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Abstract

Optical free-form surface elements have better optical characteristics than traditional optical elements, but their more complex curved structure also increases the difficulty of optical processing. Aiming at the problem of insufficient equal overlap ratio spirals, this paper proposes a polishing trajectory planning method for the optimal removal of the full surface. Based on the overlap length model of the curvature change, the polishing area, track residence time and total material removal model are established, and the processing efficiency is determined. The projection method is used to map the spatial polishing trajectory to the plane, and the mathematical relationship between the material removal amount and the trajectory spacing of the circular polishing and the projection trajectory spacing is established. This paper analyzes the influence of contact area change, geodesic radius of curvature and polishing posture on the material removal of the curved surface, as well as the change trend of the material removal contour and the contour of the overlapping part. Finally, taking the rotationally symmetrical aspherical workpiece made of K9 glass as an example, the corresponding polishing plan is formulated to verify the effectiveness of the proposed method.

Keywords

Optical surface CCOS optimal removal polish trajectory planning

Introduction

In recent years, optics has crossed and merged with other disciplines and technologies to open up new scientific research and application fields.¹ Optical curved surfaces have gradually developed from traditional flat and spherical surfaces to aspherical and free-form surfaces. According to the requirements of the optical system, free-form optics breaks the traditional optical imaging method, thus the corresponding optical curved surface is designed and constructed.² It has the advantages of simplifying the structure of the optical system, correcting aberrations, improving the quality of optical imaging, reducing the number of system units, and improving the optical design. There are many advantages such as spatial layout, reducing the cost of the optical system increasing the degree of freedom of the system, and expanding the field of view et etc. Therefore, the application and demand of optical free-form surface components are gradually increasing. However, due to the complexity and irregularity of optical free-form surfaces, it is difficult to build mathematical analysis models on free-form surfaces, and it is also more difficult to process and detect.³ As the final process of optical curved surface manufacturing, the polishing process accounts for about 60% of the entire manufacturing process time. Although the polishing efficiency is low and it takes a long time, still it has a great impact on the quality, accuracy, optical performance and service life of the optical components.

Computer Controlled Optical Surfacing, referred to as CCOS, is based on the surface inspection results to quantitatively simulate the polishing machining allowance,⁴ and use computer technology to control the small tool head to move on the surface of the workpiece with a preset trajectory. This technology determines the parameters such as polishing pressure, feed speed and residence time distribution,⁵ and use numerical control (NC) servo technology to accurately remove optical workpieces, thereby correcting surface errors and improving accuracy. Computer quantifiable control algorithms and NC machine tools have higher accuracy and repeatability, which significantly improves the surface convergence rate and processing efficiency. During optical processing, the polishing tool head needs to move along the planned polishing path. The quality of the polishing path determines the processing quality and efficiency of the workpiece. Therefore, CCOS polishing path planning is a research hotspot in the field of modern optical manufacturing.

The CCOS control algorithm directly affects the quality of optical processing, which can be divided into path planning and dwell time algorithms.⁶ Commonly used trajectory planning methods are isoparametric line method, projection method, equal residual height method, equal plane method, polyhedron method, offset surface method, equal residual height method and Peano trajectory. Isoparametric method^7,8: The mathematical model has two parameters, one parameter value is fixed, and the other parameter is changed to generate the processing trajectory. Because of the small amount of calculation, the isoparametric method can quickly generate processing trajectories, so it can improve efficiency and be widely used, but this method is only applied to workpieces with a large radius of curvature. Isoplane method^9,10: The intersection of the parallel plane and the surface of the workpiece are used as the processing track. This method is suitable for processing curved surfaces with uneven distribution of parameter lines, and can generate uniform processing trajectories, but the process of solving the intersection lines between the plane and the curved surface is very cumbersome. Parallel projection method: Project the processing trajectory of the plane to the surface of the workpiece. This method is suitable for workpieces with large changes in the curvature of the surface. The track spacing can be adjusted actively according to the curvature. It is not suitable for free-form surfaces with complex surface changes. Polyhedron method¹¹: Approximate the machined surface in the form of a polyhedron with a certain accuracy. Most polyhedrons are usually composed of triangular mesh surfaces, and then the intersection of the parallel plane and the polyhedron is used as the processing track. This method simplifies the surface structure and improves the processing efficiency, but it will reduce the machining accuracy of part of the curved surface. Equal residual height method^7,12,13: Regard the residual error between adjacent tracks as a quantification, by adjusting the track spacing, the processing accuracy and efficiency can be optimized. This method can adjust the processing trajectory according to the surface shape of the curved surface to ensure the processing quality, but the overall calculation is complicated and takes a long time. Offset surface method^14,15: Combining the characteristics of the spherical knife, offset the contact points of each tool on the machining surface to obtain the tool location point, and then obtain the tool location surface, and carry out trajectory planning on the tool location surface, and the offset distance approximately equal to the radius of the spherical knife. The processing trajectory of this method will not cause local interference, but it also increases the difficulty of solving due to the increase of the tool position surface. Peano trajectory^16,17: Peano curve is a non-differentiable curve, which is the limit of a curve sequence. It can fill up the two-dimensional or higher-dimensional space through any point in the two-dimensional area. It can give full play to the advantages of small tool head polishing and have a better anastomotic type and coverage.

However, trajectory planning research is mostly in the field of milling processing. In the polishing field, Mizaugaki et al.¹⁶ applied the Peano curve to the polishing field. Tam et al.^17,18 improved the grating path, Lissajous curve and Peano curve by using the line spacing adaptation algorithm to improve the coverage and uniformity, and transformed the Peano curve to make it more suitable for aspheric machining. The helix path is very suitable for the processing of rotating aspheric surfaces,^19,20 but the helix path may over process the workpiece in the central part. Regular polishing tracks are easy to leave regular scratches on the workpiece, which is the main source of high-frequency error of the workpiece.²¹ In order to solve this problem, Dunn and Walker²² proposed to use pseudo-random path polishing. Shi et al.²³ proposed random trajectory polishing. Deng et al.²⁴ proposed a new path planning method for off-axis aspheric NC polishing based on the idea of weighted average. Jin et al.²⁵ introduced the bisection algorithm into path planning for the optimization of line spacing.

There is still a lot of room for development in the types and maturity of trajectory planning methods. Therefore, this paper proposes a polishing trajectory planning method for the optimal removal of the entire surface. Based on the overlap length model of the curvature change, the polishing area, track residence time and total material removal model are established, and the processing efficiency is determined. The projection method is used to map the spatial polishing trajectory to the plane, establish the mathematical relationship between the material removal amount and the trajectory spacing of the annular polishing and the projection trajectory spacing, analyze the influence of the contact area change, the geodesic curvature radius and the polishing posture on the material removal of the curved surface, and the material removal change trend of contour and overlapping part contour.

The effect of track spacing on material removal

Spatial polishing trajectory and projection trajectory

The structure of optical free-form surfaces is complex and changeable, and it is difficult to directly generate tool trajectories on the surface of the workpiece. In this paper, the parallel projection method is used to combine the spatial curve trajectory of the workpiece surface with the projected trajectory in order to analyze and calculate the spatial polishing trajectory.

{\begin{matrix} F_{1} (x, y, z) = 0 \\ F_{2} (x, y, z) = 0 \end{matrix}

(1)

Draw a perpendicular line to the surface O-xy through each point on the curve C to obtain the projected plane curve:

{\begin{matrix} F (x, y) = 0 \\ z = 0 \end{matrix}

(2)

As shown in Figure 1, rotational symmetry aspheric surface with a curved surface of $z = \frac{1}{50} (x^{2} + y^{2})$ , and any point on the space curved surface has a unique point corresponding to it in the rectangular coordinate system. On the tangent plane perpendicular to the z-axis, the positional relationship between the points is constant with the plane coordinate system, that is, x and y values are the same, so the curvature of the surface only affects the z value of the polishing point. The red curve on the surface of the workpiece is a circular polishing track with a height of 5, and its plane projection is a circle.

Figure 1.

Rotational symmetry aspheric surface with a curved surface, including (a) 3D graph and (b) projection on O-xy.

The influence of polishing track spacing on material removal

When the polishing tool head is polished along the circular track, the overlapping part of the adjacent track spacing will be polished twice, thus affecting the removal of material (Figure 2) . Figure 3 is a schematic diagram of Figure 2 in the plane O-xz direction, that is, when θ = 0, h₁(ρ) changes in dθ. a_i and a_i₊₁ are the radius of the contact circle of the polishing track. Since the radius of curvature of the surface of the workpiece is much larger than the radius of the tool head, the material removal of adjacent tracks is basically the same, so the radius of the contact circle can be approximated as a fixed value, that is, a_i= a_i₊₁ = a, s_i is the distance between adjacent tracks, which can be regarded as the distance between the center points of the contact circle. Since the material removal model at the repeated polishing is approximately an axisymmetric figure, the overlap length is 2r_i. When 0 < r_i < a, adjacent tracks are over-polished.

Figure 2.

Schematic diagram of the polishing track of the contact surface.

Figure 3.

Schematic diagram of material overlap removal.

The adjacent track spacing s_i of annular polishing is

s_{i} = a_{i} + a_{i + 1} - 2 r_{i} = 2 (a - r_{i})

(3)

In the dθ plane, the relationship between the amount of material removal and the overlap length r_i is:

Δ h_{i} (ρ) = \frac{2 K R_{i}}{v_{f_{0}}} \int_{R + a - r_{i}}^{R + a} \frac{P (ρ) V (ρ)}{ρ} d ρ

(4)

In the dθ plane, the relationship between the amount of material removal and the spacing r_i is:

Δ h_{i} (ρ) = \frac{2 K R_{i}}{v_{f_{0}}} \int_{R + \frac{s_{i}}{2}}^{R + a} \frac{P (ρ) V (ρ)}{ρ} d ρ

(5)

Taking the surface shape $z = \frac{1}{50} (x^{2} + y^{2})$ as an example, the tool head and workpiece materials are polyurethane and K9 glass, and the polishing process parameters selected in the experiment are shown in Table 1.

Table 1.

Polishing process parameters.

Polishing parameters	Units	Value
Pressure (F_n)	N	10
Feed rate (v_f)	mm/s	0.1
Tool head speed(ω)	rpm	200
Polishing deflection angle (φ)	°	30
Polishing deflection angle (β)	°	45
Geodesic radius curvature (R)	mm	3

Figure 4 shows the relationship between Δh₁(ρ) and r_i based on equation (4). With the increase of overlap length r_i, Δh₁(ρ) increased nonlinearly. Because the width of the polishing path is much smaller than the radius of the small tool head and the workpiece, the relationship between the amount of material removed and the overlap length between the paths is nearly linear. Figure 5 shows the relationship between Δh₁(ρ) and s_i based on equation (5). When 0 < s_i < 2a, as s_i increases, Δh₁(ρ) decrease, that is, the amount of material removed decreases with the increase of track spacing. When s_i ≥ 2a, it means that there is no repeated polishing between adjacent tracks, and there is no material removal in the overlapping part.

Figure 4.

Δh-r_i Relationship diagram.

Figure 5.

Δh-s_i Relationship diagram.

Relationship between polishing trajectory and projection trajectory

The distance between the curved surface polishing tracks is very small, so the distance between any two points on the adjacent tracks can be regarded as a straight line distance. Points P_i and P_i₊₁ are two points on adjacent trajectories, and the coordinates are (x_i, y_i, z_i) and (x_i₊₁, y_i₊₁, z_i₊₁), then the surface trajectory spacing s_i is:

s_{i} = \sqrt{{(x_{i + 1} - x_{i})}^{2} + {(y_{i + 1} - y_{i})}^{2} + {(z_{i + 1} - z_{i})}^{2}}

(6)

In the plane of O-xz, this formula can be simplified to:

s_{i} = \sqrt{{(x_{i + 1} - x_{i})}^{2} + {(z_{i + 1} (x) - z_{i} (x))}^{2}}

(7)

The projected trajectory distance l_i= x_i+1−x_i the relationship between the projected trajectory distance l_i and the overlap length r_i is:

l_{i} = \sqrt{4 {(a - r_{i})}^{2} + {(z_{i + 1} (x) - z_{i} (x))}^{2}}

(8)

According to the actual polishing surface shape, the relationship between the material removal amount of the second polishing of the overlapping part and the projection track pitch is established. As shown in Figure 6, when r = 0.25a and r = 0.4a, along the x-axis direction, the increase in surface curvature results in a change in contact circle a. Although there is a decreasing trend in the distance between curved tracks, the change of r and curvature affects a little change, which verifies the Greenwood theoretical model. Since the curvature of the geometrical surface has a great influence on the projection of the rotationally symmetrical aspheric surface, the distance between the projected trajectories gradually decreases with the increase of the abscissa x, so the choice of the actual polishing workpiece has a great influence on the change of the projected trajectory distance.

Figure 6.

Schematic diagram of the change of the projection track pitch.

Factors affecting the material removal of the rotationally symmetric aspheric surface

Factors such as the contact area and the curvature of the workpiece affect the removal of material. Therefore, it is necessary to analyze and optimize the influence of these parameters before establishing the residence time of the rotationally symmetrical aspheric surface and the total amount of material removal.

Impact of changes in contact area on material removal

When a spherical tool head is used to polish a workpiece, the change in the surface curvature of the workpiece affects the polishing contact area, so it is necessary to analyze the relationship between the amount of material removal and the change in curvature. In the experiment, a spherical tool head of polyurethane and a rotationally symmetrical aspherical workpiece $z = \frac{1}{40} (x^{2} + y^{2})$ , $z = \frac{1}{50} (x^{2} + y^{2})$ , and $z = \frac{1}{60} (x^{2} + y^{2})$ of K9 glass were selected. The specific polishing process parameters are shown in Table 2.

Table 2.

Experimental simulation parameters.

Polishing parameters	Units	Value
Three surface radii	mm	20
Pressure (F_n)	N	10
Feed speed (v_f)	mm/s	0.1
Tool head speed(ω)	rpm	200
Spherical head radius	mm	10
Polishing deflection angle (φ)	°	30
Polishing deflection angle(β)	°	45
Geodesic radius curvature (R)	mm	Constant

Under the same experimental conditions, the polishing inclination angle, the polishing deflection angle and the geodesic curvature radius are constants, and the influence of the changes in the contact area of the three surface shapes on the amount of material removal is analyzed. Figure 7 shows the polishing results of the small tool head on three kinds of workpieces. It can be seen that the larger the radius of curvature of the workpiece surface, that is, the smaller the curvature change, the smaller the change in material removal. The change range of the material removal amount of the face workpiece $z = \frac{1}{60} (x^{2} + y^{2})$ in the interval (0, 20) is only 1.47%, and the change range of the material removal amount of the three kinds of workpieces is 0.55%–2.65%.

Figure 7.

Schematic diagram of the relationship between material removal and contact area.

The radius of curvature of the workpiece surface is larger than the radius of the tool head, and the change of the contact area affected by the change of curvature has little effect on the material removal. Therefore, when the trajectory planning is carried out in this paper, the change of the contact area is taken as the average value and taken as the fixed value.

Influence of geodesic curvature radius on material removal

As shown in Figure 8, suppose p is any polishing point of curve c on the space surface s, the unit tangent vector of curve c, k is the curvature vector, n is the unit normal vector of surface s, and v is the vector of t and n . Cross product, the projection of k on v is the geodesic curvature.

Figure 8.

Geodesic radius of curvature.

When the spherical tool head moves on the surface of the workpiece, the relative position and curvature of the polishing point P_i on the surface of the workpiece change from time to time, that is, the position and curvature of the corresponding point on the generatrix of the rotationally symmetric aspheric surface change. The relative position and curvature of the polishing point P_i determine the geodesic curvature center point O_i and the geodesic curvature radius R_i of the polishing track, so it is necessary to analyze the influence of the geodesic curvature radius on the material removal. R_i is related to the surface shape function of the actual workpiece, the relative position and curvature of the polishing point P_i. The relationship between the curvature λ and the geodesic radius of curvature R_i is:

\tan λ = \frac{\sqrt{d^{2} - R_{i}^{2}}}{R_{i}}

(9)

The shape function of the rotationally symmetric aspheric surface is z(x, y) = k(x²+y²), k is the parameter that controls the shape of the surface, and its generatrix is z(x) = kx², then the relative position of the polishing point P_i and R_i The relationship is:

R_{i} = \frac{x}{\sqrt{1 + {(2 kx)}^{2}}}

(10)

When studying the influence of geodesic curvature radius on material removal, three kinds of workpieces $z = \frac{1}{40} (x^{2} + y^{2})$ , $z = \frac{1}{50} (x^{2} + y^{2})$ and $z = \frac{1}{60} (x^{2} + y^{2})$ with the same tool head and surface shape of are still used. The contact circle radius a is taken as a fixed value, that is, a = 1.19. As shown in Figure 9, when x ≤ a, that is, when x ≤ 1.19, the material removal of the three types of workpieces changes significantly as x increases. When x > a, that is, when x > 2.38, the amount of material removal of the three kinds of workpieces decreases with the increase of curvature λ_i and h₁(ρ) decreases with the increase of x. Analyze the polishing points at relative positions 2.4, 10, and 15 mm. From 2.4 to 10 mm, h₁(ρ) changes from 2.96% to 2.98%, and from 10 to 15 mm, h₁(ρ) changes from 0.15% to 0.17%. The geodesic radius of curvature affected by curvature has a greater impact on material removal on a surface with a larger curvature change, and a smaller impact on an area with a smaller curvature change.

Figure 9.

Relationship between material removal and geodesic radius of curvature.

As the ratio of the radius of curvature of the workpiece surface to the radius of the spherical head increases, the amount of removal changes gradually decreases. When the ratio of the two is ∞, the surface of the workpiece is approximately flat. At the same time, the choice of polishing inclination angle φ and deflection angle β has a great influence on the result of removal. Therefore, it is necessary to optimize R_i,φ and φ and perform error compensation to reduce the overall change in material removal.

Polishing trajectory planning method for the optimal removal of the entire surface

After determining the influence of the contact area change, geodesic radius of curvature and polishing posture on the material removal of the curved surface, the mathematical relationship between the amount of material removal and the track spacing and overlap length of circular polishing is combined. For plane and rotationally symmetric aspheric surfaces, a planning model of concentric circle trajectories is constructed.

Application of concentric circle trajectory planning

The optical processing polishing head movement track is usually grating track and spiral track, but these two common path movement tracks are prone to iterative errors. Archimedes spiral trajectory is a commonly used processing trajectory for rotationally symmetric aspheric surfaces. As a constant velocity spiral, its trajectory is continuous and changes smoothly. However, the curvature change of the workpiece surface will affect the change of the polishing track pitch, that is, the position on the workpiece where the curvature change rate is small, the polishing track pitch is dense, and the curvature change rate position is sparse, and even the tracks do not overlap. The corrected equal overlap ratio spiral line can stabilize the polishing track distance because the overlap portion of the track pitch is equal, but for workpieces with large curvature changes, the optimal overlap ratio between the tracks on each surface is different. Therefore, the equal overlap rate method cannot cope with the problem of curvature changes, and at the same time cannot quantify the residence time distribution, the total amount of material removal and the processing efficiency. Based on this, a concentric circle trajectory planning method with variable spacing is proposed.

Concentric circles are mathematically defined as circles with the same center but different radii on the same plane. In actual polishing, the concentric circle track has the advantages of easy track pitch adjustment and smooth track, thereby improving the polishing efficiency. The equation is

{(x - a)}^{2} + {(y - b)}^{2} = R_{i} (R_{i} > 0)

(11)

Equation (11) represents any circle with a point (a, b) as the center and a positive radius. Any circle is symmetrical and continuous. Therefore, when using a concentric circle track for material removal, it is only necessary to select any direction with the center of the circle as the origin for analysis. On the plane, the point (0, 0) is taken as the center of the circle. When R_i≤ a, the periphery of the center will be over polished. When R_i > a, the material removal equation of the concentric circle track is:

H_{i} (ρ) = 2 π R_{i} h_{i} (ρ) (R_{i} > a)

(12)

The above formula is a function with ρ as the independent variable and H_i(ρ) as the dependent variable. The equation also includes parameters such as the polishing inclination angle φ, the polishing deflection angle β, and the geodesic curvature radius R_i. Therefore, in actual engineering, these parameters need to be optimized.

Figure 10 shows the polishing process of the tool head on a flat workpiece using concentric circular trajectories. Any circular polishing trajectory in the concentric circular trajectory is i, where R_i is the radius of the trajectory i, and the value range of ρ is (ρ_i_min, ρ_i_max). The distance between the polishing tracks i and i+1 is s_i, and the overlap length is 2r_i, then:

s_{i} = R_{i + 1} - R_{i} r_{i} = \frac{1}{2} (ρ_{i max} - ρ_{(i + 1) min})

(13)

Figure 10.

Material removal of flat workpieces.

r_i is a function related to the overlap length in actual polishing. When the workpiece is a flat surface, r_i is a fixed value; when the workpiece is a rotationally symmetric aspheric surface, r_i can be a linear function or a non-linear function, and the track spacing can be actively adjusted according to the curvature of each area of the workpiece. Therefore, during specific polishing, the relationship between the position x and the overlap length r_i can be established according to the track spacing with the smallest roughness at the positions with different curvatures and the processing efficiency of the polished workpiece.

Polishing trajectory planning for the optimal removal of the plane

When the workpiece is close to the plane, the geodesic radius of curvature R is equal to d, and the specific steps of concentric circle trajectory planning are: when R ≤ a, use fixed-point polishing. When R > a, the material removal model of track 1 is H₁(ρ) = 2πR₁h₁(ρ), and the material removal model of track 2 is H₂(ρ) = 2πR₂h₂(ρ). According to simulation and experimental results, at different positions on the surface of the workpiece, when the surface roughness is optimal, the overlap length r_i between the corresponding tracks is measured. The overlap length of trajectories 2 and 1 is denoted as r₁, and the overlap length between the outermost trajectories is denoted as r_n. Since the curvature is unchanged or minimal, the overlap length r_i is a constant or a linear function:

r_{i} = \frac{r_{n} - r_{1}}{n - 1} (i - 1) + r_{1}, 1 \leq i \leq n, i \in Z

(14)

The amount of material removal in the overlapping part of polishing track 2 and 1 is:

Δ H_{1} (ρ) = 2 π R_{1} Δ h_{1} (ρ)

(15)

The distance between polishing track 2 and 1 is s₁ = R₂ − R₁ = 2(a−r₁), the material removal model of polishing track i is H_i(ρ) = 2πR_ih_i (ρ), and the material removal model of polishing track i+1 is H_i+₁(ρ) = 2 πR_i+1h_i+₁(ρ). The material removal amount of the overlapping part of the polishing track i and i+1 is:

Δ H_{i} (ρ) = 2 π R_{i} Δ h_{i} (ρ)

(16)

The distance between polishing track i and 1 is:

R_{i} - R_{1} = 2 ((i - 1) a - \sum_{j = 1}^{i - 1} r_{j})

(17)

Since the overlapping length of the polishing track n+1 and n is r_n, the radius R_n₊₁ of the polishing area of the workpiece is:

R_{n + 1} = 2 (na - \sum_{i = 1}^{n} r_{i}) + R_{1}

(18)

Therefore, in determining the polishing range R_n₊₁ of the polishing workpiece, the number n of polishing tracks of the flat tool can be known. Then the theoretical material removal total of the polished workpiece is $\sum_{i = 1}^{n + 1} H_{i} - \sum_{i = 1}^{n} Δ H_{i}$ . Due to the interference of objective factors such as experimental conditions, there is an approximate proportional deviation between the experimental and theoretical results of the material removal test. In order to accurately calculate the material removal, a correction factor for the total amount of material removal is introduced:

S_{H} = α (\sum_{i = 1}^{n + 1} H_{i} - \sum_{i = 1}^{n} Δ H_{i})

(19)

Polishing trajectory planning for the optimal removal of rotationally symmetric aspheric surfaces

Figure 11 shows the polishing situation of a spherical tool head on a rotationally symmetric aspheric surface z(x, y) = k(x²+y²) with concentric trajectories. The polishing points P_i and P_i₊₁ are any points on the polishing trajectories i and i+1, respectively, and the distance from the points P_i and P_i₊₁ to the center of rotation, that is, the radius of curvature of the polishing trajectory is d_i and d_i₊₁. Because of the curvature change, make the tangent planes of points P_i and P_i₊₁ respectively, where R_i and R_i₊₁ are the geodesic curvature radii of points P_i and P_i₊₁.

Figure 11.

Schematic diagram of material removal of rotationally symmetrical aspherical workpieces.

When the workpiece is a rotationally symmetric aspheric surface, the change in curvature λ affects the material removal parameters. When R > a, the material removal model of track 1 is H₁(ρ) = 2πR₁secλ₁h₁(ρ), and the material removal model of track 2 is H₂(ρ = 2πR₂secλ₂h₂(ρ).

Due to the change of curvature, different positions on the surface of the workpiece reach the ideal surface roughness, and the overlap length r_i between the tracks of the corresponding positions is obtained. In a rotationally symmetric aspheric surface, r_i and the curvature λ function r(λ) can be established by an iterative method, or a linear or non-linear function of r_i and the number of trajectories i can be established by experimental detection.

The material removal amount of the overlapping part of polishing track 2 and 1 is:

Δ H_{1} (ρ) = 2 π R_{1} \sec λ_{1} Δ h_{1} (ρ)

(20)

The distance between polishing tracks 2 and 1 is $s_{1} = \frac{d_{2} - d_{1}}{\cos λ_{1}} = 2 (a_{1} - r_{1})$ . The material removal model of the polishing track i is H₁(ρ) = 2πR₁secλ₁h₁(ρ), then the material removal model of i+1 is H_i+₁(ρ)=2 πR_i+1secλ_i₊₁h_i+1(ρ).

The material removal amount of the overlapping part of the polishing track i and i+1 is:

Δ H_{i} (ρ) = 2 π R_{i} \sec λ_{i} Δ h_{i} (ρ)

(21)

Since the overlapping length of the polishing track n+1 and n is r_n, the radius d_n₊₁ of the polishing area of the workpiece is:

d_{n + 1} = 2 \sum_{i = 1}^{n} (a_{i} - r_{i}) \cos λ_{i} + d_{1}

(22)

When the radius of curvature of the workpiece surface is larger than the radius of the tool head, the contact circle radius a_i is approximately a, and the polishing area range d_n₊₁ of the workpiece is determined, and the number of rotationally symmetric aspheric polishing tracks n can be obtained. The total material removal model of the rotationally symmetric aspheric surface is S_H.

Trajectory planning simulation of optimal removal of rotationally symmetric aspheric surface

Archimedes spiral simulation

A spherical tool head made of polyurethane was used to polish the K9 glass plane workpiece. Under the experimental conditions in Table 3, the polishing contact circle radius a = 1.465 was obtained.

Table 3.

Contact circle radius a determines the experimental parameters.

Polishing parameters	Units	Value
Surface radius	mm	25
Pressure (F_n)	N	12
Feed rate (v_f)	mm/s	0.1
Tool head speed(ω)	rpm	300
Spherical head radius	mm	10
Polishing deflection angle (φ)	°	30
Polishing deflection angle (β)	°	45
Track spacing (s_i)	mm	2.45

Figure 12 shows the Archimedean spiral with s_i < 2a on the workpiece surface $z = \frac{1}{50} (x^{2} + y^{2})$ polishing track and the contact surface shape of the polishing point P_i of the corresponding track. A uniform spiral track is covered on a planar workpiece. On a rotationally symmetric aspheric workpiece, the density of the spiral track changes with the curvature, that is, from the center of the workpiece to the edge, the curvature gradually increases, and the track spacing changes from dense to sparse. Therefore, the overlap length r between the tracks gradually decreases, and even the outer tracks no longer do not overlap. Because the traditional spiral trajectory does not uniformly remove the material of the rotationally symmetrical aspheric surface, the surface accuracy and processing efficiency of the workpiece are reduced.

Figure 12.

Simulation diagram of Archimedes spiral.

Overlap length model based on curvature change

The h_i(ρ) and Δh_i(ρ) of the polishing track i of the rotationally symmetrical aspheric surface change with the curvature λ_i, and the ideal track spacing s_i is different in regions with different curvatures. Therefore, the r_i of the different curvature positions of the workpiece surface shape $z = \frac{1}{50} (x^{2} + y^{2})$ are determined. The lowest point of the h_i(ρ) material removal contour is h_i, and the lowest point of the Δh_i(ρ) overlapping part of the removed contour is h_i^′. The depth difference Δh_i between the two is:

Δ h_{i} = h_{i} - {h_{i}}^{'}

(23)

Experiments have found that when the overlap length r_i > 0.5a, the material removal contour Δh_i(ρ) of the overlapped part will generate a new material removal contour, and then repeat:

Δ h_{ii} = h_{i} - {h_{ii}}^{'}

(24)

As shown in Table 4, when trajectory 1 is 0.05a < r < 0.25a, the depth difference between h₁ and h₁^′ is Δh₁ < 0, that is, the lowest point h₁^′ of the overlapping part of Δh₁(ρ) removed from the contour is higher than h₁(ρ) material removed the lowest point h₁ of the contour. When 0.25a < r < 0.3a, Δh₁ changes from negative to zero and then to positive. Therefore, when the maximum material removal required by the workpiece does not exceed 0.1 mm, the ideal overlap length range between trajectories 1 and 2 is (0.25a, 0.3a). When 0.3a < r < 0.5a, Δh₁ > 0 and the value of h₁ gradually increase, that is, the amount of material removed from the secondary polishing of the overlapping part is greater than h₁(ρ) of polishing track 1, and when r = 0.45a, polishing, the right half of the removed contour of trajectory 1 basically disappears. When 0.5a < r < 0.75a, the track 1 gradually disappears, and when r = 0.45a, the track 1 disappears completely and becomes a smooth curve with the overlapping part Δh₁(ρ). Δh₁(ρ) generates a new material removal profile, then the depth difference Δh₁₁ between h₁^′ and the lowest point h₁₁^′of the new profile changes from negative to zero and then to positive. When the machining allowance of the workpiece is 0.1–0.2 mm, the best range of material removal effect is (0.55a, 0.6a), and when r = 0.75a, a new material removal profile appears.

Table 4.

Relationship between depth difference Δh_i and r.

r = ka	0.05a	0.1a	0.15a	0.2a	0.25a	0.3a	0.35a
h ₁	−0.07878	−0.07865	−0.07867	−0.07863	−0.07869	−0.07863	−0.07860
h ₁ ^′	−0.02323	−0.03847	−0.04533	−0.05382	−0.06299	−0.08599	−0.09575
Δh ₁	−0.05555	−0.04018	−0.03334	−0.02481	−0.0157	0.00736	0.01715
h_n	−0.09266	−0.09330	−0.09294	−0.09332	−0.09272	−0.09310	−0.09329
h_n ^′	−0.01141	−0.03127	−0.04277	−0.05547	−0.06643	−0.08970	−0.1034
Δh_n	−0.08125	−0.06203	−0.05017	−0.03785	−0.02629	−0.0034	0.01011
r = ka	0.40a	0.45a	0.5a	0.55a	0.6a	0.65a	0.75a
h ₁	−0.07858	−0.07857	−0.07834	−0.07591	−0.07396	−0.06625	−0.05377
h ₁ ^′	−0.1048	−0.1133	−0.1208	−0.1290	−0.1338	−0.1381	−0.1542
h ₁₁ ^′				−0.1177	−0.1425	−0.1603	−0.1992
h ₁₁₁ ^′							−0.2233
Δh ₁	0.02622	0.03473	0.04246	0.05309	0.05984	0.07185	0.10043
Δh ₁₁				−0.0113	0.0087	0.0222	0.045
h_n	−0.09331	−0.09316	−0.09361	−0.0926	−0.09035	−0.08560	−0.07297
h_n ^′	−0.1158	−0.1267	−0.1377	−0.1483	−0.1554	−0.1609	−0.1670
h_nn ^′				−0.1255	−0.1546	−0.1776	−0.2308
h_nnn ^′							−0.25
Δh_n	0.02249	0.03354	0.04409	0.0557	0.06505	0.0753	−0.09403
Δh_nn				−0.0228	−0.0008	0.0167	0.0638

The trajectory n is similar to n+1, when 0.05a < r < 0.3, the depth difference between h_n and h_n^′ is Δh_n_<0. When 0.3a < r < 0.35, Δh₁ changes from a negative value to a positive value. Therefore, when the maximum material removal required by the workpiece does not exceed 0.1 mm, the ideal overlap length range between trajectories n and n+1 is (0.3a, 0.35a). When 0.35a < r < 0.5a, Δh_n > 0, and when r = 0.5a, the right half of the removal contour of the polishing track n+1 basically disappears. When 0.5a < r < 0.75 a, the trajectory n+1 gradually disappears, and the overlapping part Δh_n(ρ) becomes a smooth curve, and the depth difference Δh_nn between h_n^′ and h_nn^′ produced by Δh_n(ρ) changes from a negative value to a positive value. When the maximum material removal of the workpiece is 0.1–0.2 mm, the material removal effect is optimal (0.6a, 0.65a), and when r = 0.75a, a new material removal profile appears.

Therefore, in actual polishing, according to the detection results of surface roughness, calculate the surface error and machining allowance of the workpiece. Combined with this experiment, when the machining allowance is about 0.1 mm, the overlap length is selected in the range of 0.25a−0.35a. When the machining allowance is about 0.2 mm, the overlap length is 0.55a–0.65a. In order to meet the surface accuracy and processing efficiency, the correct value r must be taken.

As shown in Figure 13, when the amount to be removed on the workpiece surface is about 2 mm, the concentric circle track with an overlap length of 0.65a is a schematic diagram of the polishing track along the surface O-xz. The radius is from 0 to 25 mm, and the material removal of the polishing track changes with the increase of x. And change. The red trajectory line is the polishing trajectory i, and the black contour line is the material removal contour of the overlapped part of the secondary polishing. When r > 0.5a, the secondary polishing trajectory generates a new contour, which is specifically expressed as a wave trough. Figure 14 is a 3D schematic diagram of the polishing track of Figure 13.

Figure 13.

Schematic diagram of polishing track r = 0.65a.

Figure 14.

Schematic diagram of polishing track in 3D r = 0.65a.

In actual processing, when the error between the surface of the workpiece to be polished and the theoretical model is around 0.1 mm, the r value ranges of track 1 and n+1 are (0.25a, 0.3a) and (0.3a, 0.35a), respectively. The value of r is optimized, as shown in Figures 15 and 16, when r = 0.259a and r = 0.333a, the depth difference is approximately zero.

Figure 15.

Schematic diagram of 0–5 mm with r = 0.259a.

Figure 16.

Schematic diagram of 20–25 mm with r = 0.333a.

In the surface of the workpiece $z = \frac{1}{50} (x^{2} + y^{2})$ , the area to be processed and the machining allowance both affect the number of polishing tracks. The optimal number of polishing tracks with a radius of 25 mm in the polishing area is 12. In the simulation experiment, the overlap length r_i increases approximately linearly with the addition of the trajectory i. Therefore, substituting r = 0.259a and r = 0.333a into the formula, the function of the overlap length r_i is:

r_{i} = 0.0069 a (i - 1) + 0.259 a, 1 \leq i \leq 12, i \in Z

(25)

As the area to be polished increases, such as the radius of 30, 40,and 50 mm, etc., or the curvature of the workpiece surface changes more, the advantage of this controllable track pitch will become more obvious. According to the value of r_i at different positions of the curvature, a corresponding functional relationship is established, so as to precisely control the change of the polishing track pitch and better cover the surface of the workpiece.

Trajectory planning simulation of optimal removal of rotationally symmetric aspheric surface

The concentric circle trajectory planning optimizes the polishing trajectory of the rotationally symmetrical aspheric workpiece by quantitatively controlling the coincidence length r_i, that is, the trajectory spacing s_i. Since the trajectory is based on the center of rotational symmetry, there will be no iterative error of the spiral line. According to the specific workpiece surface shape and polishing process parameters, the material removal S_H in the polishing area, the residence time t and the polishing area model can be calculated, so as to plan the polishing trajectory of the optimal removal amount of the full surface, thereby improving the polishing quality and efficiency. As shown in Figure 17, the initial track spacing si of the workpiece $z = \frac{1}{50} (x^{2} + y^{2})$ is less than 2a, where the concentric circle polishing track i and the polishing point P_i corresponding to the track.

Figure 17.

Schematic diagram of concentric circle trajectory with controllable removal.

During the polishing process, both the track pitch s_i and the overlap length r_i vary with x. As shown in Figure 18, the function relationship between overlap length r_i and x. is:

r_{i} = 0.0061 x + 0.3656, 1.465 \leq x \leq 25

(26)

Figure 18.

Change of r_i and s_i with x.

The function relationship between track spacing s_i and x is:

x_{i} = - 0.0121 x + 2.1986, 1.465 \leq x \leq 25

(27)

Since r_i increases as x increases, s_i decreases as x increases. Considering that the maximum amount of material removal does not exceed 1 mm, the overlap length r_i between tracks is small. Table 5 shows the specific values of the relative position coordinate x, the overlap length r_i and the track spacing s_i of the polishing track i.

Table 5.

The specific parameters of r_i, s_i, R and λ.

i	1	2	3	4	5	6	7	8	9	10	11	12
x	2.271	4.388	6.430	8.382	10.24	11.99	13.64	15.20	16.68	18.06	19.38	20.62
r_i	0.379	0.390	0.400	0.410	0.420	0.430	0.440	0.450	0.460	0.470	0.480
s_i	2.171	2.151	2.131	2.110	2.090	2.070	2.050	2.030	2.010	1.990	1.969	1.949
R	2.271	4.388	6.430	8.382	10.24	11.99	13.64	15.20	16.68	18.06	19.38	20.62
λ	5.191	9.955	14.42	18.54	22.20	25.62	28.62	31.30	33.71	35.84	37.78	39.52

The polishing tool head moves on a rotationally symmetrical aspherical workpiece, and the curvature λ_i of the polishing point p_i changes with x. Figure 19 is the generatrix $z = \frac{1}{50} (x^{2})$ of the curved surface $z = \frac{1}{50} (x^{2} + y^{2})$ , and the relationship between the curvature λ_i of the track i and the position coordinate x. Table 5 shows the specific values of the curvature λ_i of the trajectory i and the geodesic curvature radius R_i

Figure 19.

Change of λ with x.

As shown in Figure 20, the curvature λ affects the material removal of the polishing track, the curvature λ and the track spacing s_i affect the material removal of the overlapping part between the tracks, h_i(ρ) decreases with the increase of λ, and Δh_i(ρ) increases with r_i and As λ increases, the corresponding h_i(ρ) and Δh_i(ρ) of the polishing track i are shown in Table 6.

Figure 20.

Change of h_i and Δh_i with x.

Table 6.

The specific parameters of r_i, s_i, R, and λ.

i	1	2	3	4	5	6	7	8	9	10	11	12
r_i	0.26a	0.27a	0.27a	0.28a	0.29a	0.29a	0.3a	0.3a	0.31a	0.32a	0.33a	0.34a
x	2.271	4.388	6.43	8.382	10.24	11.99	13.64	15.2	16.68	18.06	19.38	20.62
h ₁	0.005	0.006	0.006	0.007	0.007	0.008	0.008	0.008	0.009	0.009	0.010
h	0.078	0.07	0.069	0.06881	0.0687	0.0686	0.0685	0.0684	0.0683	0.0682	0.0681	0.0679
H ₁	0.0675	0.163	0.27	0.386	0.51	0.642	0.782	0.929	1.084	1.244	1.412

Many factors such as curvature λ, polishing track radius R_sc and geodesic curvature radius R_i affect the material removal amount H_i(ρ) of polishing track i. The $\sum_{i = 1}^{n + 1} H_{i}$ of polishing track i in the polishing area of the workpiece is:

\sum_{i = 1}^{n + 1} H_{i} = \sum_{i = 1}^{12} H_{i} = \sum_{i = 1}^{12} 2 π R_{i} \sec λ_{i} h_{i} (ρ)

(28)

The $\sum_{i = 1}^{n} Δ H_{i}$ of the overlapping part of the polishing track i in the polishing area of the workpiece is:

\sum_{i = 1}^{n + 1} Δ H_{i} = \sum_{i = 1}^{12} Δ H_{i} = \sum_{i = 1}^{12} 2 π R_{i} \sec λ_{i} Δ h_{i} (ρ)

(29)

Figure 21 is a schematic diagram of the material removal amount H_i(ρ) of the workpiece $z = \frac{1}{50} (x^{2} + y^{2})$ surface polishing track i and the material removal amount ΔH_i (ρ) of the overlapping part. H_i (ρ) increases with the increase of the track radius R_sc, and ΔH_i(ρ) increase as the overlap length r_i between tracks increases. According to Table 6, the value of $\sum_{i = 1}^{n + 1} H_{i}$ is 74.037 mm and the value of $\sum_{i = 1}^{n} Δ H_{i}$ is 7.489 mm. Then the total theoretical material removal of the polished workpiece is:

S_{H} = \sum_{i = 1}^{n + 1} H_{i} - \sum_{i = 1}^{n} Δ H_{i} = 66.548 mm

(30)

Figure 21.

Change of H_i and ΔH_i with x.

The t shown in Figure 22 is the corresponding residence times of the polishing track i of the workpiece $z = \frac{1}{50} (x^{2} + y^{2})$ surface, and t gradually increases with the increase of the polishing track radius R_sc. As shown in Table 7, the theoretical residence time value $\sum_{i = 1}^{n + 1} t$ of the polished workpiece is 154.2 min, which is 2.57 h. According to the total amount of material removal, the residence time of each trajectory and the amount of material removal, the polishing trajectory planning for the optimal removal of the entire surface is carried out.

Figure 22.

Schematic diagram of residence time of polishing track.

Table 7.

The specific parameters of r_i, s_i, R, and λ.

i	1	2	3	4	5	6	7	8	9	10	11	12
T (s)	142.7	275.7	404	526.7	643.4	753.4	857	955	1048	1135	1218	1296
T (min)	2.378	4.595	6.733	8.778	10.72	12.56	14.28	15.92	17.47	18.91	20.29	21.59

Optical surface polishing and testing experiment

In order to verify the accuracy of the method proposed in this paper, an optical surface polishing and testing experiment is carried out. The experimental and testing equipment include LPS250 five-axis linkage free-form surface polishing machine (Figure 23), 2207 surface profile measuring instrument (Figure 24), which can measure and analyze the surface profile and geometric morphology of the workpiece, and OLYMPUS digital microscope DSX500 (Figure 25), which can measure the surface profile of the workpiece in 2D and 3D, convert the physical image in the microscope to digital-to-analog and directly display the enlarged picture on the display in the form of electronic signal.

Figure 23.

Five-axis linkage free-form surface polishing machine.

Figure 24.

2207 surface profile measuring instrument.

Figure 25.

OLYMPUS Digital microscope.

The material used in the polishing experiment is an K9 optical blank glass workpiece, as shown in Figure 26 which is approximately cylindrical in shape and about 21 mm thick. One side of the two surfaces of the workpiece is a circular plane with a diameter of about 60.5 mm, and the other side is a rotationally symmetric aspheric surface with an aspheric ring with a diameter of about 50 mm.

Figure 26.

K9 Optical glass blank workpiece: (a) circular plane and (b) rotationally symmetric aspheric surface.

Use 2207 surface profiler to measure several points on the surface of optical glass blank workpiece. As shown in Figure 27, within the sampling length, the arithmetic average deviation of contour Ra is 2.1753 μm, and the maximum peak-valley height of roughness Rz is 15.2208 μm. Within the evaluation length, the vertical distance between the maximum peak and valley Ri is 16.8149 μm. These parameters are used as raw data, and the detection results of polishing areas are compared to check the polishing effect.

Figure 27.

K9 Surface roughness of glass blank workpiece.

Experiment of polishing material removal in planar annular track

The distance between polishing tracks s_i is 1.45a, and the radii of two polishing tracks R_sc are 15 and 16.726 mm. Polishing process parameters: polishing force F_n of 5 and 10 N, rotating speed ω of 600 rpm, polishing angle φ of 30°, and residence time t of 80 min. Figure 28 shows the trajectory of the spherical tool head in the ring polishing experiment. Figure 29 is the residence diagram of a certain position in the circular trajectory polishing material removal experiment. During the processing, the polishing solution can not only cool the polishing area to ensure a constant temperature, but also the polishing particles can adhere to the polyurethane surface, reducing tool head wear and auxiliary material removal.

Figure 28.

Tool Trajectory of LPS250 Machine Tool.

Figure 29.

Experimental process of annular polishing material removal.

Figure 30 shows the circular polishing track of the glass plane. In the polishing experiment, the material removal in the vertical direction of the circular polishing track is approximately proportional to the polishing force, and the surface profile of any vertical direction of the circular track is approximately the same. The polishing effect of F_n= 10 N is more intuitive than F_n= 5 N.

Figure 30.

Experimental results of glass ring polishing: (a) F_n= 5 N and (b) F_n= 10 N.

Figure 31 shows the surface profile of the circular polishing track in any vertical direction. The surface roughness of the polishing area gradually decreases from the edge to the center, and the polishing effect of the overlapping polishing area between tracks is better.

Figure 31.

Details of two places of glass planar annular polishing: (a) F_n= 5 N and (b) F_n= 10 N.

Polishing experiment of circular trajectory of rotationally symmetric aspheric surface

In the experiment of circular trajectory polishing of rotationally symmetric aspheric surface, the pitch of polishing trajectory s_i is 1.4a, the radii of two polishing trajectories R_sc are 15 and 16.429 mm, and the heights z are 4.5 and 5.398 mm respectively. Polishing process parameters: polishing force F_n is 10 N, tool head rotation speed ω is 600 rpm, polishing angle φ is 0°, and residence time t is 80 min. Figure 32 shows the circular polishing track of glass. As shown in Figure 33, under OLYMPUS digital microscope DSX500, the details of material removal in any vertical direction of the circular polishing track of rotationally symmetric aspheric glass are measured.

Figure 32.

Removal of curved annular polishing material.

Figure 33.

Detail of curved surface ring polishing.

Figure 34 shows the surface profile of the circular polishing track of glass. Figure 35 shows the surface roughness of the polishing area, with parameters R_a of 0.0165 μm, R_z of 0.1240 μm, and R_t of 0.1676 μm, and the polishing area is only 0.76%, 0.81%, and 1.00% of the blank surface.

Figure 34.

Surface profile of circular polishing trajectory.

Figure 35.

Schematic diagram of surface roughness of polishing area.

Comparison with other method

In order to verify the effectiveness of the proposed method, a comparative polishing experiment by using equidistant trajectory planning method is carried out. The workpiece material is still K9 optical rough glass workpiece. The polished workpiece is a rotationally symmetric aspheric surface. The diameter of the aspheric ring is about 50 mm, and the fixed row spacing is 1.2 mm. The polishing experiment results are shown in the Figure 36.

Figure 36.

Experiment results of equidistant trajectory planning method.

Figure 37 shows the surface profile of the circular polishing track of glass, with parameters R_a of 0.1341 μm, R_z of 0.5878 μm, and R_t of 0.7377 μm, and the polishing area is only 6.16%, 3.86%, and 4.39% of the blank surface. The comparison results with the method proposed in this paper are shown in the Table 8. It can be concluded that the surface quality after polishing with the trajectory planning method proposed in this paper is better, which proves the superiority of this method.

Figure 37.

Surface profile of circular polishing trajectory of equidistant trajectory planning method.

Table 8.

Comparison results.

Trajectory planning method	Unit	R_a	R_z	R_t
Proposed method	μm	0.0165	0.1240	0.1676
Equidistant trajectory planning method	μm	0.1341	0.5878	0.7377

Conclusions

This paper uses the projection method to map the spatial polishing trajectory, analyzes the mapping relationship between the polishing trajectory and the projection trajectory, and calculates the mathematical relationship between the amount of material removal and the track spacing and overlap length of circular polishing. It also analyzes the influence of contact area change, geodesic radius of curvature and polishing posture on the material removal of the curved surface. For the lack of equal overlap ratio spiral lines, a concentric circle trajectory plan with a controllable removal amount is carried out. Based on the overlap length model of the curvature change, the polishing area, the trajectory residence time and the total material removal model are obtained to determine the processing efficiency. For flat and rotationally symmetrical aspherical workpieces, formulate corresponding polishing schemes. Take the rotationally symmetrical aspheric surface as an example, when R ≤ a, fixed-point polishing was used; when R > a, circular polishing with gradual track pitch was used. By analyzing the relationship between the material removal contour and the contour change of the overlapping part, this paper solves the linear function of the overlap length, calculate the total material removal, the material removal amount of each circular track and the residence time, and determine the processing efficiency.

Footnotes

Handling Editor: Chenhui Liang

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: This research was funded by National Natural Science Foundation of China, grant number 51135006.

ORCID iD

Yuanxun Cao

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Polishing trajectory planning method for optimal removal of full surface

Abstract

Keywords

Introduction

The effect of track spacing on material removal

Spatial polishing trajectory and projection trajectory

The influence of polishing track spacing on material removal

Relationship between polishing trajectory and projection trajectory

Factors affecting the material removal of the rotationally symmetric aspheric surface

Impact of changes in contact area on material removal

Influence of geodesic curvature radius on material removal

Polishing trajectory planning method for the optimal removal of the entire surface

Application of concentric circle trajectory planning

Polishing trajectory planning for the optimal removal of the plane

Polishing trajectory planning for the optimal removal of rotationally symmetric aspheric surfaces

Trajectory planning simulation of optimal removal of rotationally symmetric aspheric surface

Archimedes spiral simulation

Overlap length model based on curvature change

Trajectory planning simulation of optimal removal of rotationally symmetric aspheric surface

Optical surface polishing and testing experiment

Experiment of polishing material removal in planar annular track

Polishing experiment of circular trajectory of rotationally symmetric aspheric surface

Comparison with other method

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References