# A \"Double Accuracy Theory\" and Experimental Research on Precision Grinding
## 2 Theoretical Analysis of Grinding Machine Accuracy  
Before analyzing grinding machine accuracy, multi-body system theory must be introduced for analysis [16]. The grinding machine itself is a multi-body mechanical system. Multi-body system theory is the basic theoretical basis for studying multi-body systems. It has good systematics and versatility and has been effectively applied in motion analysis of complex mechanical systems such as robots, coordinate measuring machines, machine tools, etc. Using multi-body system theory to analyze the errors of machine tool process system can not only meet the needs of machine tool error research, but also continuously expand the application field of multi-body system theory [17]. The topology description methods of multi-body systems mainly include direct path method, correlation matrix method and low-order body array method [18]. Compared with other methods, the low-order volume array method has simpler expression and higher computational efficiency and is currently a commonly used description method [19]. The type of grinding machine to be analyzed in this paper is Kellenbergaer universal high accuracy grinding machine from Switzerland, and its grinding accuracy is 0.1 \(\upmu\)m. In the later part of the article, this grinding machine is also used for processing. The multi-body system diagram of the grinding machine after labeling is shown in Figure 1.  
According to Figure 1 and the structure of Kellenbergaer universal high-precision grinding machine, it is divided into four moving bodies. These four moving bodies are respectively represented: grinding machine bed, C-axis and guide rail components, B-axis and grinding wheel rotating components. At that same time, as can be seen from Figure 1, each object has at least one adjacent low order body. If B0 is the lower order number object of B1, let Bj be any typical body in the multi-body system, and define Bi as the \(n\)-order lower order body of Bj, then there are:  
\[L^{n}(j)=i;\;L^{0}(j)=j;\;L^{n}(0)=j \tag{1}\]  
When body Bj and body Bi are adjacent low order bodies, there is:  
\[L(j)=i \tag{2}\]  
In the meantime, the \(N\)-order low-order matrix column of Bj defined according to Figure 1 is shown in Table 1.  
According to Table 1, the low-order body array of the multi-body system topology can connect the individuals in the multi-body system. For any object in the system, it can be traced back to the base coordinate system through the array, so it is widely used in practice. Meanwhile, the Kellenbergaer grinding machine is decomposed into motion coordinate system, and its purpose is to obtain the motion mathematical model of the grinding machine, so as to analyze the geometric error of the grinding machine and carry out comprehensive modeling. The grinding machine motion coordinate decomposition is shown in Figure 2.  
According to Figure 2, the Kellenbergaer grinding machine can be equipped with three grinding wheels on the B axis in the figure. One of the grinding wheels is driven by a motorized spindle with a maximum rotation speed of 60,000 \(r\)/min. The B axis can rotate 360 degrees and move back and forth; the C axis is the workpiece axis; and the C axis can move left and right along with the guide rail. According to Figure 2, coordinate system \(\sum O_{00};\sum O_{10};\sum O_{20};\sum O_{30};\sum O_{40}\) is the coordinate system of grinding machine, guide rail movement, C-axis rotation, B-axis movement, and B-axis \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**j** & **1** & **2** & **3** & **5** \\ \hline L\({}^{0}\)(j) & 1 & 2 & 3 & 4 \\ L\({}^{1}\)(j) & 0 & 1 & 1 & 3 \\ L\({}^{2}\)(j) & 0 & 0 & 0 & 1 \\ L\({}^{3}\)(j) & 0 & 0 & 0 & 0 \\ L\({}^{4}\)(j) & 0 & 0 & 0 & 0 \\ L\({}^{5}\)(j) & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grinding machine multi-body system low sequence body array. Figure 1: Multi-body system diagram of grinding machine. rotation, respectively.