Research on the dynamic tracking error compensation method of the linear axis of an ultraprecision lathe based on a piezo nanopositioning platform

Abstract

Single-point diamond turning technology has been widely used in processing microstructures. The accuracy of ultraprecision cutting machine tools affects the performance of the microstructure. By improving the structure of the machine tool itself, the machining accuracy of ultraprecision diamond lathes has almost become optimized. Therefore, this research identifies and analyzes the linear axis tracking error through the dynamic modeling of a macro/micro cutting system. Furthermore, the impact of the tracking error on machining accuracy is obtained. Based on the macro/micro cutting system, a servo tracking error compensation method is proposed, and the effectiveness of this error compensation strategy is verified by simulation. The proposed experimental approach includes cutting experiments of tracking error compensation for a hyperbolic sine wave surface structure, verifying the surface profile accuracy of the workpiece with and without tracking error compensation. Additionally, this study proposes a profile evaluation method for microstructure. Experimental results show that the proposed tracking error compensation strategy effectively reduces the tracking error of ultraprecision cutting machine tools. Additionally, the proposed approach significantly improves the microstructure machining profile accuracy and can be used for ultraprecision lathes with high precision.

Appendix 1

The profile evaluation method of microstructures is shown as follows:

• Topography measurement: The hyperbolic sine microstructure workpiece П (x, y) is measured and leveled using a white-light interferometer. x and y are the two-dimensional coordinates.

• Rotation angle removal between the theoretical topography and the measured topography: Determining the rotation angle between the theoretical topography and the measured topography. An orthogonal transformation is used to derive the theoretical topography formula after rotation. Thus, the rotation angle between the theoretical topography and the measured topography is matched. The primary expression of the ideal topography is [28, 29]:

  $$z\left(u,v\right)=A\mathrm{sin}\left(2\pi \cdot u/\lambda \right)+B\mathrm{cos}\left(2\pi \cdot v/\lambda \right)$$

  (17)

where A and B are the peak values of the waveform, and λ is the period length.

The gravity method shown in Eq. (18) is used to calculate the peak or valley position (x_i, y_i) in each periodic subregion D_i of the measured topography П (x, y). Then, fitting the line for the corresponding group, the rotation angle θ between the measured topography and the theoretical topography is obtained, as shown in Fig. 

Fig. 25

Diagram of the rotation angle calculation

25.

$${x}_{i}=\frac{\sum \limits_{\left(x,y\right)\epsilon {D}_{i}}x\prod \left(x,y\right)}{\sum \limits_{\left(x,y\right)\epsilon {D}_{i}}\prod \left(x,y\right)},{y}_{i}=\frac{\sum \limits_{\left(x,y\right)\epsilon {D}_{i}}y\prod \left(x,y\right)}{\sum \limits_{\left(x,y\right)\epsilon {D}_{i}}\prod \left(x,y\right)}$$

(18)

Next, the rotation transformation shown in Eq. (19) is performed:

$$\begin{bmatrix}x\\y\\\prod \nolimits^{'}\end{bmatrix}=\begin{bmatrix}\cos\;\theta&\sin\;\theta&0\\-\sin\;\theta&\cos\;\theta&0\\0&0&1\end{bmatrix}\begin{bmatrix}u\\v\\z\end{bmatrix}$$

(19)

The ideal topography after rotation is obtained as follows:

$$\begin{array}{c}\prod\nolimits^{'}\left(x,\;y\right)=z\left(u,\;v\right)\\=z\left(x\;\cos\;\theta-y\;\sin\;\theta,\;x\;\sin\;\theta+y\;\cos\;\theta\right)\\=Asin\left[2\pi\left(x\;\cos\;\theta-y\;\sin\;\theta\right)/\lambda\right]+B\;cos\left[2\pi f\left(x\;\sin\;\theta+y\;\cos\;\theta\right)/\lambda\right].\end{array}$$

(20)

• Topography alignment: The two-dimensional cross-correlation method shown in Eq. (21) is used to align the rotated theoretical topography П'(x, y) with the measured topography П (x, y)

  $$\mathrm{corr}\left(p,q\right)=\prod \nolimits^{'}\left(x,y\right)\otimes\prod\left(x,y\right)$$

  (21)

  . where $\otimes$ is the cross-correlation symbol and (p, q) is the correlation displacement.

The peak of Eq. (21) is (p₀, q₀). Therefore, the translation of the ideal topography relative to the measured topography is determined as follows:

$$\Delta x={p}_{0}-m,\Delta y={q}_{0}-n$$

(22)

where m and n are the numbers of pixels in the horizontal and vertical directions of the image, respectively. The ideal and measured topography after translation can be characterized under the same field of view.

Figure 

Fig. 26

a Topography processing results before error compensation. b Topography processing results after error compensation

26(a) and (b) show the topography processing results before and after error compensation, respectively. The ideal topography and the measured topography are well aligned after matching the rotation angle and translation to analyze a direct comparison. The above method does not change the measurement results of each point on the surface and can avoid errors introduced by discrete point interpolation.