Measurement of Contact Pressure Distribution Map at Workpiece/tin Lap Interface and Process Parameters Optimization During Full-Aperture Polishing With Tin Lap

The polishing experiments were conducted on the KPJ-1200 full-aperture polishing machine tool (Beijing Weina Precision Machinery Co., Ltd, China), as schematically shown in Fig. 1. The granite platen was covered with a tin lap (φ1200 mm $\times$ 12 mm in thickness, Hyprez, Engis, USA). The workpiece (430 mm $\times$ 430 mm $\times$ 40 mm, fused silica) to be polished was clamped by a special designed fixture mounted on the work shaft. Before polishing, the lap's surface shape error was measured with the method proposed by Liao et al. and conditioned with a diamond turning tool fixed on the conditioning shaft to improve the surface shape error [4]. The translation of the work shaft and conditioning shaft along the gantry was driven by a servo motor with ball screw. Additionally, radial and circumferential grooves (2°, 2 mm in width, 50 mm spacing) were cut onto the tin lap surface with the diamond turning tool. During polishing, the platen and the workpiece was driven by servo motors to rotate anticlockwise. Simultaneously the abrasive slurry (CeO₂, ∼0.5 μm, universal photonics, New York, America; flow rate: 1.6 l min⁻¹) was injected onto tin lap surface and transported to polishing site by the slurry channels. The temperature and the humidity at the ambient were 21.3 °C and 65.2% RH respectively. Subsequently, the workpiece's surface figure was measured on the laser interferometer (32'', Zygo, USA) at 21.7 °C and 37.2% RH.

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After the tin lap was conditioned by the diamond turning tool, a novel measurement system was developed to measure the contact pressure P(x_p, y_p) at the workpiece/tin lap interface under given polishing conditions. This system used a pressure transducer (piezoelectric, resolution: 1.3 Pa, bonded with a small fused silica block as detection head, φ15 mm $\times$ 1 mm in thickness) fixed into the granite transducer fixture (120 mm $\times$ 120 mm $\times$ 31.4 mm, a hole 25 mm in diameter $\times$ 31.4 mm in thickness), as shown in Fig. 2. It's worthwhile noticing that the thickness of the granite transducer fixture was designed to produce an equal theoretical contact pressure at its bottom surface to that at workpiece. Besides, the bottom surface of the granite transducer fixture was pre-polished to a peak to valley (PV) < 0.5 μm with polyurethane polishing pad. Therefore, the measured contact pressure by the pressure transducer can be regarded as the interface contact pressure P(x_p, y_p) when the workpiece was contacting with the same area on the tin lap. The ball spring screws mounted on the limit stop were used to frictionlessly support the side face of the granite transducer fixture to avoid overturning moment. And the limit stop was fixed into the conditioning shaft by the fastening screw.

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A Cartesian coordinate system is fixed on the tin lap surface, where the origin O is set at the geometric center of tin lap surface. The z axis is oriented from the origin and vertical to the tin lap surface. Further, the x axis is oriented from the origin and parallel to the gantry's side surface. Then, the y axis is oriented from the origin and the direction is determined by right-hand rule. During measurement, the tin lap was anticlockwise rotating around its geometric center O with constant angular velocity w₁ (π/3) and the granite transducer fixture was sinusoidally reciprocating along the radial direction of the tin lap. Hence, the instantaneous coordinate (x_p, y_p) of the measured point in the tin lap coordinate system can be expressed as:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{x}_{p}=\left(r-l-vt-A\,\sin \left({w}_{i}t\right)\right)\cos \left(\alpha -{w}_{1}t\right)\\ {y}_{p}=-\left(r-l-vt-A\,\sin \left({w}_{i}t\right)\right)\sin \left(\alpha -{w}_{1}t\right)\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where the r is the radius of the tin lap (600 mm), l is the distance between the center of the pressure transducer and the outer edge, v is translation speed of the granite transducer fixture (2 mm s⁻¹), A is the amplitude of sinusoidal reciprocating motion (5 mm), w_i represents the angular velocity of the ith measurement, which is:

$$\begin{eqnarray}{w}_{i}={\left(-1\right)}^{i}\displaystyle \frac{{w}_{1}}{{\rm{quotient}}\left({\rm{i}}+1,2\right)}\begin{array}{c}\left(i\in \left[1,8\right]\&\,i\in {N}^{* }\right)\end{array}\end{eqnarray} \tag{ 3 }$$

and α is the initial angle between the starting location of the ith measurement and the x axis in the tin lap coordinate system.

During its measurement, the detection head of the pressure transducer was always contacting with the asperities on the tin lap surface and acquiring the contact pressure P(x_p, y_p) at the local contact area. Synchronously, the measured data of contact pressure P(x_p, y_p) was wirelessly transferred to PC by LAN. Next, the acquired data was filtered to eliminate abnormal value and averaged at the same location in the tin lap coordinate system. Finally, the contact pressure distribution map was solved by interpolation algorithm.

As shown in Fig. 3, a Cartesian coordinate system is fixed on the workpiece polishing surface, of which the origin O_w is set at the geometric center of the workpiece polishing surface. The x_w (y_w) axis is oriented from the origin O_w and parallel to the x (y) axis in tin lap coordinate system described in Fig. 2, respectively. Assumed there is any arbitrary point P₁ on the workpiece, where the initial polar angle and radius in the workpiece coordinate system are β and r₀. The distance between the O and O_w is e. During the polishing, the tin lap and workpiece is continuously rotating around O and O_w with constant angular velocity w₁ and w₂, and the workpiece is reciprocating concurrently along the radial direction of tin lap in sinusoidal mode. The instantaneous coordinate (x, y) of specific point P₁ in tin lap coordinate system can be formulated as:

$$\begin{eqnarray}&&\left\{\begin{array}{l}x=(e+A\,\sin (wt+{\varphi }_{0}))\cos \,({w}_{1}t)+{r}_{0}\,\cos (\beta +{w}_{2}t-{w}_{1}t)\\ y=-(e+A\,\sin (wt+{\varphi }_{0}))\sin \,({w}_{1}t)+{r}_{0}\,\sin (\beta +{w}_{2}t-{w}_{1}t)\end{array}\right.\end{eqnarray} \tag{ 4 }$$

where w and φ₀ represent the angular velocity and initial phase angle of the sinusoidal reciprocating motion.

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The total material removal amount h(x, y) of the point P₁ within the polishing time T can be obtained from Eq. 1:

$$\begin{eqnarray}&&h(x,y)=k\displaystyle {\int }_{0}^{T}p(x,y,t)v(x,y,t)dt\end{eqnarray} \tag{ 5 }$$

When the polishing time T is discretized with a miniature interval (Δt), the Eq. 5 can be written as:

$$\begin{eqnarray}&&h(x,y)=k\displaystyle \sum _{i=1}^{N}p(x({t}_{i}),y({t}_{i}),{t}_{i})v(x({t}_{i}),y({t}_{i}),{t}_{i}){\rm{\Delta }}t\end{eqnarray} \tag{ 6 }$$

And

$$\begin{eqnarray}&&v\left(x({t}_{i}),y({t}_{i}),{t}_{i}\right){\rm{\Delta }}t={\rm{\Delta }}s\left(x({t}_{i}),y({t}_{i}),{t}_{i}\right)\end{eqnarray} \tag{ 7 }$$

If the discretized time interval is minimum enough, the arc length Δs(x_w, y_w, t_i) can be derived as:

$$\begin{eqnarray}&&{\rm{\Delta }}s(x({t}_{i}),y({t}_{i}),{t}_{i})=\sqrt{{(x^{\prime} ({t}_{i}))}^{2}+{(y^{\prime} ({t}_{i}))}^{2}}{\rm{\Delta }}t\end{eqnarray} \tag{ 8 }$$

where x'(t_i) and y'(t_i) are the first derivative of the instantaneous coordinate (x(t_i), y(t_i)) of specific point P₁ in tin lap coordinate system. In terms of instantaneous pressure p(x(t_i), y(t_i), t_i), the polishing time t_i is substituted into Eq. 4 to solve the instantaneous coordinate (x(t_i), y(t_i)) of P₁ in tin lap coordinate system. Then, the instantaneous contact pressure p(x(t_i), y(t_i), t_i) can be sought out according to the measured contact pressure distribution map . The material removal coefficient k was assumed to be an constant, which was calculated by averaging the values of five samples (fused silica, 100 mm in diameter and 10 mm in thickness) with Eq. 1 under given polishing conditions. Consequently, the material removal amount h(x, y) of given point within polishing time T can be obtained.

To improve the convergence efficiency and accuracy of workpiece's surface figure, the process parameters are optimized with an artificial intelligence algorithm.

Genetic algorithms (GA)

GA is inspired by the process of natural evolution and commonly used to search optimal solutions for multiple influencing factors problem. Figure 4 depicts the flow chart of operation processes for GA. Firstly, the random individuals v₁,v₂,...,v_n are coded into chromosomes composing a specific population g(i), and these individuals must satisfy the given constraints. Then, the fitness fit(v_i), which is calculated by the defined fitness function F(v_i), of each chromosome in the specific population g(i) is used to evaluate each individual:

$$\begin{eqnarray}&&fit({v}_{i})=\displaystyle \frac{F({v}_{i})}{\displaystyle {\sum }_{i=1}^{N}F({v}_{i})}\end{eqnarray} \tag{ 9 }$$

Next, a new population g(i+1) is generated by bio-inspired operators such as selection, crossover and mutation to avoid falling into local optimal solution.¹⁹ Finally, if there is any chromosome g(i_o, j_o) approaching the optimization target or reaching maximum iteration times i_max, the chromosome g(i_o, j_o) is selected as the optimum individual. Or the new population g(i+1) is used for the next iteration until meeting any cycle termination conditions.

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Multi-objective optimization based on GA

GA optimization was adopted to search optimal combination of process parameters in this research. The GA started with a population of random individuals in the process parameters of e, A, w, w₁, w₂, φ₀ and θ₀, which is the angle between the x_w axis in the workpiece coordinate system and x axis in tin lap coordinate system. The problem of this optimization is how to minimize the PV of workpiece's surface figure and maximum the convergence efficiency. Accordingly, the optimization objective function was:

$$\begin{eqnarray}&&{O}_{1}=\,\min \left\{{f}_{{\rm{w}}}(e,A,w,{w}_{1},{w}_{2},{\phi }_{0},{\theta }_{0})\right\}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{O}_{2}=\,\max \left\{{f}_{{\rm{c}}}(e,A,w,{w}_{1},{w}_{2},{\phi }_{0},{\theta }_{0})\right\}\end{eqnarray} \tag{ 11 }$$

where the O₁ and O₂ are the minimum and maximum values of the workpiece's surface figure and convergence efficiency.

In this research, the multi-objective optimization problem is simplified to a single objective optimization problem by applying the weight sum method. In the practical full-aperture polishing process, the surface figure is more important than convergence efficiency.so the weighting factors of the surface figure and convergence efficiency are 3/4 and 1/4, respectively. The objective function of the simplified single objective optimization problem is given by:

$$\begin{eqnarray}&&U\left(e,A,w,{w}_{1},{w}_{2},{\varphi }_{0},{\theta }_{0}\right)=\displaystyle \frac{3}{4}{O}_{1}-\displaystyle \frac{1}{4}{O}_{2}\end{eqnarray} \tag{ 12 }$$

and the value scopes of the e, A, w, w₁, w₂, φ₀ and θ₀ are listed in Table I.

Table I.  Value scopes of process parameters.

Variables	Scope
e (mm)	[−350, −250]
A (mm)	[−40, 40]
w (rad s−1)	[π/60, π/10]
w1 (rad s−1)	[π/15, π]
w2 (rad s−1)	[π/6, 2π/3]
φ0 (rad)	[−π, π]
θ0 (rad)	[−π, π]

Before polishing, the surface figure of the workpiece to be polished was measured on the laser interferometer. Taking the lowest point p_w as the reference point, the expected material removal amount m(x, y) at any point on the workpiece is equal to the height relative to the point p_w. Theoretical value of the material removal amount h(x, y) at the same point can be calculated by the proposed model . As a result, the O₁ is:

$$\begin{eqnarray}&&{O}_{1}=\,{\rm{\min }}\left\{{\rm{\max }}\left[m\left(x,y\right)-h\left(x,y\right)\right]-\,{\rm{\min }}\left[m\left(x,y\right)-h\left(x,y\right)\right]\right\}\end{eqnarray} \tag{ 13 }$$

and O₂ is:

$$\begin{eqnarray}&&{O}_{2}=\,\max \left\{\displaystyle \frac{\left[\max \left(m\left(x,y\right)\right)-\,\min \left(m\left(x,y\right)\right)\right]-\left\{\max \left[m\left(x,y\right)-h\left(x,y\right)\right]-\,\min \left[m\left(x,y\right)-h\left(x,y\right)\right]\right\}}{T}\right\}\end{eqnarray} \tag{ 14 }$$

The optimization of process parameters using GA is implemented on the MATLAB 2014a with programmed codes, where the setting conditions are as follows: 100 generations, population size of 180 with roulette selection function, scattered crossover probability of 0.7 and Gaussian mutation probability of 0.02.