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Abstract

The optical performance of the off-axis three-mirror imaging system can be greatly improved using freeform surfaces. This article focuses on the polishing of the primary mirror and tertiary mirror in an off-axis three-mirror imaging system. The primary mirror and tertiary mirror are fabricated on one monolithic substrate and described by non-uniform rational B-spline–based freeform surfaces. The separated and integrated polishing strategies are presented for polishing the two mirrors on the four-axis computer numerical control polishing platform. A tool path generation approach is proposed for polishing of the non-uniform rational B-spline–based freeform surface. Three kinds of the tool paths are given for ultra-precision polishing of the primary mirror and tertiary mirror with the freeform surfaces. The concentric circle path and the approximately concentric circle path are generated for polishing two mirrors separately, while the spiral path is calculated for integrated polishing of two mirrors simultaneously. The polishing tool posture along the planned tool paths is also analyzed. The ultra-precision polishing experiments of the primary mirror and tertiary mirror on the four-axis computer numerical control polishing platform are performed to verify the proposed approach for tool path generation.

Keywords

Tool path generation ultra-precision polishing freeform optical surface off-axis three-mirror imaging system

Introduction

Freeform optical surfaces are widely used in modern optical systems because of many advantages, such as flexible optical design, excellent optical performance and simplified system structure. The application of the freeform surface in the off-axis three-mirror imaging system can obtain higher optical performance and system specifications.¹ The freeform surfaces have been defined as surfaces with no axis of rotational invariance within or beyond the part.² Many description methods of the freeform surfaces have been used in freeform surface modeling, such as elementary functions, general XY polynomials, Zernike polynomials and non-uniform rational B-splines (NURBSs). The demand of high-precision freeform optical surfaces with submicron form accuracy and nanometric surface finish offers new challenges for optical manufacturing, measurement and testing.

Several different manufacturing processes have been developed for fabrication of the freeform surfaces, such as the slow slide servo and fast tool servo for diamond turning,^3–6 ultra-precision milling,⁷ fly-cutting⁸ and ultra-precision grinding.^9,10 Although the above processes can fabricate high-precision freeform optical surfaces, there still exist some defects on the machined surface, such as tool marks after diamond turning or raster milling, surface texture and subsurface cracks after grinding. To obtain the freeform optical surfaces with sub-micrometer form accuracy and nanometer surface finish, ultra-precision polishing as a post process is used to remove these surface defects and correct the form deviation.^11–15

In ultra-precision freeform polishing, the spinning polishing tool moves along the tool path and contacts with the freeform surface with a constant normal pressure. According to Preston’s law,¹⁶ the material removal is proportional to the polishing pressure and the relative velocity between the polishing tool and workpiece. The material removal along the direction of the tool path can be controlled by adjusting the feed speed of the polishing tool, while the material removal perpendicular to the tool path is determined by the overlapping of the tool paths. The dwell time map can be calculated by solving a set of mathematical equations about the tool influence function (TIF) and error map obtained by measurement. The form deviation all over the freeform surface can be corrected by multi-passes of polishing and measurement.

Because of the complexity of the freeform surface and the contact condition, it is very difficult to ensure a deterministic material removal and accurate form error correction during polishing. There are several factors affecting the ultra-precision polishing process, such as the polishing path, the polishing slurry, the polishing strategy, TIF and surface measurement. Walker et al.^17,18 developed the Precessions™ polishing process for aspheric surface form correcting based on the IRP200 computer numerical control (CNC) polishing machine, which has the ability to both polish precision ground surfaces and control form down to the optical diffraction limit. Cheung and colleagues^19,20 studied the factors affecting surface generation in ultra-precision freeform polishing and found that surface generation is significantly affected by the influence function and the desired surface roughness can be achieved by appropriate combinations of mechanical polishing and fluid jet polishing. Fan et al.²¹ predicted the material removal in freeform surface polishing by modeling the local and the global polished profiles. Pan et al.²² established the motion model of particle trajectories on polishing contact region by analyzing the relative motion between the bonnet tool and workpiece and proposed a vertical continuous precession polishing method for aspheric lenses, by which the texture on the contact zone was approximately random and uniform.

The tool path for ultra-precision polishing directly impacts the uniform coverage of surface as well as the residual error after polishing. Furthermore, the tool path used for freeform polishing must be acceleration-smooth enough to accommodate the dynamic performance of CNC machine tool. Many kinds of polishing paths have been presented by researchers, such as scanning path,^23–26 peano-like path²⁷ and pseudorandom path.²⁸ However, these paths are limited to flat or simple surface polishing and need complicated machine tool movement, which is not practical for freeform surface polishing.

In this article, the tool path generation approach for polishing of the primary mirror and tertiary mirror in an off-axis three-mirror system is proposed for constructing the iterative polishing loop. The two mirrors are fabricated on one monolithic substrate and described by NURBS-based freeform surfaces. The polishing strategies and iterative polishing process for the freeform mirrors are presented in section “Ultra-precision polishing of freeform mirrors.” The transformation from the parametric domain of the NURBS surfaces to Cartesian coordinate system is described in section “Transformation of NURBS-based freeform surfaces.” In section “Tool path generation for freeform mirror polishing,” three kinds of tool paths are calculated for the freeform mirrors according to the proposed polishing strategies. The polishing tool posture along the tool paths is also analyzed in section “Determination of polishing tool posture and the inverse kinematics.” The generated tool paths are verified by the polishing experiments in section “Tool paths and polishing experiments.” The conclusions are summarized in section “Conclusion.”

Ultra-precision polishing of freeform mirrors

Polishing strategies

An off-axis three-mirror imaging system investigated in this article is constructed by integrated optical design method and consists of two separated NURBS-based optical surfaces (as shown in Figure 1), which can improve the optical performance in theory. But the separated layout of the primary mirror and tertiary mirror on one monolithic substrate brings more difficulties for polishing.

Figure 1.

The 3D model of the freeform mirrors.

There are two strategies for polishing of the freeform mirrors. The first one is separated polishing, and the second is integrated polishing. For the separated polishing, the two optical surfaces can be polished sequentially, during which the polishing tool contacts with the workpiece surface continuously except shift between two surfaces. For the integrated polishing, the two optical surfaces are polished simultaneously, by which the same polishing conditions for the two surfaces can be maintained.

In this article, a four-axis CNC polishing platform with the flexible polishing tool is used for polishing of the freeform mirrors under the two polishing strategies. As shown in Figure 2, the four-axis CNC polishing platform is composed of two translational axes (X-axis and Z-axis) and two rotational axes (B-axis and C-axis). The workpiece is mounted on the C-axis and can move along the X-axis. The polishing spindle is mounted on the B-axis and can move with the Z-axis.

Figure 2.

Configuration of the four-axis CNC polishing platform.

According to the configuration of the four-axis CNC polishing platform, the freeform mirror can be polished in the cylindrical coordinate system. The separated polishing strategy is accomplished by the sway motion of the C-axis and the translation of the X- and Z-axes, whereas the integrated polishing strategy is carried out by a continuous rotation of the C-axis with the translating of the X- and Z-axes. The force-controlled polishing spindle can maintain a constant polishing posture relative to the mirror surface by B-axis rotation. As the shape of freeform mirror is concave, the spherical polishing tool is used to comply with the curved surface.

Iterative polishing process

In this article, the iterative polishing loop consisting of optical design module, freeform polishing module and surface measurement module is built for polishing of the primary mirror and tertiary mirror in an off-axis three-mirror system. As shown in Figure 3, in the optical design module, the shape of the freeform surface is determined and two NURBS-based freeform surfaces are constructed. The freeform mirror is fabricated by CNC milling based on the results of optical design. In the freeform polishing module, the tool paths are generated according to the proposed polishing strategies. The concentric circle path (CCP), the spiral path (SP) and the approximately concentric circle path (ACCP) are generated, respectively, for comparison, which is the focus of this article. Then, the dwell time, defined by the feed speed of the polishing tool along the polishing path, is calculated by solving a set of mathematic equations constructed on the basis of material removal function. The polishing operation is then carried out according to the planned tool path and feed speed on the polishing platform, which is followed by an online measurement process. According to the error map obtained by measurement, if the form error $ε$ is larger than the upper limit $ε_{0}$ , the tool path and feed speed have to be modified for the next polishing process. After multi-iterative passes of polishing and measurement, the form error all over the freeform surface can be corrected into the tolerance range.

Figure 3.

Iterative polishing loop of freeform optical surfaces.

Tool path generation is an important and fundamental part in the overall iterative polishing loop. This article focuses on tool path generation for polishing of the primary mirror and tertiary mirror. The issues on dwell time calculation and freeform polishing process in the iterative polishing loop will be discussed in the future work.

Transformation of NURBS-based freeform surfaces

In freeform optical surface machining, NURBS provides a much more flexible and elegant method with advantages such as easy data exchange and precision persistence to describe the freeform surface than other description methods.²⁹ However, the parametric expression of NURBS surface cannot be used for tool path generation directly. In this study, the polishing operation of the freeform mirror is carried out in the cylindrical coordinate system, so the translation from parametric domain ( $u, v$ ) of the NURBS surface to the Cartesian coordinate system ( $x, y, z$ ), and ultimately to the cylindrical coordinate system ( $r, θ, z$ ), is necessary.

The general expression of NURBS surface is³⁰

S (u, v) = \frac{\sum_{i = 0}^{m} \sum_{j = 0}^{n} ω_{i, j} d_{i, j} N_{i, p} (u) N_{j, q} (v)}{\sum_{i = 0}^{m} \sum_{j = 0}^{n} ω_{i, j} N_{i, p} (u) N_{j, q} (v)}

(1)

where $u$ and $v$ are the independent parameters, $ω_{i, j}$ is the weight factor, $d_{i, j}$ is the control point, $N_{i, p} (u)$ is the B-spline basis function of degree $p$ in u-direction and $N_{j, q} (v)$ is the B-spline basis function of degree $q$ in v-direction. The knot vector with the length of $m + p + 2$ in u-direction is

U = \underset{p + 1}{\underset{︸}{{0, \dots, 0}}, u_{p + 1}, \dots u_{m}, \underset{p + 1}{\underset{︸}{1, \dots, 1}}}

The knot vector with the length of $n + q + 2$ in v-direction is

V = \underset{q + 1}{\underset{︸}{{0, \dots, 0}}, u_{q}, \dots u_{n}, \underset{q + 1}{\underset{︸}{1, \dots, 1}}}

According to Krishnamurthy et al.,³¹ when the parametric domain of the NURBS surface is meshed into regular distributed points, the points in Cartesian coordinate system corresponding to the regular distributed points can be calculated by the direct evaluation method. First, the parametric domains (0, 1) in u- and v-directions are divided into $k - 1$ and $l - 1$ segments, respectively. So there are $k \times l$ evaluation points regularly distributed in the parametric space.

For the evaluation point ( $u, v$ ), the corresponding point ( $x, y, z$ ) in Cartesian coordinate system can be calculated by the following steps:

Find the span of $[u_{i}, u_{i + 1})$ and $[v_{j}, v_{j + 1})$ in the knot vectors, in which the current evaluation point ( $u, v$ ) is located.

Compute the non-zero basis functions $N_{i - p, p} (u), \dots, N_{i, p} (u)$ and $N_{j - q, q} (v), \dots, N_{j, q} (v)$ along the two parameter directions by the De Boor³² recursive expression

\begin{matrix} N_{i, p} (u) = \frac{u - u_{i}}{u_{i + p} - u_{i}} N_{i, p - 1} (u) \\ + \frac{u_{i + p + 1} - u}{u_{i + p + 1} - u_{i + 1}} N_{i + 1, p - 1} (u) \end{matrix}

(2)

N_{i, 0} (u) = {\begin{matrix} 1, & if u_{i} ⩽ u < u_{i + 1} \\ 0, & otherwise \end{matrix}

(3)

The recursive expression in v-direction has the same format as equations (2) and (3).

Multiply the non-zero basis functions with their corresponding control points and sum the results; the point $P (x_{s}, y_{s}, z_{s})$ in Cartesian coordinate system can be expressed as

P (x_{s}, y_{s}, z_{s}) = \sum_{s = i - p}^{i} \sum_{t = j - q}^{j} N_{s, p} (u) N_{t, q} (v) d_{s, t} (x_{d}, y_{d}, z_{d})

(4)

For the NURBS surface of the primary mirror, the B-spline basis functions in the two directions have the same degree of 5 and all the weight factors are 1 ( $ω_{i, j} = 1$ ). The numbers of control points $d_{i, j}$ both in u-direction and v-direction are 6 ( $m = n = 6$ ). The knot vectors with the length of 12 are

U = {0, 0, 0, 0, 0, 0, 140.4687, 140.4687, 140.4687, 140.4687, 140.4687, 140.4687}

and

V = {0, 0, 0, 0, 0, 0, 139.3719, 139.3719, 139.3719, 139.3719, 139.3719, 139.3719}

The evaluation points in Cartesian coordinate system are calculated using this direct evaluation method. Similarly, the NURBS surface of the tertiary mirror can be translated into the Cartesian coordinate system. Figure 4 shows the control points of the original NURBS-based freeform optical surfaces. Figure 5 shows the regular evaluated points on the freeform surfaces in Cartesian coordinate system.

Figure 4.

Control points of the NURBS-based freeform optical surfaces on the primary mirror and tertiary mirror.

Figure 5.

Regular evaluated points on the freeform optical surfaces on the primary mirror and tertiary mirror.

Tool path generation for freeform mirror polishing

CCP and SP

As shown in Figure 6, the part coordinate system (O-XYZ) is constructed by setting the origin O on the center of part bottom, the Y-axis of the part coordinate system is parallel to the long edge of part bottom while the X-axis of the part coordinate system is parallel to the short edge. The tool path is generated in the machining coordinate system ( $o - r θ z$ ), which shares the same origin with part coordinate system but it is a cylindrical coordinate system, in which the Z-axis coincides with the C-axis of the polishing machine. Because the tool path is generated in the cylindrical coordinate system, the transformation of NURBS surface in parametric domain to Cartesian coordinate system is carried out followed by the transformation of Cartesian coordinate system to the cylindrical coordinate system. The tool path generation process starts with the generation of driven path in driven plane ( $r - θ$ plane) in cylindrical coordinate system and the tool contact path on the freeform surface is obtained by interpolation. Then, the tool path is calculated by offsetting the tool contact point in the normal direction.

Figure 6.

Part coordinate system (O-XYZ) and machining coordinate system ( $o - r θ z$ ).

The driven path for the CCP is a set of concentric circles, which can be expressed as

{\begin{matrix} x = (r_{0} + i \cdot Δ r) \cos θ \\ y = (r_{0} + i \cdot Δ r) \sin θ \end{matrix}, θ \in [0, 2 π)

(5)

The driven path for SP is a spiral expressed by

r = r_{0} + \frac{Δ r}{2 π} θ

(6)

where $r_{0}$ is the radius of the first driven path. $Δ r$ is the radial increment which determines the tool path spacing. The range of $Δ r$ is usually described as

0 < Δ r < r_{patch}

(7)

where $r_{patch}$ is the radius of polishing contact patch.

Because the freeform surface has been decomposed into discrete points ( $x_{s}, y_{s}, z_{s}$ ) in equation (4), the tool contact point ( $x_{c}, y_{c}, z_{c}$ ) on the freeform surface corresponding to the driven point $(x, y)$ is generated by the bicubic interpolation method. The obtained tool contact points on the freeform surface are distributed on the concentric circles or spiral from the top viewpoint, which is not suitable for normal vectors’ calculation directly. Figure 7 is the sketch map of the tool contact points in CCP from the top viewpoint. The mesh grid is generated by projecting the $x_{c}$ and $y_{c}$ coordinates of the tool contact points onto the X- and Y-axes, respectively. The corresponding mesh grid on the freeform surface can be obtained by the interpolation method described above. The mesh grid construction method for SP is the same as that for CCP. After a bicubic fit of all the mesh points on the freeform surface, the diagonal vectors at each mesh point are computed and crossed to form the normals. The normal vector $\vec{n}$ at the tool contact point $O_{c} (x_{c}, y_{c}, z_{c})$ can be found among the normal vectors of the mesh points. The tool location point $O_{t} (x_{t}, y_{t}, z_{t})$ is based on the offsetting of tool contact point $O_{c}$ with a defined distance in the normal direction, that is

(x_{t}, y_{t}, z_{t}) = (x_{c}, y_{c}, z_{c}) + D_{off} \cdot \vec{n}

(8)

where $D_{off}$ is the distance between the tool contact point and tool location point and $\vec{n}$ is the normal vector at point $O_{c}$ on the contact path.

Figure 7.

Mesh grid on driven plane of CCP from the top viewpoint.

ACCP

For CCP and SP, the tool contact path spacing on the freeform surface is uneven because the same increment $Δ r$ is adopted in the driven paths. The uneven tool contact path results in the non-uniform residual error after polishing and the inexact solution of mathematical problem in dwell time calculation. In this section, the ACCP with equal spacing between the tool contact paths is generated.

The equal spaced tool contact path is generated by a searching algorithm. The initial circle tool contact path passing through two corners of the design surface is generated as shown in Figure 8(a). The polar radius for this path in the cylindrical coordinate system is given by

r_{0} = \sqrt{x_{s 0}^{2} + y_{s 0}^{2}}

(9)

where ( $x_{s 0}, y_{s 0}$ ) is the coordinate of one corner of the freeform surface. The polar angles on this path have the equal interval of $Δ θ$ , by which the freeform surface is divided into section curves. The tool contact points are the intersections of the tool contact paths and the section curves.

Figure 8.

Calculation of tool contact paths with equal spacing for ACCP on the primary mirror (a) The tool contact path and (b) The section curve.

When the section curve at polar angle $θ_{i}$ is considered, the $z_{0}$ coordinate of the first tool contact point indicated by the intersection between the initial path and the section curve at polar angle $θ_{i}$ is calculated by interpolation method. The other contact points on this section curve are calculated by the iterative method. As shown in Figure 8(b), $C_{j - 1} (r_{j - 1}, z_{j - 1}) |_{j = 1}$ is the intersection point of the initial path and section curve at $θ_{i}$ . $d$ is the tool contact path spacing. The next point $C_{j}$ on this section curve is calculated based on the former point $C_{j - 1}$ . The calculation process can be described in the following steps:

Step 1. Let j = 1.

Step 2. Let $Δ r = d$ , then calculate the point $C'_{j} (r_{j - 1} + Δ r, z'_{j})$ by cubic interpolation.

Step 3. Calculate the distance of $C_{j - 1} C'_{j}$ , that is

C_{j - 1} C'_{j} = \sqrt{Δ r^{2} + (z'_{j}^{2} - z_{j - 1}^{2})}

(10)

Step 4. If $C_{j - 1} C'_{j} > d$ , decrease $Δ r$ by $δ$ and return to Step 3, or

C_{j} = (r_{j - 1} + Δ r, z'_{j})

(11)

Step 5. If $r_{j} + d < r_{max}$ , let j = j + 1 and return to Step 2, or end the process.

In this searching algorithm, if $δ$ is small enough, the tool contact spacing can be regarded as constant. The tool contact points on other section curves can be calculated using the same method by varying the angle $θ_{i}$ with a defined increment $Δ θ$ . After the tool contact path with equal spacing is generated, the tool path can be obtained by offsetting of the tool contact point with a defined distance in the normal direction.

Determination of polishing tool posture and the inverse kinematics

The tool used in freeform surface polishing is usually flexible in the material to comply with the curved surface. The spherical tool made of solid elastic material or bonnet inflated by air pressure is the most common polishing tool for freeform surface polishing.^33,34 As shown in Figure 9, the tool location point $O'$ is the center of the polishing tool. The tool path represents the trajectory of polishing tool center when it moves on the freeform surface. The tool contact point P is the center of contact patch, which is generated by the contacting between the flexible polishing tool and the workpiece surface with a reasonable posture and pressure. The tool contact path represents the trajectory of the tool contact patch on the workpiece surface. As the normal vector $\vec{n}$ at tool contact point P always passes the center of spherical polishing tool, the tool path is actually on an offset surface in the normal direction. To obtain a stable material removal profile in freeform surface polishing, the axis of polishing spindle H should maintain a certain inclination angle $λ$ relative to the normal vector at each tool contact point.

Figure 9.

Geometrical illustration for polishing of the primary mirror and tertiary mirror.

In Figure 9, the polishing posture is described by the polishing inclination angle $λ$ . $β$ is the rotation angle of the B-axis. By changing the included angle $β$ between the polishing spindle and Z-axis, the polishing inclination angle $λ$ can be kept constant at each tool contact point. $Z_{B} (0, 0, z_{B})$ is the intersection point of the axis of polishing spindle with the Z-axis, which is also the rotation center of the B-axis. The direction vector of the Z-axis is {0, 0, 1} and the coordinates of point $O'$ are { $x_{O'}, y_{O'}, z_{O'}$ }. The coordinates of $\vec{O' Z_{B}}$ can be expressed as { $- x_{O'}, - y_{O'}, z_{B} - z_{O'}$ }. As $β$ is the included angle between the Z-axis and $\vec{O' Z_{B}}$ , $\cos β$ can be represented as

\begin{matrix} \cos β = \frac{(- x_{O'}, - y_{O'}, z_{B} - z_{O'}) (0, 0, 1)}{1 \cdot \sqrt{x_{O'}^{2} + y_{O'}^{2} + (z_{B} - z_{O'})^{2}}} \\ = \frac{z_{B} - z_{O'}}{\sqrt{x_{O'}^{2} + y_{O'}^{2} + (z_{B} - z_{O'})^{2}}} \end{matrix}

(12)

If the unit normal vector at point P is { $x, y, z$ }, $\cos λ$ can be represented as

\cos λ = \frac{(- x_{O'}, - y_{O'}, z_{B} - z_{O'}) (x, y, z)}{\sqrt{x^{2} + y^{2} + z^{2}} \cdot \sqrt{x_{O'}^{2} + y_{O'}^{2} + (z_{B} - z_{O'})^{2}}}

(13)

where $\sqrt{x^{2} + y^{2} + z^{2}} = 1$

From equations (12) and (13), it is derived as

\frac{\cos β}{\cos λ} = \frac{z_{B} - z_{O'}}{- {xx}_{O'} - {yy}_{O'} + z (z_{B} - z_{O'})}

(14)

that is

z_{B} - z_{O'} = \frac{\cos β ({xx}_{O'} + {yy}_{O'})}{z \cos β - \cos λ}

(15)

From equations (12) and (15), a quadratic equation about $\cos β$ can be obtained by

\begin{matrix} [({xx}_{O'} + {yy}_{O'})^{2} + z^{2} (x_{O'}^{2} + y_{O'}^{2})] \\ \cos^{2} β - 2 z \cos λ (x_{O'}^{2} + y_{O'}^{2}) \cos β \\ + \cos^{2} λ (x_{O'}^{2} + y_{O'}^{2}) - ({xx}_{O'} + {yy}_{O'})^{2} = 0 \end{matrix}

(16)

The value of $β$ can be determined according to the solution of equation (16). During the polishing process, the displacement increments of the X/Z-axis are determined by the length of polishing spindle $L$ , the polishing tool posture and the coordinates of tool position. As shown in Figure 10, M ( $c_{1}, x_{1}, z_{1}$ ) and N ( $c_{2}, x_{2}, z_{2}$ ) are the two adjacent tool position points on the tool path, $β_{1}$ and $β_{2}$ are the rotation angles of the B-axis and the displacement increment of X-axis $dx$ can be represented as

dx = x_{2} - a - L \times \sin β_{2}

(17)

where

a = x_{1} - L \times \sin β_{1}

(18)

Figure 10.

Schematic diagram of inverse kinematics for polishing of the primary mirror and tertiary mirror.

From equations (17) and (18), it is derived as

\begin{matrix} dx = x_{2} - x_{1} + L \times \sin β_{1} - L \times \sin \\ β_{2} = (x_{2} - x_{1}) + L (\sin β_{1} - \sin β_{2}) \end{matrix}

(19)

The displacement increment of Z-axis $dz$ can be represented as

\begin{matrix} dz = z_{2} + L \times \cos β_{2} - z_{1} - L \times \cos \\ β_{1} = (z_{2} - z_{1}) + L (\cos β_{2} - \cos β_{1}) \end{matrix}

(20)

Tool paths and polishing experiments

Generated tool paths

According to the approach to generate the tool paths for freeform mirror polishing proposed in this article, the actual CCP, ACCP and SP are calculated using the MATLAB software package. Figures 11 –13 show the generated tool contact paths and tool paths for polishing of the two freeform mirrors, where the spacing of the tool paths has been assigned a large value for illustration purpose. Figure 11 illustrates the CCP for the separated polishing strategy. The driven path spacing $Δ r$ is 10 mm, and the angle increment $Δ θ$ is 2°. Using CCP, the primary mirror is polished first, and then the tertiary mirror is polished after the C-axis is rotated with 180°. Figure 12 gives the SP for the integrated polishing strategy. The pitch of driven spiral $Δ r$ is 10 mm, and the angle increment $Δ θ$ is 2°. The primary mirror and tertiary mirror are polished simultaneously using SP. Figure 13 shows the ACCP with constant tool contact path spacing for the separated polishing strategy. The tool contact path spacing $d$ is set to 4 mm and the angle increment $Δ θ$ is 2°. For the polishing inclination angle $λ$ of 15°, the calculated angles of the B-axis corresponding to the paths given in Figure 13 are shown in Figure 14.

Figure 11.

Concentric circle path (CCP) for the primary mirror and tertiary mirror.

Figure 12.

Spiral path (SP) for the primary mirror and tertiary mirror.

Figure 13.

Approximately concentric circle path (ACCP) for the primary mirror and tertiary mirror.

Figure 14.

Angles of B-axis along ACCP.

Polishing experiments and results

The experiments for polishing of the primary mirror and tertiary mirror have been carried out to verify the generated CCP, SP and ACCP on the four-axis CNC polishing platform, respectively. The spherical polishing tool (as show in Figure 15) used in the experiment is made of polyurethane and covered with the polishing cloth. The workpiece material is 6061 aluminum alloy and alumina with the sizes of 5, 10 and 15 μm in water-based solvent is used as the polishing slurry. As the polishing process is very time-consuming, the generated CCP, SP and ACCP are verified in the fast task mode. The practical polishing experiment is conducted on the primary mirror using CCP with path spacing $Δ r$ of 1 mm and angle increment $Δ θ$ of 2°. Table 1 shows the process parameters for polishing of the primary mirror. The feed speed along the tool path is determined by solving a set of mathematic equations constructed on the basis of TIF and removal depth. Before polishing, the freeform surface is machined by CNC milling and the roughness of primary mirror before polishing is Ra 0.5 μm. Figure 16 shows the comparison of the polished primary mirror and the unpolished tertiary mirror. The surface roughness of the polished surface on the primary mirror is measured in the direction perpendicular to polishing path by Talysurf PGI 1240 (Taylor-Hobson Ltd, UK). As shown in Figure 17, after 19 passes of iterative polishing, the surface roughness of primary mirror has been decreased to Ra 28 nm. To improve the surface finish to a high level, further polishing passes are still needed.

Figure 15.

Spherical polishing tool.

Table 1.

Polishing process parameters.

Polishing force F	10 N
Radius of spherical polishing tool R	15 mm
Spindle revolutions n	500 min⁻¹
Angle increment $Δ θ$	2°
Path spacing $Δ r$	1 mm
Preston coefficient k	$7.52 \times 10^{- 6}$

Figure 16.

A comparison of the polished primary mirror and the unpolished tertiary mirror.

Figure 17.

Surface roughness of the primary mirror.

Conclusion

Tool path generation is a crucial step in the iterative freeform surface polishing process. This article investigates the tool path generation approach for polishing of the primary mirror and tertiary mirror in the freeform off-axis three-mirror imaging system. The separated and integrated polishing strategies and the iterative polishing process are presented for polishing of the primary mirror and tertiary mirror, which are fabricated on one monolithic substrate and described by NURBS-based freeform surfaces. The NURBS surface in the parametric domain is transformed to Cartesian coordinate system and cylindrical coordinate system for convenience of tool path generation.

In particular, this article focuses on tool path generation for polishing of the primary mirror and tertiary mirror in an off-axis three-mirror imaging system. The CCP driven by a set of concentric circles on driven plane is calculated for polishing the primary mirror and tertiary mirror separately, while the SP driven by a spiral is calculated for polishing the primary mirror and tertiary mirror simultaneously. In consideration of the arbitrary shape of freeform surface and the ill-conditioned equations in dwell time calculation, the ACCP is calculated based on a searching algorithm, by which the equal spaced tool contact path on the freeform surface can be generated. In order to keep a certain polishing posture, the angular position of the B-axis along the tool path is determined by geometrical analysis.

The polishing experiments for the primary mirror and tertiary mirror are performed using the generated CCP, SP and ACCP on the four-axis CNC polishing platform respectively, which confirms the validity of the tool path generation approach for freeform surface polishing proposed in this article. After 19 passes of iterative polishing using CCP, the surface roughness of the primary mirror has been reduced to Ra 28 nm.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful to the financial support from the National Key Basic Research and Development Program (973 program) of China (Grant No. 2011CB706702).

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Tool path generation for ultra-precision polishing of freeform optical surfaces in an off-axis three-mirror imaging system

Abstract

Keywords

Introduction

Ultra-precision polishing of freeform mirrors

Polishing strategies

Iterative polishing process

Transformation of NURBS-based freeform surfaces

Tool path generation for freeform mirror polishing

CCP and SP

ACCP

Determination of polishing tool posture and the inverse kinematics

Tool paths and polishing experiments

Generated tool paths

Polishing experiments and results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References