## 2 Experiments
### Coordinate Offset  
The first experiment is to simulate the machine tool moved with a fixed (constant) positioning error. To simulate this situation, we give all measurement points of a constant offset, as seen in Figure 4a, where \(\varepsilon x\), \(\varepsilon y\) and \(\varepsilon z\) represent the amounts of the given coordinate offset. To easily observe the effect, we gave a large offset value for all points. For example, as seen in Table 2, we gave \(-1\) mm offset for \(z\)-axis denoted of \"z-axis, \(-1\) mm\". The original coordinate of the LaserTRACER is (653.60, \(-27.13\), 262.18) in unit of mm. After we gave the non-zero coordinate offset to different axes, the determined coordinate differences are listed in Table 2. From the results, we can see that the coordinate differences are almost exactly equal to the offset values that were given.  
Because the coordinate difference is the negative of the given offset value, the coordinate offset can be cancelled. For example, assuming the actual position of the LaserTRACER is (100, 50, 30) mm, and the determined coordinate is (99, 50, 30) mm, when we give \"\(z\)-axis, \(+1\) mm\", the target points for \(x\)-axis testing will be:  
\[\left(\begin{array}{cccc}99+k\times\Delta\mathrm{s}+\mathrm{s}_{\mathrm{x}}+ \mathrm{\varepsilon}_{\mathrm{x}}+\mathrm{\sigma}_{\mathrm{x}},&50,&30\\ \end{array}\right), \tag{5}\]  
Figure 4: Illustration of the simulation for machine tool moved with positioning error: (**a**) coordinate offset; and (**b**) proportional error. (Note: vector \(\overrightarrow{o}=\left[\mathrm{\sigma}_{x}\,\mathrm{\sigma}_{y}\,\mathrm{ \sigma}_{z}\right]^{T}\) represents the coordinate offset vector; \(t_{i}\) represents the \(i\)-th target point; \(k\) represents the proportional value; and \(t_{i}^{\prime}\) represents the \(i\)-th target point with coordinate offset or proportional error).  
where \(k\) (= 1, 2,..., \(n\), \(n\) represents the number of target points that is to be measured) denote the measurement point number, and; \(\Delta s\) denotes the interval distance of each point; \(s_{x}\) denotes an offset distance along \(x\)-axis that was applied to avoid the reflector crashing the ATLI; \(\varepsilon_{x}\) represents the coordinate offset error (\(\varepsilon_{x}\) = 1); and \(\sigma_{x}\) represents the positioning error, including the repeatability. Thus, the measurement difference for each point is:  
\[\begin{array}{l}\left(\begin{array}{cccc}99+k\times\Delta{\rm s}+{\rm s} _{\rm x}+\varepsilon_{\rm x}+\sigma_{\rm x},&50,&30\end{array}\right)-\left( \begin{array}{cccc}100+k\times\Delta{\rm s}+{\rm s}_{\rm x},&50,&30\end{array} \right)\\ =\left(\begin{array}{cccc}\sigma_{\rm x},&0,&0\end{array}\right)\end{array} \tag{6}\]  
Assuming the testing axis yield the \(y\)-axis, the measurement points will be:  
\[\left(\begin{array}{cccc}99+\varepsilon_{\rm x},&50+k\times\Delta{\rm s}+{ \rm s}_{\rm y}+\sigma_{\rm y},&30\end{array}\right) \tag{7}\]  
Thus, the measurement difference for each point is:  
\[\begin{array}{l}\left(\begin{array}{cccc}99+\varepsilon_{\rm x},&50+k \times\Delta{\rm s}+{\rm s}_{\rm y}+\sigma_{\rm y},&30\end{array}\right)- \left(\begin{array}{cccc}100,&50+k\times\Delta{\rm s}+{\rm s}_{\rm y},&30 \end{array}\right)\\ =\left(\begin{array}{cccc}0,&\sigma_{\rm y},&0\end{array}\right)\end{array} \tag{8}\]  
From the results we can know that although the constant offset of the machine tool causes coordinate determined error, it does not affect the ISO 230-2 and -6 test results as well as the positioning error and repeatability of the testing axis can still be observed.