinding Tests
Parallel grinding tests of K9 glass with a diameter of Φ400 mm were performed on a large grinding machine (CM1500, Xi’an Jiaotong University, China), as shown in Figure 17. The maximum machinable diameter of the machine tool is ≥ Φ1500 mm. The strokes of the X and Zaxes were 1800 and 400 mm, respectively. Linear encoders (LC183, HEIDENHAIN, Germany) with a resolution of 10 nm were used in all three linear axes to provide a high positioning resolution. A highstiffness hydrostatic spindle with a runout of ≤ 0.2 µm was used to achieve high machining precision. The geometries of the workpiece and grinding wheel were consistent with those used in the simulation. The grinding process comprised three stages: rough grinding, semifinishing grinding, and fine grinding.
The process parameters for all the grinding stages are listed in Table 1. A D151 metalbonded diamond grinding wheel with a grain dimension of 127–160 μm was used in the rough grinding stage to ensure a high material removal rate. A D46 metalbonded diamond wheel with a grain dimension of 40–50 μm was used in the semifinishing stage. A shallower grinding depth and a lower feed rate were adopted in this stage to reduce the form error to less than 5 μm. A D15A resinbonded diamond wheel with a grain dimension of 10–15 μm was used in the fine grinding stage to reduce the surface roughness and subsurface damage depth. A cup truer provided by Guo et al. was used to dress the arcprofile diamond wheels. The dressing principle and conditions are described in Refs. [36, 37]. A form error of 3.4 μm PV was achieved after error compensation using the process parameters listed in Table 1, as shown in Figure 18.
Conclusions
In this study, analytical and numerical form error models of aspherical optical elements induced by wheel setting errors were established for both parallel and cross grinding. The effects of the direction of the wheel setting error on the shape and sensitivity of the form error were analyzed. The main conclusions are as follows.

(1)
The expressions of workpiece profiles induced by the wheel setting error were piecewise functions in most cases. Their forms were independent of the grinding mode. However, compared with parallel grinding, the ground form error for cross grinding was more sensitive to the wheel setting error in the circular arc segments of the profiles.

(2)
The form error was more sensitive to the feeddirection wheel setting error than the lateral wheel setting error during the grinding of largediameter aspherical surfaces. In addition, the form error in the central region was more sensitive to the wheel setting error in the axial direction of the grinding wheel than that in the tangential direction.

(3)
The maximum form error induced by the feeddirection setting error was recorded at the rim of the workpiece. It exhibited a bilinear relationship with the wheel setting error and the distance to the optical axis. By contrast, the maximum form error caused by the lateral wheel setting error was recorded at the center of the workpiece.

(4)
Based on the relationship between the wheel setting error and form error, the setting error was determined and used to accurately align the wheel during the grinding tests. An error compensation method integrating tilt correction, filtering, and partial compensation was proposed to efficiently reduce the form error. The results showed that a form error of 3.4 μm PV was achieved for a Φ400 mm elliptical K9 glass surface.
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Funding
Supported by Fellowship of China National Postdoctoral Program for Innovative Talents (Grant No. BX20200268), Research Project of State Key Laboratory of Mechanical System and Vibration (Grant No. MSV202103), National Natural Science Foundation of China (Grant No. 51720105016), and Higher Education Discipline Innovation Project (Grant No. B12016).
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Contributions
CL was in charge of the theoretical analysis, experiments and manuscript writing. LS took part in the experimental design and grinding tests. ZC and JC participated in measuring the workpiece form error. QL, JD and ZJ assisted with the modeling and data analysis. All authors read and approved the final manuscript.
Authors’ Information
Changsheng Li, born in 1989, is currently an assistant professor at School of Mechanical Engineering, Xi’an Jiaotong University, China. He received his BSc, MSc and Ph.D. from Xi’an Jiaotong University, China, in 2012, 2015 and 2019. His research interests include precision/ultraprecision machining and precision assembly.
Lin Sun, born in 1989, is currently an assistant professor at School of Mechanical Engineering, Xi’an Jiaotong University, China. He received his BSc in Northeastern University, China, in 2012, and Ph.D. degree in ultraprecision grinding from Xi’an Jiaotong University, China, in 2019.
Zhaoxiang Chen, born in 1988, is currently an assistant engineer and a master candidate at State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, China. He received his BSc in mechanical engineering from Politecnico di Torino, Italy, in 2013.
Jianfang Chen, born in 1985, is currently an engineer at Shaanxi Qinchuan Precision CNC Machine Tool Engineering Research Co., China.
Qijing Lin is currently an associate research fellow at School of Mechanical Engineering, Xi’an Jiaotong University, China. He received his BSc, MSc and Ph.D. degree from Xi’an Jiaotong University, China. His research interests include micro/nanofabrication and micro/nano sensors.
Jianjun Ding is currently a professor at School of Mechanical Engineering, Xi’an Jiaotong University, China. He received his BSc and MSc degree from Central South University, China, and Ph.D. degree from Xi’an Jiaotong University, China.
Zhuangde Jiang is a distinguished professor in the field of MEMS, nanotechnology and precision engineering. He is an academician of Chinese Academy of Engineering and a professor at Xi’an Jiaotong University, China. In addition, Professor Jiang is affiliated with a number of committees at national levels. He is the director of Strategic Steering Committee under the State Council, the Head of Mechanical Discipline Assessment of National Science and Technology Award Committee, Vice Chairman of Chinese Society of MicroNano Technology, and the President of Shaanxi Provincial Association of Science and Technology.
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Li, C., Sun, L., Chen, Z. et al. Wheel Setting Error Modeling and Compensation for Arc Envelope Grinding of LargeAperture Aspherical Optics. Chin. J. Mech. Eng. 35, 108 (2022). https://doi.org/10.1186/s10033022007825
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DOI: https://doi.org/10.1186/s10033022007825
Keywords
 Aspherical optics
 Wheel setting
 Arc envelope grinding
 Form error
 Large aperture