###### Abstract
[16] established a five-degree-of-freedom dynamic model of the aerostatic spindle and studied the spindle vibration under the action of the cutting force and its influence on the generation of ultra-precision diamond turning surfaces. In summary, there are relatively few studies on the relationship between aerostatic guideways and machining topography errors. In this study, a diamond micro-planing experiment was carried out on a self-developed three-axis machine tool. The BRUKER white-light interferometer was used to measure the surface of the workpiece, and the spatial power spectral density distribution of the surface error was analyzed. Combined with the space and time domain errors in the vertical direction of the machine tool, the contour error was analyzed and identified. We found that the low-frequency vibration characteristic of the vacuum-preloaded aerostatic bearing is the main factor that causes planing surface contour errors. Dynamic Characteristics Analysis and Vibration Characteristics of Machine Tool in the Vertical Direction  
The three-axis machine tool includes three linear axes in total, including the \(X\)-axis and \(Y\)-axis, which move horizontally, and the \(Z\)-axis, which moves vertically. The three linear axes all are porous aerostatic guideways. The basic arrangement of the \(X\)-axis and \(Y\)-axis is shown in Figure 1. The bottom pad of the \(X\)-axis is a single block vacuum-preloaded (VPL) porous aerostatic bearing; its bearing surface is directly supported on a precision marble plane, and two symmetrically arranged porous aerostatic bearings support the bottom surface of the \(Y\)-axis. The pressure bearing is on the same precision marble plane. The left and right bearings of the \(X\)-axis are preloaded with each other. Linear motors drive both the \(X\)-axis and the \(Y\)-axis, and there are no other intermediate force transmission links. The aerostatic guideway moves under the suspension support of the high-pressure air film, and the bottom VPL porous aerostatic bearing only supports the \(X\)-axis slide in the vertical direction. Now, the \(X\)-axis slide is considered a rigid body, and the dynamic characteristics of the slide in the vertical direction can be studied. In order to facilitate the analysis and calculation, the vertical dynamic model of the \(X\)-axis slide was simplified to a two-degree-of-freedom system composed of rigid rods, springs, and dampers, as shown in Figure 2. Then, the kinetic equation matrix can be expressed as:  
\[ \begin{bmatrix}m&0\\ 0&I\end{bmatrix}\begin{bmatrix}\ddot{x}_{c}\\ \ddot{\theta}_{c}\end{bmatrix} + \begin{bmatrix}c_{A}+c_{B}&c_{B}{}_{B}{}_{B}{}_ {c}-c_{A}l_{Ac}\\ c_{B}{}_{Bc}-c_{A}l_{Ac}&c_{B}{}_{B}{}_{Bc}{}^{2}+c_{A}l_{Ac}\end{bmatrix} \begin{bmatrix}\dot{x}_{c}\\ \dot{\theta}_{c}\end{bmatrix} + \begin{bmatrix}k_{A}+k_{B}&k_{B}{}_{B}{}_{Bc}-k_{ A}l_{Ac}\\ k_{B}{}_{Bc}{}^{2}+k_{A}l_{Ac}\end{bmatrix}\begin{bmatrix}x_{c}\\ \theta_{c}\end{bmatrix} = \begin{bmatrix}F_{c}\\ M_{c}\end{bmatrix}  \tag{1}\]  
In the figure, A and B are points A and B of the simplified porous aerostatic bearing back force point; C is the center of mass of the slide; \(k_{A}\) and \(c_{A}\) are the equivalent stiffness and damping of the porous aerostatic bearing at point A; \(k_{B}\) and \(c_{B}\) are the equivalent stiffness and damping of the porous aerostatic bearing at point B; \(F_{c}\) and \(M_{c}\) are the interference force and the interference moment reduced to the center of mass; and \(x_{c}\) and \(\theta_{c}\) are the vibration displacement and rotation angle of the slide at the cutting point. Figure 1: The basic arrangement of the \(X\)-axis and \(Y\)-axis of a three-axis machine tool. Figure 2: Dynamic model of vertical direction of \(X\)-axis slide. Since the damping of the air film was small, and the processing method used in this study was one-way planing processing, the dynamic equation was approximated and simplified.