## 2 Research Methodology
### Machine Tool Kinematic Model  
The fiducial is used to obtain the probing information containing errors resulting from both the integrated on-machine measurement system and the machine tool itself. The relationship between the tip of the probe and tip of the tool has been established in the previous work of the authors [18]. As mentioned earlier, the repeatability of measurement of the probing system is relatively low at 0.24 \(\upmu\)m. It is reasonable to assume that the main contributor of the error sources is the machine tool. As a result, the integrated error effect can be synthesized to find the difference between the actual position and probing position of each fiducial in the working volume. It is not necessary to measure and model all the errors separately. More attention should be focused on the synthesized influence of the measurement task. In the present work, the position errors of the fiducials are mainly considered. The ultra-precision raster milling machine tool has a wBZ  
\[{}^{x}{}_{T}{}_{Y}=\left[ \begin{array}{cc}0\\ I_{3\times 3}&y\\ 0\\ 0_{1\times 3}&1\end{array} \right] \tag{7}\]  
\[{}^{Y}{}_{T}{}_{C}=\left[ \begin{array}{cc}0\\ R_{C}&0\\ 0\\ 0_{1\times 3}&1\end{array} \right]R_{C}=\left[ \begin{array}{cc}\cos c&-\sin c&0 \\ \sin c&\cos c&0\\ 0&0&1\end{array} \right] \tag{8}\]  
\[{}^{C}{}_{T}{}_{P}=\left[ \begin{array}{cc}p_{x}+\Delta_{px}\\ I_{3\times 3}&y_{y}+\Delta_{py}\\ p_{z}+\Delta_{pz}\\ 0_{1\times 3}&1\end{array} \right] \tag{9}\]  
\[{}^{F}{}_{T}{}_{P}=\left[ \begin{array}{cc}x+R_{C}(p_{x}+\Delta_{px})\\ R_{C}&y+R_{C}(p_{y}+\Delta_{py})\\ R_{C}(p_{z}+\Delta_{pz})\\ 0_{1\times 3}&1\end{array} \right] \tag{10}\]  
The three-workpiece branch sub-matrices are given as follows:  
\[{}^{F}{}_{T}{}_{Z}=\left[ \begin{array}{cc}0\\ I_{3\times 3}&0\\ z\\ 0_{1\times 3}&1\end{array} \right] \tag{11}\]  
\[{}^{Z}{}_{T}{}_{B}=\left[ \begin{array}{cc}R_{B}&0\\ R_{B}&0\\ 0&0\\ 0_{1\times 3}&1\end{array} \right]R_{B}=\left[ \begin{array}{cc}\cos b&0&\sin b \\ 0&1&0\\ -\sin b&0&\cos b\end{array} \right] \tag{12}\]  
\[{}^{B}{}_{T}{}_{w}=\left[ \begin{array}{cc}w_{i,x}+\Delta_{wix}\\ R_{w}&w_{i,y}+\Delta_{wiy}\\ w_{i,z}+\Delta_{wiz}\\ 0_{1\times 3}&1\end{array} \right] \tag{13}\]  
\[{}^{F}{}_{T}{}_{w}=\left[ \begin{array}{cc}R_{B}(w_{i,x}+\Delta_{wix})\\ R_{B}&R_{B}(w_{i,y}+\Delta_{wiy})\\ z+R_{B}(w_{i,z}+\Delta_{wiz})\\ 0_{1\times 3}&1\end{array} \right] \tag{14}\]  
where \(x,\ y,\ z,\ b\) and \(c\) are the machine axes positions for each fiducial. \(p_{x}\), \(p_{y}\) and \(p_{z}\) are the probe center coordinates and \(\Delta_{px}\), \(\Delta_{py}\) and \(\Delta_{pz}\) are the setup errors of the probe. (\(w_{i,x}\),\(w_{i,y}\),\(w_{i,z}\)) is the position of the \(i\)th fiducials relative to the last axis (B-axis) and \(\Delta_{wix}\), \(\Delta_{wiy}\), \(\Delta_{wiz}\) are the fiducial \(i\)'s setup errors. \(R_{w}\) is the rotation matrix and it is usually given as follows in most studies. \[R_{w}=I_{3\times 3} \tag{15}\]  
Since there is no need to rotate the spindle in the probing process, the errors resulting from spindle are not included in the above equation model. It is noted that many commercial multi-axis machine tools are not equipped with an on-machine probing system, which means that the developed on-machine probe may have a different coordinate frame from that of the tool tip. Although the established fiducial-aided geometry is likely to minimize the impact of the error source in the measuring procedure, the transformation relationship has to be established before the actual machining process. The deviation between the probe tip center and the \(i\)th fiducial \(\delta_{i}\) is described as:  
\[\delta_{i}-\left[ \begin{array}{c}x\\ y\\ -z\end{array} \right]-R_{C}\left[ \begin{array}{c}p_{x}\\ p_{y}\\ p_{z}\end{array} \right]+R_{B}\left[ \begin{array}{c}w_{i,x}\\ w_{i,y}\\ w_{i,z}\end{array} \right]=R_{C}\left[ \begin{array}{c}\Delta_{px}\\ \Delta_{py}\\ \Delta_{pz}\end{array} \right]-R_{B}\left[ \begin{array}{c}\Delta_{mix}\\ \Delta_{uriy}\\ \Delta_{wiz}\end{array}