Predictive models of the local and the global polished profiles in deterministic polishing of free-form surfaces

Abstract

The deterministic polishing process is usually used as the final step to produce part surfaces of high surface finish and form accuracy. This article presents the models of the local and the global polished profiles for the deterministic polishing process when a surface is polished by a spherical polisher. The local polished profile is modeled by integrating the index of material removal, which is defined as the polished depth at unit length of the polishing path, at each tool–surface contact region. According to the model, the local polished profile is determined by the process parameters, the tool attitude, the measuring angle and the geometrical/mechanical properties of workpiece and tool. The linear algebraic expression of the global polished profile is derived by convoluting the local polished depth at each dwell point of the polishing process. The polishing experiments are conducted to verify the proposed model in the different polishing conditions. Moreover, simulation results are given to illustrate the application of the proposed model in optimizing the feed rate to minimize the surface form error in the deterministic polishing process.

Introduction

Recently, the needs for free-form part surfaces have been widely recognized in optoelectronic and communication industries. The components with free-form surfaces not only occupy superior optical properties but also reduce the volume and weight for the optical products. The fabrication of high-quality and precision part surface is becoming more and more stringent in manufacturing. The manufacturing process for high-precision part surfaces typically involves a series of steps including generating, fine grinding, polishing and corrective polishing by sub-aperture tool. ¹ As the final step for the fabrication, the polishing process is essential to the quality and duration of the part surfaces. A polishing process is considered to be deterministic if the material removal behavior during the polishing process is predictable whenever the processing parameters are set to fixed values. ² Different from the traditional polishing process which merely aimed at reducing the roughness of the part surface, the deterministic polishing is regarded as an error compensation process to eliminate the form error left by the precision grinding or diamond turning process.^2,3 In recent years, much research has been performed on the deterministic polishing process, focusing on the design of polishing system,^4–6 polishing mechanism,^7–9 prediction of roughness, ¹⁰ tool path planning^11,12 and so on. Apart from the studies discussed above, efforts are also devoted to the material removal during the polishing process. It is essential to model and analyze the material removal of polishing in a quantitative manner so as to enhance the reproducibility and controllability of the deterministic polishing process.

The orbital motion is commonly used to polish large optics, and the material removal function of the orbital motion was studied and optimized by Jones.^13,14 The polishing of small surface components usually employs a rotational spindle at specific angular velocity with a small spherical or cylinder polisher. To model the material removal during the polishing, one should get the knowledge about the wear rate and tool–surface contact during the polishing. Preston’s equation is widely used to model the wear rate during the polishing, which indicates that the material removal is proportional to the contact pressure, sliding velocity and dwell time. ¹⁵ Based on Preston’s equation, Kim and Kim ¹⁶ presented a model of material removal function, which is also called time influence function, by assuming the distribution of pressure in the contact region to be Gaussian. Similar model was also developed by Cheung et al. ¹⁷ Due to the geometric properties of part surface, the size of tool–surface contact region is different from point to point. According to the study by Song et al., ¹⁸ the pressure distribution at the contact agrees with the Hertzian contact theory, and the rotational speed, polisher decrement and curvature radius of surface all have effects on the material removal. Yang and Lee ¹⁹ modeled the local material removal rate by assuming the contact between the polisher and surface to be Hertzian. According to their study, the material removal rate during the polishing is not invariant due to the curvature effect. However, the above listed models of material removal can only be used to predict the removal depth and profile when the tool polishes on a single point. In order to control the amount of material removed precisely by using these models, the tool should be commanded to dwell at each dwell point on the tool path according to the computed dwell time and then move to the next point. The dwell-and-go mode makes the machine tool start and stop frequently, which is time-consuming and hits the acceleration and deceleration limits of the machine tool.

In an earlier research by Zhang et al., ²⁰ the polished profile and depth were directly linked to the feed rate along the tool path by assuming that the wear rate during the polishing follows Archard law. According to their study, the polished profile orthogonal to the tool path is parabolic, and the polished depth depends on the tool–surface geometry and the process parameters (normal force, feed rate, spindle velocity, etc.). If the part surface is axisymmetric and the concentric tool path is used, the measuring direction of form error is naturally along the meridian of part surface, thus the model of polished profile orthogonal to the tool path can be used to optimize the feed rate.^21,22 However, a more common condition that the part surface is nonaxisymmetric and the tool path is not concentric makes the measuring direction not orthogonal to the tool path, as shown in Figure 1. Therefore, as the polisher moves along the tool path on a nonaxisymmetric free-form surface, a model related to the polished profile in the measuring direction is needed. Such a model is useful for the process planning of the deterministic polishing.

Figure 1.

Schematic diagram of polishing along a tool path.

In this article, the polished profile is defined as the depth of material removal along the measuring direction, as shown in Figure 1. In section “Local polished profile,” the local polished profile is derived by integrating the removal index along the tool path at the tool–surface contact patch. The expression of the global polished profile along the measuring direction is developed in section “Global polished profile,” by superimposing the local polished profile at each dwell point. The validity of the presented model for polished profile is verified by experiments in section “Experimental verification and discussion.” The application of the proposed model in the process planning is illustrated in section “Application.” Findings are summarized in section “Conclusion.”

Local polished profile

As shown in Figure 2, as the tool moves across the point P, the feed rate of the tool along the tool path is v_a and the angular spindle velocity is ω. Establish a local coordinate system {o; x, y, z } at P such that the z-direction is normal to the surface and the y-direction is tangential to the tool path. The coordinate system {o; x ′, y ′, z ′} is also established at o(P) such that the x′-direction is along the measuring path. In this article, the local polished profile is defined as the material removal depth along the measuring direction for a single tool path. Thus, the local polished profile is expressed in the coordinate frame x′oy′. The included angle between x′ and x is defined as the measuring angle α here. The coordinate transformation between frames xoy and x′oy′ is given by

$$\left[\begin{array}{c}\hfill x\hfill \\ \multicolumn{1}{c}{y}\end{array}\right]=\left[\begin{array}{cc}\hfill \cos \alpha \hfill & \hfill -\sin \alpha \hfill \\ \multicolumn{1}{c}{\sin \alpha }& \hfill \cos \alpha \hfill \end{array}\right]\left[\begin{array}{c}\hfill x\prime \hfill \\ \multicolumn{1}{c}{y\prime }\end{array}\right]$$

(1)

Figure 2.

Definition of the local coordinate systems.

During the polishing, a contact patch is formed between the spherical tool and part surface, within which the material removal takes place. The Hertzian contact theory is widely used to model the distribution of pressure in the contact patch.^18–20 According to Hertzian contact theory, in the coordinate frame x′oy′, the distribution of the contact pressure is given by^23,24

$$p(x\prime ,y\prime )=-\frac{p_0}{a}\sqrt{a^2-{x\prime }^2-{y\prime }^2}$$

(2)

over the contact circle

$$x{\prime }^2+y{\prime }^2=a^2$$

(3)

where a is the radius of the contact patch which is determined by the polishing normal force and the material properties of the workpiece and the polisher, as indicated in Appendix 2. In equation (2), p ₀ is the pressure at the center of the contact circle which is given by

$$p_0=\frac{3F_n}{2\pi a^2}$$

(4)

where F_n is the normal polishing force.

Figure 3 shows the contact circle between the polisher and part surface. Let H be an arbitrary point on the x-axis. When the tool feeds along the polishing path, the contact patch is also moving. The material removal at H is accumulated as the contact patch feeding across H, thus L ₁ L ₂ in Figure 3 can be regarded as the polishing belt for the material removal at H. The index of material removal at H

$$w_{h/l}\equiv \frac{\mathrm{dh}}{\mathrm{dl}}$$

(5)

is defined as the polished depth at unit length of polishing path. Thus, the polished depth at H can be calculated by integrating w_h/l along the tool path from L ₂ to L ₁, which gives

$$h_H=\underset{L_2}{\overset{L_1}{\int }}{w}_{h/l}\mathrm{dL}$$

(6)

Figure 3.

Geometric entities in the contact patch.

According to Archard wear law, the index of material removal is derived by Zhang et al. ²⁰ as

$$w_{h/l}=k\frac{p{v}_s}{H_v{v}_a}$$

(7)

where k is the wear coefficient, p is the contact pressure, v_s and v_a correspond to the relative sliding velocity and feed velocity, respectively, and H_v is the hardness of the part surface.

Let $H\prime (x_{H\prime },0)$ be an arbitrary point on the x′-axis. If the changes in the process parameters and the local curvature of the part surface are very small in the vicinity of point o(P), the polished depth at H′ is the same as that at H. In the coordinate frame x′oy′, the line segment L ₁ L ₂ can be expressed in a parametric form as

$$\{\begin{array}{c}x\prime =t\hfill \\ \multicolumn{1}{c}{y\prime =\cot \alpha t-\cot \alpha x_{H\prime }}\end{array}$$

(8)

By integrating w_h/l along L ₁ L ₂ in the coordinate frame x′oy′, the polished depth at H′ can be obtained, which gives

$$h_{H\prime }=\underset{L_2}{\overset{L_1}{\int }}{w}_{h/l}\mathrm{dL}=\underset{{x\prime }_a}{\overset{{x\prime }_b}{\int }}{w}_{h/l}\sqrt{1+{\cot }^2\alpha }\mathrm{dt}$$

(9)

where $x{\prime }_a$ and $x{\prime }_b$ correspond to the coordinate values of L ₂ and L ₁ on the x′-axis, respectively. In order to get the values of $x{\prime }_a$ and $x{\prime }_b$ , substitute equation (8) into equation (3), which gives

$$x{\prime }^2+{(\cot \mathrm{ax}-\cot \alpha x_H)}^2=a^2$$

(10)

from which

$$x{\prime }_{a,b}=\frac{x_H{\cot }^2\alpha \pm \sqrt{{\cot }^4\alpha x_H^2-{\csc }^2\alpha ({\cot }^2\alpha x_H^2-a^2)}}{{\csc }^2\alpha }$$

(11)

Substituting equations (7), (8) and (11) into equation (9), the polished depth at H′ can be expressed as

$$h_H(x_H)=\frac{k}{H_v{v}_a}\underset{{x\prime }_a(x_H)}{\overset{{x\prime }_b(x_H)}{\int }}p(x\prime (t),y\prime (t))v_s(x\prime (t),y\prime (t))·\sqrt{1+{\cot }^2\alpha }\mathrm{dt}$$

(12)

in which v_a is assumed to be constant since the contact patch is small. As the value of $x_{H\prime }$ varies from −a/cos α to a/cos α, the local polished profile can be calculated by equation (12). Thus, the local polished profile can be further expressed as

$$h(x\prime )=\frac{k}{H_v{v}_a}\underset{{x\prime }_a(x\prime )}{\overset{{x\prime }_b(x\prime )}{\int }}p(x\prime (t),y\prime (t))v_s(x\prime (t),y\prime (t))·\sqrt{1+{\cot }^2\alpha }\mathrm{dt}-a\cos \alpha \le x\prime \le a\cos \alpha$$

(13)

As indicated by equation (13), the distribution of the pressure and the relative sliding velocity in the contact patch is essential to the computation of the local polished profile. The velocity distribution in the polishing contact patch can be predigested in Figure 4. In Figure 4, the local coordinate system {o; x, y, z } is established fixed to the part surface at point o which is the same as that in Figure 2. Plane xoy is the tangent plane of the surface at point o. The point O_t is the center of the spherical polisher and Z_t is the unit vector of the tool axis. To avoid the zero velocity of the center point on the contact patch, the tool is tilted by an angle σ to the z-direction. As shown in Figure 4, the inclination angle σ is the included angle between vector z and Z_t , and the declination angle λ is the included angle between x and $\mathit{Z}{\prime }_t$ which is the projection of Z_t on the plane xoy. The polishing attitude is determined by the inclination angle σ and declination angle λ in the local coordinate system {o; x, y, z }. The angular spindle velocity ω can be expressed in a vector form as

$$\mathbf{\omega }\mathbf{=}\mathbf{\omega }\mathbf{(}\mathbf{cos}\mathbf{\lambda }\mathbf{sin}\mathbf{\sigma }\mathbf{,}\mathbf{sin}\mathbf{\lambda }\mathbf{sin}\mathbf{\sigma }\mathbf{,}\mathbf{cos}\mathbf{\sigma }\mathbf{)}$$

(14)

where ω = | ω |. The point P(x_p, y_p, z_p ) is an arbitrary point in the contact patch and r is a vector from the point O_t (0, 0, R_p ) to the point P, which gives

$$\mathit{r}=(x_p,y_p,z_p-R_p)$$

(15)

where

$$R_p\approx \sqrt{R_t^2-a^2}$$

(16)

Figure 4.

Schematic diagram of polishing attitude and relative sliding velocity in the contact patch.

The sliding velocity ${\mathit{v}}_s^p$ due to the angular spindle velocity at the point P is calculated as the cross product of angular velocity ω and the length vector r , which gives

$${\mathit{v}}_s^p=\omega \times \mathit{r}=\omega \left[\begin{array}{c}\hfill -R_p\sin \lambda \sin \sigma -\cos \sigma y_p\hfill \\ \multicolumn{1}{c}{\cos \sigma x_p+R_p\cos \lambda \sin \sigma }\\ \multicolumn{1}{c}{\cos \lambda \sin \sigma y_p-\sin \lambda \sin \sigma x_p}\end{array}\right]$$

(17)

On the other hand, the feed velocity at P can be expressed as a vector by

$${\mathit{v}}_a^p=(v_{\mathrm{ax}}^p,v_{\mathrm{ay}}^p,v_{\mathrm{az}}^p)$$

(18)

For the spherical tool polishing on a surface, only the tangential components of the velocity have effect on the material removal in the contact patch. Thus, the relative sliding velocity at point P(x_p, y_p, z_p ) is

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathit{v}}_s=\sqrt{{(v_{\mathrm{sx}}^p+v_{\mathrm{ax}}^p)}^2+{(v_{\mathrm{sx}}^p+v_{\mathrm{ay}}^p)}^2}\hfill \\ \multicolumn{1}{c}{}& =\sqrt{{(-R_p\omega \sin \lambda \sin \sigma -\omega \cos \sigma y_p+v_{\mathrm{ax}}^P)}^2+{(\omega \cos \sigma x_p+R_p\omega \cos \lambda \sin \sigma +v_{\mathrm{ay}}^p)}^2}\hfill \end{array}$$

(19)

In the actual polishing process, the feed rate ${\mathit{v}}_a^p$ is very small compared to the sliding velocity ${\mathit{v}}_s^p$ , thus the relative sliding velocity of point P can be approximated as

$$\begin{array}{cc}\multicolumn{1}{c}{}& v_s\approx \sqrt{{(v_{\mathrm{sx}}^p)}^2+{(v_{\mathrm{sy}}^p)}^2}\hfill \\ \multicolumn{1}{c}{}& =\sqrt{\begin{array}{c}\multicolumn{1}{c}{{(-R_p\omega \sin \lambda \sin \sigma -\omega \cos \sigma y_p)}^2}\\ \multicolumn{1}{c}{+{(\omega \cos \sigma x_p+R_p\omega \cos \lambda \sin \sigma )}^2}\end{array}}\hfill \end{array}$$

(20)

Equation (20) is the expression of the relative sliding velocity in the coordinate frame xoy. By substituting equation (1) into equation (20), the distribution of the relative sliding velocity in the coordinate frame x′oy′ can be obtained, which gives

$$v_s=\omega \sqrt{{[-R_p\sin \lambda \sin \sigma -\cos \sigma (x\prime \sin \alpha +y\prime \cos \alpha )]}^2+{[\cos \sigma (x\prime \cos \alpha -y\prime \sin \alpha )+R_p\cos \lambda \sin \sigma ]}^2}$$

(21)

Further substituting equations (2) and (21) into equation (13), the local polished profile in x′-direction can be expressed by

$$h(x\prime )=-\frac{k{F}_n\omega \sqrt{1+{\cot }^2\alpha }}{2\pi a^3{H}_v{v}_a}\underset{{x\prime }_a(x\prime )}{\overset{{x\prime }_b(x\prime )}{\int }}f(t)\mathrm{dt}-a\cos \alpha \le x\prime \le a\cos \alpha$$

(22)

where

$$f(t)=\sqrt{\left\{\begin{array}{c}\hfill {[-R_p\sin \lambda \sin \sigma -\cos \sigma (x\prime (t)\sin \alpha +y\prime (t)\cos \alpha )]}^2\hfill \\ \multicolumn{1}{c}{+{[\cos \sigma (x\prime (t)\cos \alpha -y\prime (t)\sin \alpha )+R_p\cos \lambda \sin \sigma ]}^2}\end{array}\right\}·(a^2-x\prime {(t)}^2-y\prime {(t)}^2)}$$

(23)

where x′(t) and y′(t) are given by equation (8), and $x{\prime }_a$ and $x{\prime }_b$ are given by equation (11). As indicated in equation (22), the polished depth is inversely proportional to the feed rate v_a . Therefore, if removal depth has been planned before the polishing process and the parameters such as F_n, ω, σ and λ remain constant during a polishing cycle, the designed feed rate v_a can be calculated by equation (22). For the convenience of calculating the designed feed rate, the removal rate function is defined as

$$R(x\prime )=-\frac{k{F}_n\omega \sqrt{1+{\cot }^2\alpha }}{2\pi a^3{H}_v}\underset{{x\prime }_a(x\prime )}{\overset{{x\prime }_b(x\prime )}{\int }}f(t)\mathrm{dt}$$

(24)

which represents the polished depth at the unit feed rate.

Global polished profile

The interaction of the tool path and measuring path is illustrated in Figure 5. During a polishing cycle, the part surface is entirely covered by the tool paths, and the spherical polisher moves along the tool paths. And the form accuracy of the part surface is measured by a certain instrument along the measuring path. The intersections of the tool paths and the measuring path are the tool dwell locations, which are defined as the dwell points $x_i^s(i=1,2,\dots ,N_{\tau })$ . And $x_j^p(j=1,2,\dots ,N_r)$ are defined as the control points, which are the sampling points for the measurement and characterization of the surface error. The global polished profile is defined as the removal depth for all the tool paths along the measuring path in the global coordinate frame XOZ. It is determined by the superposition of the local polished profiles at all the dwell points along the measuring path.

Figure 5.

Dwell points and control points on the measuring path.

As shown in Figure 6, suppose that the current dwell point is $x_i^s$ and the material removal takes place within the contact region, then the polished depth at the control point $x_j^p$ can be expressed by

$$h(x_i^s,x_j^p)=\frac{1}{v(x_i^s)}·R_i(x_j^p-x_i^s)=R_i(x_j^p-x_i^s)·\tau (x_i^s)$$

(25)

where $\tau (x_i^s)$ is the inverse of the feed rate at $x_i^s$ , and $R_i(x_j^p-x_i^s)$ is the removal rate function that represents the polished depth at control point $x_j^p$ when the polisher passes the dwell point $x_i^s$ at the unit feed rate. If the control point $x_j^p$ lies inside the contact circle of the tool located at the dwell point $x_i^s$ , R_i assumes a value that depends on the value of $x_j^p-x_i^s$ ; otherwise, R_i is set to zero since the material removal at $x_j^p$ is not influenced by tool dwells at $x_i^s$ . Thus, the removal rate function R_i can be expressed as

$$R_i(x_j^p-x_i^s)=\{\begin{array}{cc}\hfill \frac{3k\omega F_n\sqrt{1+{\cot }^2{\alpha }_i}}{2H_V{a}_i^3}\underset{x{\prime }_a(x_j^p)}{\overset{x{\prime }_b(x_j^p)}{\int }}f(x_j^p,t)\mathrm{dt}\hfill & \hfill |x_j^p-x_i^s|\le a_i\cos {\alpha }_i\hfill \\ \multicolumn{1}{c}{0}& \hfill |x_j^p-x_i^s|>a_i\cos {\alpha }_i\hfill \end{array}$$

(26)

where α_i is the measuring angle and a_i is the contact radius at dwell point $x_i^s$ .

Figure 6.

Schematic diagram of the global polished profile.

If the polisher has visited all the dwell points on the measuring path, the actual depth of material removal at $x_j^p$ is superposed by the polished depth at each dwell point. Therefore, the total polished depth at the control point $x_j^p$ can be expressed as

$$H_j=H(x_j^p)=\sum _{i=1}^{N_{\tau }}{R}_i(x_j^p-x_i^s)·\tau (x_i^s)$$

(27)

where H_j is the polished depth at jth control point. The total polished depth along the measure path can be expressed in a vector form by

$$\mathit{H}={[\begin{array}{cccccc}\hfill H_1\hfill & \hfill H_2\hfill & \hfill \dots \hfill & \hfill H_j\hfill & \hfill \dots \hfill & \hfill H_{N_r}\hfill \end{array}]}^T$$

(28)

Finally, the global polished profile is described in the discrete form as

$$\mathit{H}={\left[\begin{array}{ccccc}\hfill R_1(x_1^p-x_1^s)\hfill & \hfill \cdots \hfill & \hfill R_i(x_1^p-x_i^s)\hfill & \hfill \cdots \hfill & \hfill R_{N_{\tau }}(x_1^p-x_{N_{\tau }}^s)\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{R_1(x_j^p-x_1^s)}& \hfill \hfill & \hfill R_i(x_j^p-x_i^s)\hfill & \hfill \hfill & \hfill R_{N_{\tau }}(x_j^p-x_{N_{\tau }}^s)\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{R_1(x_{N_r}^p-x_1^s)}& \hfill \cdots \hfill & \hfill R_i(x_{N_r}^p-x_i^s)\hfill & \hfill \cdots \hfill & \hfill R_{N_{\tau }}(x_{N_r}^p-x_{N_{\tau }}^s)\hfill \end{array}\right]}_{N_r\times N_{\tau }}{\left[\begin{array}{c}\hfill {\tau }_1\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\tau }_i}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\tau }_{N_{\tau }}}\end{array}\right]}_{N_{\tau }}$$

(29)

Equation (29) can be abbreviated as

$$\mathit{H}=\mathit{R}·\tau$$

(30)

where R is an N_r ×N_τ matrix and τ is a column vector with N_τ dimension.

For a given set of tool dwell locations, the influence matrix R can be calculated by equation (26). Thus, if the feed rates at all the dwell points have already been planned, the global polished profile can be obtained by using equation (29). However, for the planning of the deterministic polishing process to correct the surface error, the elements in H are determined through the surface measurement. Suppose D = [d ₁, d ₂, … , d_Nr ] ^T is the designed polished depth at N_r equally spaced control point along the measuring path. D is obtained from the surface measurement, which depends on the amount of material one wants to remove from the surface in the cycle of surface correction. Since the numbers of dwell points and measuring points are not equal, the influence matrix R is not square and the direct inverse of R does not exist. In order to determine the effect of feed rates at the dwell points on the surface error reduction, a cost function is defined as

$$J(\tau )=\frac{1}{2}\Vert \mathit{R}\tau -\mathit{D}{\Vert }^2$$

(31)

where ||·|| denotes the two-norm of a vector or matrix. The basic optimization problem of deterministic polishing planning can be formulated as

$$minJ(\tau )$$

(32)

Since the values in τ should be nonnegative, the method of nonnegative least square (NNLS) is used to solve the optimal problem in equation (32).

Experimental verification and discussion

The focus of this section is to validate the model of the polished profile as suggested in equation (22) and show how the shape of the polished profile varies with the measuring angle α. The polishing experiments are conducted on a five-axis polishing machine that is composed of three translational axes (X-axis, Y-axis and Z-axis) and two rotational axes (B-axis and C-axis), as shown in Figure 7(a). The piezoelectric force sensor is mounted on the end effector of the polishing spindle, which is used as a feedback sensor to control the polishing force. The polishing normal force F_n is set at 4 N in the experiments. The spherical polisher is made of polyurethane with the radius of 4 mm. The angular spindle velocity is set to 104.7 rad/s in the experiments. The flat specimens of glass ceramics are used for the polishing experiments. In order to get the mechanical properties of the glass ceramics, the nano-indentation experiment is made, in which the hardness H_v = 8300 MPa. The polishing slurry used in the experiments is a mixture of CeO₂ powder and water. The volume concentration of the polishing slurry is 0.5% and average radius of the CeO₂ particles is 0.83 µm. The polished profiles are measured by Form Talysurf PGI 1240 (Taylor Hobson Ltd, UK) along the measuring direction, which is illustrated in Figure 7(b). The radius of the measuring stylus is 0.3 mm. The measurements are conducted in the waviness mode to suppress the high-frequency noise of the profiles.

Figure 7.

(a) Structure of the polishing machine tool and (b) measuring system and polished workpiece.

Figure 8 shows the planning of the polishing experiments. Experiments are performed by feeding the polisher along a straight path across the measuring path at v_a = 0.1 mm/s. The attitude of the polishing tool is also important for the polishing experiments. By rotating the polishing spindle around the B-axis, the inclination angle σ is adjustable. As shown in Figure 8(a), in the first group of experiments, the inclination angle σ is fixed at 45°. The included angle of the tool feed direction and measurement direction varies from 90° to 30°, which corresponds to the angle α varying from 0° to 60°, respectively. The setting of the second group of experiments is illustrated in Figure 8(b). The inclination angle σ = 15°, and the measuring angle α varies from 10° to 50°. However, the declination angle λ is not constant in these experiments, which varies with different tool feed directions. As indicated in Figure 8, the declination angle λ has the same value as α since the $\mathit{Z}{\prime }_t$ -direction is parallel to the measurement direction in the experiments.

Figure 8.

Planning of the polishing experiments: (a) σ = 45° and α varies from 0° to 60°; (b) σ = 15° and α varies from 10° to 50°.

The measured data are shown in Figures 9 and 10. In these figures, the dotted lines are the experimental results and the solid lines are the simulation results calculated by equation (22). The wear coefficient k in equation (7) is obtained from trial and error by fitting the theoretical polished depths to the experimental ones. The wear coefficient k is approximately 0.00624 in the experiments. The experimental and theoretical polished profiles show good agreement in Figures 9 and 10, which validates the proposed model in this article. In Figures 9 and 10, with the increase in α, both the simulations and the experimental results show that the width of the polished profile increases, while the depth decreases slightly. As indicated in equation (22), the width of the local polished profile equals to 2acos α, thus the increase in α may increase the width of the profile. On the other hand, the declination angle λ changing with α makes the polished depth decrease with the increase in α in these experiments. The contrast between Figures 9 and 10 shows that the polishing depth decreases with the decreasing inclination angle σ. It is attributed that increasing decreasing inclination angle would enlarge the relative sliding velocity in the contact path; hence, large amount of material can be removed. Moreover, in Figure 10, the polished profiles bias away from the center of the profiles. Comparing Figure 10 with Figure 9, it can be found that the polished profile tends to show larger deviation for a smaller inclination angle σ. Decreasing the inclination angle σ would increase the deviation of the relative sliding velocity in the contact area, which results in the large deviation of the local polished profile.

Figure 9.

(a) Theoretical local polished profiles for the first group of experiments; experimental and theoretical profiles for (b) α = 0°, (c) α = 30° and (d) α = 60°.

Figure 10.

(a) Theoretical local polished profiles for the second group of experiments; experimental and theoretical profiles for (b) α = 10°, (c) α = 30° and (d) α = 50°.

Application

The model-based prediction of the local polished profile has been verified by the experiments in the last section. According to the proposed model, the local polished profile is essentially determined by process parameters, mechanical/geometric properties of part surface and polisher, polishing attitude and the measuring angle and so on. Equation (29) extends the local polished profile to the global polished profile, which is useful for the planning of the deterministic polishing process.

An ellipsoid part surface is selected to illustrate the computation of the polished profiles. A spherical polisher with radius of 4 mm is used. The mechanical properties of the polisher are the same as those in the last section. During the polishing process, the polishing attitude is set at σ = 30° and λ = 0°. The other settings of process parameters are the same as those in the last section.

The ellipsoid surface with radii of 20, 50 and 10 mm is given by

$$\left[\begin{array}{c}\hfill x\hfill \\ \multicolumn{1}{c}{y}\\ \multicolumn{1}{c}{z}\end{array}\right]=\left[\begin{array}{c}\hfill 20\cos \frac{\pi u}{2}\cos \pi v\hfill \\ \multicolumn{1}{c}{50\cos \frac{\pi u}{2}\sin \pi v}\\ \multicolumn{1}{c}{10\sin \frac{\pi u}{2}}\end{array}\right]$$

(33)

for 0 < u, v≤ 1, and all units are in millimeters. As shown in Figure 11, the tool paths are along constant values of u, which also can be obtained by intersecting the surface with the planes parallel to the XY plane. And the measuring paths are obtained by intersecting the surface with planes perpendicular to the XY plane. As the polisher feeds along the tool path across the measuring path, the measuring angle α is not constant. As shown in Figure 12(a), the measuring angle α varies from 90° to 0° along the measuring path. Moreover, the local radius of curvature of the surface is not constant along the measuring path, thus the radius of contact circle as the polisher feeds across the measuring path is different, as shown in Figure 12(b). Figure 13 shows the theoretical local polished profile along the measuring path of y = 10 mm as v_a = 1 mm/s. Due to the different measuring angles and contact radii, both the widths and depths of the local polished profiles at different locations along the measuring path are different.

Figure 11.

Tool paths and measuring paths on the ellipsoid surface.

Figure 12.

(a) Measuring angle and (b) contact radius varying along the measuring path.

Figure 13.

Local polished profile along the measuring path of y = 10 mm.

The deterministic polishing process is usually used to correct the surface form. According to the theory of Fourier transformation, the profile of the form error can be viewed as the combination of a uniform function and many sinusoidal functions with different wavelengths. ²⁵ For the ellipsoid surface finishing example, assume that the designed depth function along the measuring path (y = 10 mm) may have a mathematical form of

$$D(x)=a_1+a_2·(\sin \frac{2\pi x}{{\lambda }_1}+\sin \frac{2\pi x}{{\lambda }_2})$$

(34)

where λ ₁ and λ ₂ are the wavelengths of the sinusoidal error profile and a ₁ and a ₂ are two constants. Here, λ ₁ = 6 mm, λ ₂ = 12 mm, a ₁ = 1.6 × 10⁻³ mm and a ₂ = 1.6 × 10⁻⁴ mm. Thus, the designed removal depth distribution matrix D follows equation (34), which is shown in Figure 14(a).

Figure 14.

Polishing of an ellipsoid surface: (a) designed removal depth and actual polished depth, (b) dwell points and feed rates and (c) residual error.

In the computation, the interval of dwell point is 0.4 mm and the interval of control point is 0.1 mm. Since the contact radius and measuring direction have been calculated as shown in Figure 12, the local removal function R(x) in equation (26) can be calculated and the influential matrix of material removal R in equation (29) is obtained. As the matrix R is not square, there is no direct inverse matrix for R . As indicated by equation (31), since the influential matrix R and the removal depth distribution matrix D are known, the optimal solution of vector τ can be obtained by minimizing the residual error. The vector τ should not be negative [τ_i ≥ 0, i = 1, 2, …, N_τ ], thus the algorithm of NNLS is used to pursue an optimal solution of τ . The computation is carried out using the optimization toolbox of MATLAB^®. Based on the proposed computation, the final polished depth is shown in Figure 14(a), the dwell point distributions and corresponding feed rates are shown in Figure 14(b) and the residual errors at the control points are given in Figure 14(c). According to the simulation results, the final polished depth is close to the designed removal depth and the residual error is acceptable compared to the original designed removal depth.

Conclusion

The models of the local and the global polished profiles for deterministic polishing process are presented in this article. The local polished profile is modeled by integrating the index of material removal along the tool path at each contact patch between spherical polisher and part surface. The algebraic expression of global polished profile is derived by convoluting the local polished depth at each dwell point of polishing process. According to the proposed model, the polished profile is determined by the process parameters (normal polishing force, angular spindle velocity and feed rate), mechanical properties of workpiece (workpiece hardness, Young’s modulus and Poisson’s ratio), geometric properties of part surface and polisher (local principle radius of the surface, radius of polisher, etc.), polishing attitude (inclination angle and declination angle) and the measuring angle.

In particular, this article focuses on the effects of the measuring angle and the polishing attitude upon the polished profile. The polishing experiments are conducted by varying the measuring angle. The experimental results agree well with the ones obtained by theoretical calculation, which confirms the validity of the proposed model. According to the experimental and theoretical results, the width of the local polished profile increases with the increasing measuring angle, and the depth is affected by the inclination angle and declination angle. Based on the derivation of the global polished profile, the application of the proposed model is illustrated to remove a specified amount of material from an ellipsoid surface along the measuring path. The feed rates along the tool paths at every dwell point are obtained by solving an optimization problem with nonnegative constraint. The NNLS method is used to determine the feed rate distribution of the tool. The result shows that the method is potentially useful for the process planning for the deterministic polishing process to correct the form error of the part surface.