2.1.

Principle of Online Monitoring

The continuous polishing machine consists of a massive rotary table with an annular polishing pad. On one side is a large circular conditioning plate, which is substantially wider than the annulus. On the remaining portions of the lap are two to three rings whose inside diameters match the annulus width. A phenolic septum within these rings holds the workpieces, as shown in Fig. 1.

The surface shape of the polishing pad can be adjusted by moving the conditioning plate radially outward or inward. If the conditioning plate is moved radially outward, the resulting increased overhang will cause an increase in the moment, leading to an increase in the linear pressure gradient. This pressure increase at the outer radii will cause the lap to become more convex. An inward movement will similarly result in a moment that tends to make the lap become concave.¹⁰ When the center of the conditioning plate is located near the central region of the belt, the entire surface of the polishing pad will be removed approximately uniformly, and the surface shape of the polishing pad will be unchanged; this state of the system is the equilibrium state. When the workpiece is polished under the equilibrium state, the surface shape of the polishing pad is insensitive to the initial surface shape of the workpiece. If the polishing pad is damaged by the workpiece, it can be completely repaired by the conditioning plate so that the surface shape of the workpiece can stably converge into an ideal shape. This is the ideal working state of continuous polishing.^6,10 To achieve the equilibrium state, reliable online surface shape monitoring is absolutely necessary. Direct surface measurement during operation is not technically feasible due to the presence of slurry on the surface and the relative motion of the pad and workpiece. Therefore, the surface shape of the polishing pad should be monitored indirectly through the measurement of a process variable that is related to the surface shape of the polishing pad and can be measured directly. According to the inherent characteristic of the continuous polishing technology, any two parts in uniform contact at all locations and having angular orientations with respect to each other must be spherical, and the curvature may be positive, negative, or zero.^10,11 It can be considered that the three surfaces of the conditioning plate, polishing pad, and workpiece are spherical with the same radius of curvature.¹² Here, the workpiece is considered as a monitoring element, and the polishing system can be simplified to the model shown in Fig. 2.

The symbols are defined as follows:

The positive and negative angles correspond to the convex and concave surface shapes of the polishing pad. Based on the geometric relations of mathematics, the relationship between $\mathrm{\Delta }h$ and $\alpha$ can be given as

Considering that $\mathrm{\Delta }h\ll R$ and $\sin \alpha \approx \alpha$, the equation can be expressed as

It can be seen from Eq. (2) that if the horizontal tilt angle of the conditioning plate can be accurately measured online, the surface of the bottom surface of the monitoring element can be detected online; thus, the surface shape of the polishing pad can be detected indirectly.

The polishing pad deforms with the weight of the conditioning plate. Owing to the variables between the inside and outside of the polishing pad, the conditioning plate tilts, and the tilting angles vary with the position of the conditioning plate. Therefore, a quantitative analysis of the variables of the pad is necessary to figure out the effects of the angles caused by the deformation of the asphalt. ANSYS is an important tool for performing structural static analysis. With computer numerical calculation and image display, ANSYS can conduct numerical simulations on a system containing relevant physical phenomena, such as stress and deformation. The parameters used in the analysis are shown in Table 1.

Table 1

Characteristic parameters of the experiment device.

The systematic geometric model and the contact element generated by meshing are shown in Fig. 3(a). The calculations after applying the loads and constraints are shown in Fig. 3(b). The calculated results indicate that the deformation at the outer and inner edges of the polishing pad is larger, decreasing from the outside and the inside of the pad to the center of the belt, and the deformation distribution is relatively uniform at the center of the belt. The reason is that the diameter of the conditioning plate is larger than the width of the polishing belt, the inner and outer protruding portions cause an increase in the moment, resulting in an increase in the pressure at the edges. The length is taken as 5 mm, and an analysis of the distributions of the variables on the surface of the polishing pad caused by the differences in the positions of the conditioning plate is conducted. The variables of the tilting angles of the conditioning plate are shown in Fig. 3(c).

The results show that the maximum tilting angle is $<1.7\times {10}^{-3}\mathrm{arc}\sec$. According to Eq. (2), it can be concluded that, for a workpiece with a caliber of 100 mm, the error caused by the deformation of the asphalt is $<7.2\times {10}^{-5}\lambda$, and the error can be ignored.

A schematic diagram of the experimental device is shown in Fig. 4. A reflector with a diameter of 100 mm ($\mathrm{PV}<0.1\lambda$) and having a two-dimensional (2-D) adjustment frame was fixed at the center of the conditioning plate. A high-precision goniometer, connected with the light source through a light guide fiber, was fixed above the reflector using a mechanical support. When there is a slight change in the angle at the reflector, the reflected light also produces a difference angle, and the image spot on the photosensitive surface of the charge coupled device detector generates an offset. Thus, the real-time variation in the inclination angle of the reflector can be measured and recorded in real time by using the computer analysis software.

The goniometer used in the experiment was independently developed by our researchers. The picture and main performance parameters are shown in Fig. 5 and Table 2, respectively. When the conditioning plate is located near the center of the polishing annulus ($e=0.2275\mathrm{m}$ and the diameter of the monitor element $D=100\mathrm{mm}$), the theoretical calculation shows that the accuracy of PV value $\mathrm{\Delta }h$ can be better than $0.1\lambda$.

Table 2

Performance parameters of high-precision goniometer.

The goniometer can only measure the relative change in the angle instead of the absolute value of the angle. The angle measured by the goniometer does not truly reflect the surface shape of the polishing pad; therefore, the system needs to be calibrated before use. In this study, the Zygo interferometer is used as a standard equipment to measure the actual surface shape of the monitoring element. The goniometer measures the corresponding angle value to obtain the calibration curve, which can be used as the basis of measurement. The angle measured by the goniometer can reflect the surface of the polishing pad accurately after calibration.

2.2.

Adjustment Rules of Surface Shape and Method of Automatic Control

The automatic control of the surface shape of the polishing pad includes the adjustment of the surface shape and the long-term maintenance of the ideal surface shape. The adjustment of the surface shape in continuous polishing is achieved by moving the conditioning plate. The surface figure of the pad is determined by the material removal distribution; the material removal in continuous polishing has been historically described by the widely used Preston equation ¹³

where $H$ is the change in thickness value of the polishing pad at a given point in a cycle, $C$ is a proportionality constant, $P$ is the pressure over the contact area, $V$ is the relative velocity, $t$ is the conditioning time, $S$ is the relative sliding distance on the polishing pad, and $T_n\sim T_{n+1}$ is the cycle, respectively. The relative sliding distance can be calculated by using geometric relations. The Winkler’s hypothesis can be used to determine the contact pressure, owing to the small thickness and the grooves in the surface of the lap, it can be written as follows:¹⁴ where $P$ is the contact pressure, $Z$ is the deformation in the thickness of the lap, and $k$ is the elastic coefficient, which can be expressed as the ratio of the Young modulus to the thickness of the lap.

Taking into account the force balance and torque balance of the conditioning plate and the workpiece, the distribution of contact pressure can be calculated according to Eq. (4); then, the material removal in the polishing lap surface can then be established. To simplify the calculation, the contact pressure is assumed to be remain constant in a cycle, and the surface shape is modified at the end of the cycle. The contact pressure is then updated according to Eq. (4). The amount of the material removed in each cycle is calculated by repeating the Eqs. (3) and (4).

Figure 6 shows the simulation results (using computer) based on the above model. The horizontal axis represents the distance between the center of the conditioning plate and the polishing lap, whereas the vertical axis represents the change speed of the surface shape of the polishing pad (with a change to a concave surface when the axis is positive and a change to convex surface when it is negative). According to the calculation results, the basic rules of the surface adjustment of the disk surface are obtained as follows. (1) If the conditioning plate is not moved, the surface PVs of the polishing lap will change at a constant speed. (2) An equilibrium state exists in the system, indicating that the surface shape could remain unchanged, and the surface shape could be flat or spherical with a small curvature.¹⁵ In the equilibrium state, the position corresponding to the conditioning plate is called the equilibrium position of the system, that is, the surface change speed of the polishing pad in the figure is zero. However, it is worth noting that the equilibrium position of the system is different under different working conditions, as can be seen in Fig. 6 ($Rw$ in the figure denotes the radius of workpiece). (3) The characteristic curve has the same shape under different conditions; however, a certain shift occurs in the direction of the vertical axis. Therefore, only the relationship between the change rate of polishing and the eccentricity in a working condition needs to be measured experimentally and saved into the computer. When the working condition changes, the surface change rate of the polishing pad can be measured using the online monitoring system under the new working condition. The new characteristic curve under the new working condition can be deduced and the new equilibrium position of the system under the new conditions can also be established.

A computer is used to control the surface of the polishing pad according to the program flowchart (shown in Fig. 7), which is based on the above rules. Here, $\alpha (t)$ denotes the horizontal inclination angle of the conditioning plate, $\beta$ denotes the allowable error, $e$ denotes the eccentricity of the conditioning plate, $e_1$ and $e_2$ represent the innermost and outermost positions of the conditioning plate on the radially adjustable range of the polishing pad, and $e_{eq}$ represents the equilibrium position of the system, respectively. When the system operating parameters of the system are set correctly, the horizontal inclination of the calibration plate is detected in real time. If $|\alpha (t)|$ is greater than the allowable error $\beta$, and the angle changes evenly over time, it means that the surface of the polishing pad, conditioning plate, and the workpiece has been matched with each other. However, the surface figure does not meet the requirement of processing accuracy yet; therefore, we need to calculate the rate of change of the inclination angle and the change rate of the surface shape under such conditions. The equilibrium position $e_{\mathrm{eq}}$ of the system can be obtained according to the basic characteristics of the polishing system. When the angle is equal to zero, it indicates that the surface shape of the polishing pad has attained a good value; the conditioning plate should be pushed to equilibrium position of the system to keep the ideal shape for a long time. Otherwise, the conditioning plate would be pushed to the outermost or innermost side in the radial direction area of the adjustable range by the automatic actuator, depending on whether the surface shape is concave or convex.