Wheel Setting Error Modeling and Compensation for Arc Envelope Grinding of Large-Aperture Aspherical Optics

Abstract

Precision grinding is a key process for realizing the use of large-aperture aspherical optical elements in laser nuclear fusion devices, large-aperture astronomical telescopes, and high-resolution space cameras. In this study, the arc envelope grinding process of large-aperture 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 feed-direction 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 peak-to-valley (PV) for a Φ400 mm elliptical K9 glass surface is achieved.

1 Introduction

Large ultra-precision aspheric optical elements are highly demanded for the development of laser nuclear fusion devices [1], large-aperture astronomical telescopes [2, 3], high-resolution 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 42-m European Extremely Large Telescope (E-ELT) [2] consist of 492 and 798 quasi-hexagonal 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 large-aperture 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 ultra-precision grinding of large-aperture 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 6.5-m primary mirror of the Magellan telescope [14] and 8.4-m 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 ultra-precision 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 1/10⁶ PV, respectively. Subsequently, another large ultra-precision grinding machine, Big OptiX [17, 18], was developed to achieve high-efficiency and low-damage grinding of hard and brittle materials. By employing a novel R-theta grinding mode with an inclined toroidal shape diamond wheel, a high material removal rate of 187.5 mm³/s was achieved when grinding the 1.45-m Zerodur mirror [18] used for the E-ELT 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 ULTRASONIC100-5 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 high-order off-axis 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 ultra-precision grinding of large-aperture optics in recent years. For example, researchers at Tsinghua University developed a six-axis, large-scale 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 ultra-precision 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 four-axis aspheric machine tools, namely, the 800-mm FSGJ-1 [21], 1.2-m FSGJ-2 [22], and 2-m FSGJ-3. These machines integrated the functions of rough grinding, fine grinding, polishing, and online measurement. Using the computer-controlled optical surfacing technique, a form precision of 12 nm RMS for a 1-m mirrors was achieved. In recent years, Jiang et al. at Xi’an Jiaotong University developed two ultra-precision aspheric grinding machines with maximum machinable diameters of Ф900 mm [23, 24] and Ф1500 mm [25, 26]. The achievable form error for a 400-mm 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 Z-directions must be adjusted during the grinding process. The tool setting in the X- and Y-directions ensures the lowest point of the grinding wheel coincides with the axis of the turntable. The tool setting in the Z-direction 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 X-direction, it approaches the outer cylindrical surface from both the positive and negative directions of the X-axis. The X-coordinates of the machine tool in the contact state are recorded as x₁ and x₂. The middle point between x₁ and x₂, i.e., x₀ =  (x₁ + x₂)/2, is the zero point of the workpiece in the X-direction. 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 contact-type tool setting methods include on-machine touch probes [28] and force sensors [29, 30], whereas examples of non-contact 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 V-shaped and Λ-shaped 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.

2 Modeling of Aspherical Surface Generation

2.1 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 Z-axis can be expressed as

$$z\left( h \right) = \frac{{ch^{2} }}{{1 + \sqrt {1 - (1 + k)c^{2} h^{2} } }} + \sum {A_{m} h^{2m} } ,$$

(1)

where $c = {1 \mathord{\left/ {\vphantom {1 R}} \right. \kern-\nulldelimiterspace} R}_{0}$ is the vertex curvature; R₀ 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 high-order terms of the aspherical surface, where m is an integer.

Figure 1

Surfaces corresponding to different conic constants k

Eq. (1) is reduced to a quadratic aspheric surface if higher-order terms are absent. It can be rewritten as

$$\frac{{\left( {z - \frac{1}{{\left( {1 + k} \right)c}}} \right)^{2} }}{{\left( {\frac{1}{{\left( {1 + k} \right)c}}} \right)^{2} }} + \frac{{h^{2} }}{{\left( {\frac{1}{{\sqrt {1 + k} c}}} \right)^{2} }} = 1.$$

(2)

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.,

$$\left\{ {\begin{array}{*{20}l} {x = h\cos \theta ,} \hfill \\ {y = h\sin \theta ,} \hfill \\ {z = \frac{{ch^{2} }}{{1 + \sqrt {1 - (1 + k)c^{2} h^{2} } }},} \hfill \\ \end{array} } \right.$$

(3)

Figure 2

Coordinate system of aspheric workpiece

where θ is the angle between the positive direction of the X-axis and the projected vector of $\overrightarrow {OP}$ on the XOY plane, as shown in Figure 2.

2.2 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′.

Figure 3

Coordinate system of grinding wheel (Only region of Y′ ≥ 0 is shown, since wheel is symmetrical about X′O′Z′-plane)

In the O′X′Y′Z′ coordinate system, the wheel surface can be described by the following equations:

$$\begin{gathered} \left\{ \begin{gathered} x^{\prime} = l\cos \theta^{\prime}, \hfill \\ y^{\prime} = r\sin \gamma^{\prime}, \hfill \\ z^{\prime} = l\sin \theta^{\prime}, \hfill \\ \end{gathered} \right. \hfill \\ l = a + r\cos \gamma^{\prime}, \hfill \\ \end{gathered}$$

(4)

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.

2.3 Modeling of Aspherical Surface Generation

During the modeling of the generation process of the aspherical surface, the X-, Y-, and Z-axes 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:

$$\left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x^{\prime}} \\ {y^{\prime}} \\ {z^{\prime}} \\ \end{array} } \right] + {\varvec{B}},$$

(5)

Figure 4

Coordinate systems for grinding of aspheric surface using arc wheel

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.

$${\varvec{B}} = \left[ {\begin{array}{*{20}c} {X_{s} } \\ {Y_{s} } \\ {Z_{s} } \\ \end{array} } \right].$$

(6)

In the initial grinding state, the X- and Y-components of B₀ 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 Z-axes, 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 Z-axes, 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.

Figure 5

Parallel grinding and cross grinding of aspherical surfaces using arc grinding wheel

The grinding point is the intersection point between the workpiece and wheel surfaces. Combining Eqs. (3)‒(5) yields

$$\left[ {\begin{array}{*{20}c} {h\cos \theta } \\ {h\sin \theta } \\ {\frac{{ch^{2} }}{{1 + \sqrt {1 - (1 + k)c^{2} h^{2} } }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {(a + r\cos \gamma^{\prime})\cos \theta^{\prime}} \\ {r\sin \gamma^{\prime}} \\ {(a + r\cos \gamma^{\prime})\sin \theta^{\prime}} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {X_{s} } \\ {Y_{s} } \\ {Z_{s} } \\ \end{array} } \right].$$

(7)

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

$$\vec{n}_{s} = \left[ {\begin{array}{*{20}c} {\frac{\partial (y,z)}{{\partial (h,\theta )}}} \\ {\frac{\partial (z,x)}{{\partial (h,\theta )}}} \\ {\frac{\partial (x,y)}{{\partial (h,\theta )}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - \frac{{{\text{d}}z}}{{{\text{d}}h}}h\cos \theta } \\ { - \frac{{{\text{d}}z}}{{{\text{d}}h}}h\sin \theta } \\ h \\ \end{array} } \right],$$

(8)

where

$$\begin{aligned} \frac{{{\text{d}}z}}{{{\text{d}}h}} = z^{\prime}\left( h \right) = \frac{ch}{{1 + \sqrt {1 - \left( {k + 1} \right)c^{2} h^{2} } }} \hfill \\ \left( {2 + \frac{{\left( {k + 1} \right)c^{2} h^{2} }}{{\left( {1 + \sqrt {1 - \left( {k + 1} \right)c^{2} h^{2} } } \right)\sqrt {1 - \left( {k + 1} \right)c^{2} h^{2} } }}} \right). \hfill \\ \end{aligned}$$

(9)

Similarly, the normal vector at point P′(γ′, θ′) on the grinding wheel can be written as

$$\vec{n}_{w} = \left[ {\begin{array}{*{20}c} {\frac{\partial (y,z)}{{\partial (\gamma^{\prime},\theta^{\prime})}}} \\ {\frac{\partial (z,x)}{{\partial (\gamma^{\prime},\theta^{\prime})}}} \\ {\frac{\partial (x,y)}{{\partial (\gamma^{\prime},\theta^{\prime})}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {r\left( {a + r\cos \gamma^{\prime}} \right)\cos \gamma^{\prime}\cos \theta^{\prime}} \\ {r\left( {a + r\cos \gamma^{\prime}} \right)\sin \gamma^{\prime}} \\ {r\left( {a + r\cos \gamma^{\prime}} \right)\sin \theta^{\prime}\cos \gamma^{\prime}} \\ \end{array} } \right].$$

(10)

The normals of the workpiece and wheel surfaces at the grinding point are collinear. Therefore, the following equations can be obtained:

$$\begin{aligned} \frac{{ - \frac{{{\text{d}}z}}{{{\text{d}}h}}h\cos \theta }}{{r\left( {a + r\cos \gamma^{\prime}} \right)\cos \gamma^{\prime}\cos \theta^{\prime}}} &= \frac{{ - \frac{{{\text{d}}z}}{{\text{d}}h}h\sin \theta }}{{r\left( {a + r\cos \gamma^{\prime}} \right)\sin \gamma^{\prime}}} \hfill \\ &= \frac{h}{{r\left( {a + r\cos \gamma^{\prime}} \right)\sin \theta^{\prime}\cos \gamma^{\prime}}}. \hfill \\ \end{aligned}$$

(11)

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:

$$\left[ {\begin{array}{*{20}c} {h\cos \theta } \\ {h\sin \theta } \\ {z\left( h \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {(a + r\cos \gamma^{\prime})\cos \theta^{\prime}} \\ {r\sin \gamma^{\prime}} \\ {(a + r\cos \gamma^{\prime})\sin \theta^{\prime}} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {X_{s} } \\ {Y_{s} } \\ {Z_{s} } \\ \end{array} } \right].$$

(12)

If the grinding path is known, then the expression of $z\left( h \right)$ can be obtained by solving Eqs. (11) and (12).

3 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 arc-shaped 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.

3.1 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:

$$z_{1} \left( h \right) = \left\{ {\begin{array}{*{20}l} {\frac{{c\left( {h - \Delta h} \right)^{2} }}{{1 + \sqrt {1 - \left( {k + 1} \right)c^{2} \left( {h - \Delta h} \right)^{2} } }}{,}\;h \ge h_{1} ,} \hfill \\ {\frac{{c_{s} \left( {h - \Delta h} \right)^{2} }}{{1 + \sqrt {1 - c_{s}^{2} \left( {h - \Delta h} \right)^{2} } }}{,}\;h < h_{1} ,} \hfill \\ \end{array} } \right.$$

(13)

Figure 6

Effect of wheel setting error along outward feed direction on workpiece profile during parallel grinding ($\Delta h$ = 0.4 mm)

where $\Delta h$ is the wheel setting error along the feed direction. The value of h₁ 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 ≤ h₁. 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 arc-shaped 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 ≥ h₁. The form error at 400 mm induced by a wheel setting error of 0.4 mm was − 0.0446 mm during parallel grinding.

Figure 7

Evolution of workpiece form error with radial distance to optical axis for $\Delta h$ = 0.4 mm

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:

$$z_{1} \left( h \right) = \left\{ {\begin{array}{*{20}c} {\frac{{c\left( { - h - \Delta h} \right)^{2} }}{{1 + \sqrt {1 - \left( {k + 1} \right)c^{2} \left( { - h - \Delta h} \right)^{2} } }}{,}\;h < h_{2} ,} \\ {\frac{{c_{s} \left( { - h - \Delta h} \right)^{2} }}{{1 + \sqrt {1 - c_{s}^{2} \left( { - h - \Delta h} \right)^{2} } }}{,}\;h_{2} \le h < h_{3} ,} \\ {\frac{{c\left( {h - \Delta h} \right)^{2} }}{{1 + \sqrt {1 - \left( {k + 1} \right)c^{2} \left( {h - \Delta h} \right)^{2} } }}{,}\;h \ge h_{3} .} \\ \end{array} } \right.$$

(14)

Figure 8

Effect of inward wheel setting error on workpiece profile during parallel grinding ($\Delta h$ =  −0.4 mm)

The values of c_s, c, and k are known for a specified wheel and aspheric surface. Therefore, the values of h₂ and h₃ 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 ≥ h₃. 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.

Figure 9

Evolution of workpiece form error with radial distance to optical axis during parallel grinding ($\Delta h$ = − 0.4 mm)

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.

Figure 10

Slope of workpiece form error as a function of wheel setting error in feed direction

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:

$$e_{z} = z_{1} \left( h \right) - z\left( h \right) \approx - z^{\prime}\left( h \right)\Delta h,$$

(15)

where z₁ and z are the actual and ideal profiles, respectively. e_z for a quadratic aspheric surface is expressed as

$$\begin{gathered} e_{z} \approx - \frac{ch}{{1 + \sqrt {1 - \left( {k + 1} \right)c^{2} h^{2} } }} \hfill \\ \left( {2 + \frac{{\left( {k + 1} \right)c^{2} h^{2} }}{{\left( {1 + \sqrt {1 - \left( {k + 1} \right)c^{2} h^{2} } } \right)\sqrt {1 - \left( {k + 1} \right)c^{2} h^{2} } }}} \right)\Delta h. \hfill \\ \end{gathered}$$

(16)

$h \ll R_{0}$ and $\left( {k + 1} \right)c^{2} h^{2} \ll 1$ because h ≤ 400 mm and R₀ = 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.

3.2 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:

$$z_{1} \left( h \right) = \frac{{c_{s} \left( {h - \Delta l} \right)^{2} }}{{1 + \sqrt {1 - c_{s}^{2} \left( {h - \Delta l} \right)^{2} } }}.$$

(17)

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.

Figure 11

Effect of lateral wheel setting error on workpiece profile for parallel grinding ($\Delta l$= 0.4 mm)

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.

Figure 12

Workpiece profiles and form error as a function of h for cross grinding with lateral wheel setting error of $\Delta l$ = 0.4 mm

In summary, the form error was more sensitive to the feed-direction wheel setting error than the lateral wheel setting error during the grinding of large-aperture aspherical surfaces. The maximum form error induced by the feed-direction 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.

4 Error Compensation and Grinding Tests

4.1 Error Compensation

Because the form error was insensitive to the wheel setting error along the lateral direction, only the feed-direction wheel setting error was considered during the grinding of the large-aperture 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.

Figure 13

Theoretical and measured error curves caused by wheel setting error

In this study, off-line 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 large-aperture 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 large-aperture 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 low-pass 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).

Figure 14

Error compensation during grinding

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(h₁, z₁) 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(h₂, z₂) 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 Z-direction 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.

Figure 15

Error curves of workpiece generatrix before and after tilt correction

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.

Figure 16

Error curves of workpiece generatrix before and after compensation

4.2 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 Z-axes 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 high-stiffness 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, semi-finishing grinding, and fine grinding.

Figure 17

Grinding of Φ400 mm K9 glass blank on CM1500 large grinding machine

The process parameters for all the grinding stages are listed in Table 1. A D151 metal-bonded 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 metal-bonded diamond wheel with a grain dimension of 40–50 μm was used in the semi-finishing 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 resin-bonded 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 arc-profile 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.

Table 1 Process parameters

Figure 18

Error curve of aspherical surface after grinding. Surface roughness was removed using a low-pass filter

5 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 feed-direction wheel setting error than the lateral wheel setting error during the grinding of large-diameter 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 feed-direction 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.