## 2 Features of the Machine Tool Taken into Consideration in Volumetric Verification
### Structural Configuration and Kinematic Model of the Machine Tool
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Definition** & **Symbol** & **Definition** \\ \hline CSO & Coordinate system of the machine tool (MT) & \(\overline{X}\) & Linear error vector in the \(X\)-axis \\ CSY & Coordinate system of the \(Y\)-axis & \(\overline{Y}\) & Linear error vector in the \(Y\)-axis \\ CSZ & Coordinate system of the \(Z\)-axis & \(\overline{Z}\) & Linear error vector in the \(Z\)-axis \\ CSC & Coordinate system of the \(C\)-axis & \(\overline{C}\) & Linear error vector in the \(C\)-axis \\ CSB & Coordinate system of the \(B\)-axis & \(\overline{B}\) & Linear error vector in the \(B\)-axis \\ CSX & Coordinate system of the \(X\)-axis & \(\overline{T_{LT}}\) & Translation vector CS0-CSLT \\ CSLT & Coordinate system of the laser tracker (LT) & \(\overline{X_{LT}}\) & Coordinates of the LT \\ \(\overline{R\left(x\right)}\) & Rotational error matrix of the \(X\)-axis & \(\overline{T}\) & Offset of the tool \\ \(\overline{R\left(y\right)}\) & Rotational error matrix of the \(Y\)-axis & \(\overline{R\left(c\right)}\) & Rotational error matrix of the C-axis \\ \(\overline{R\left(z\right)}\) & Rotational error matrix of the \(Z\)-axis & \(\overline{R\left(b\right)}\) & Rotational error matrix of the B-axis \\ P & Nominal MT coordinates & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature of kinematic chain of Zayer TB 5000 machine tool (MT). Figure 1: Kinematic chain and kinematic model of Zayer TB 5000 machine tool (MT). \(\overline{X}\) represents the linear error vector in the \(x\)-axis of the milling machine. \[\overline{X}=\left( \begin{array}{c}-x+\ \delta_{x}\left(x\right)\\ \delta_{y}\left(x\right)\\ \delta_{z}\left(x\right)\end{array} \right) \tag{4}\]  
\(\overline{Y}\) represents the linear error vector in the \(y\)-axis of the milling machine. \[\overline{Y}=\left( \begin{array}{c}\delta_{x}\left(y\right)-y\cdot\epsilon_ {xy}\\ y+\delta_{y}\left(y\right)\\ \delta_{z}\left(y\right)\end{array} \right) \tag{5}\]  
\(\overline{Z}\) represents the linear error vector in the \(z\)-axis of the milling machine. \[\overline{Z}=\left( \begin{array}{c}\delta_{x}\left(z\right)-z\cdot\epsilon_ {xz}\\ \delta_{y}\left(z\right)-z\cdot\epsilon_{yz}\\ z+\delta_{z}\left(z\right)\end{array} \right) \tag{6}\]  
where \(\epsilon_{x}\left(k\right),\ \epsilon_{y}\left(k\right),\ \text{and}\ \epsilon_{z}\left(k\right)\) are three rotation errors of an axis \(k=x,y,z\); \(\delta_{k}\left(k\right)\) is the position error of the axis \(k=x,y,z\); \(\delta_{k}\left(j\right)\) with \(k\
eq j\) is the straightness error in the \(k\) direction; and \(\epsilon_{xy},\ \epsilon_{xz},\ \text{and}\ \epsilon_{yz}\) are squareness errors. \(\overline{T_{LT}}\) represents the translation vector between the coordinate systems of the machine, CSO, and laser tracker, CSLT. \[\overline{T_{LT}}=\left( \begin{array}{c}oX_{LT}\\ oY_{LT}\\ oZ_{LT}\end{array} \right) \tag{7}\]  
\(\overline{R_{LT}}\) represents the rotation matrix \(\theta\) between CSLT and CSO around a unitary vector \(u=\left(u_{x},\ u_{y},\ u_{z}\right)\), where \(u_{x}^{2}+u_{x}^{2}+u_{x}^{2}=\ 1\). \(\overline{X_{LT}}\) represents the coordinates of a machine point measured with an LT.  
\[\overline{X_{LT}}=\left( \begin{array}{c}X_{LT}\\ Y_{LT}\\ Z_{LT}\end{array} \right) \tag{8}\]