## 3 ISO 230-2 and ISO 230-6 Tests Using LaserTRACER
### LaserTRACER Coordinate Determination  
The coordinate of the LaserTRACER on the machine tool is determined through six-point measurement, as seen in Figure 10, and the following Equations:  
\[\left\{ \begin{array}{l}\left(x_{1}-x_{\text{t}}\right)^{2}+\left(y_{1}-y_{ \text{t}}\right)^{2}+\left(z_{1}-z_{\text{t}}\right)^{2}=\left(L_{0}+\Delta L _{1}\right)^{2}\\ \left(x_{2}-x_{\text{t}}\right)^{2}+\left(y_{2}-y_{\text{t}}\right)^{2}+ \left(z_{2}-z_{\text{t}}\right)^{2}=\left(L_{0}+\Delta L_{2}\right)^{2}\\ \left(x_{3}-x_{\text{t}}\right)^{2}+\left(y_{3}-y_{\text{t}}\right)^{2}+ \left(z_{3}-z_{\text{t}}\right)^{2}=\left(L_{0}+\Delta L_{3}\right)^{2}\\ \left(x_{4}-x_{\text{t}}\right)^{2}+\left(y_{4}-y_{\text{t}}\right)^{2}+ \left(z_{4}-z_{\text{t}}\right)^{2}=\left(L_{0}+\Delta L_{4}\right)^{2}\\ \left(x_{5}-x_{\text{t}}\right)^{2}+\left(y_{5}-y_{\text{t}}\right)^{2}+ \left(z_{5}-z_{\text{t}}\right)^{2}=\left(L_{0}+\Delta L_{5}\right)^{2}\\ \left(x_{6}-x_{\text{t}}\right)^{2}+\left(y_{6}-y_{\text{t}}\right)^{2}+ \left(z_{6}-z_{\text{t}}\right)^{2}=\left(L_{0}+\Delta L_{6}\right)^{2}\\ \end{array} \right., \tag{11}\]  
\[\Delta L_{i}=L_{i}-L_{0},\quad i=1,2,\ldots,6, \tag{12}\]  
Figure 9: The ATLI must be placed inside the working area of machine tools: (**a**) the ATLI placed inside the working area (\(xy\) plane view); (**b**) when ATLI placed outside of the working area in \(y\)-direction (\(xy\) plane view); (**c**) the ATLI placed outside of the working area in \(x\)-direction (\(xy\) plane view); and (**d**) the ATLI placed outside of the working area in \(x\)-direction (\(xz\) plane view). where (\(x_{i}\)\(y_{i}\)\(z_{i}\)) is the six stop points of the cat's eye reflector, and these points are independent and should be given; (\(x_{i}\)\(y_{i}\)\(z_{i}\)) is the coordinate of the LaserTRACER, which is to be determined; \(L_{0}\) is the initial distance from the LaserTRACER to the cat's eye reflector, which is an unknown value; and \(\Delta L_{i}\) is the measured distance deviation from the LaserTRACER. Thus, the coordinate of the ATLI related to [R] can be determined by moving the reflector of six individual spatial points. If we define a symbol \(\Psi\) as below:  
\[\psi={x_{i}}^{2}+{y_{i}}^{2}+{z_{i}  
If the determined coordinate and actual position of the LaserTRACER are different, the \(\alpha\) error is involved which is caused by misalignment. Please notice that the essence of misalignment of LaserTRACER is different to laser interferometer since the \(\alpha\) error is a function of measurement length. As seen in Figure 5, \(\Delta x\), \(\Delta y\) and \(\Delta z\) represent the determined LaserTRACER coordinate error in \(x\)-, \(y\)- and \(z\)-direction, respectively. Measuring length difference between the ideal path (represented \(d[k]\)) and the actual path (represented \(\hat{d}[k]\)) for \(k\)-th target point of \(x\)-axis test is:  
\[d\left[k\right]=\hat{d}\left[k\right]\cos\left[k\right]-\Delta x, \tag{17}\]  
\[\alpha\left[k\right]=\tan^{-1}\left(\frac{\sqrt{\Delta y^{2}+\Delta z^{2}}}{ \Delta x+d\left[k\right]}\right), \tag{18}\]  
\[\Delta d\left[k\right]=\left(\hat{d}\left[k\right]-\hat{d}\left[n\right] \right)-\left(d\left[k\right]-d\left[n\right]\right)=\hat{d}\left[n\right] \left(\cos\alpha\left[n\right]-1\right)+\hat{d}\left[k\right]\left(1-\cos \alpha\left[k\right]\right), \tag{19}\]  
in which \(k=1,2,\,\ldots,n,\,n\) is the number of the target points, and \(d[n]\) and \(\hat{d}[n]\) are the distance from LaserTRACER to \(n\)-th target point (_i.e._, \(t_{n}\)). Since ISO 230-2 and -6 results are calculated by the relative change in distance, and \(\alpha\) changes convergence (far smaller than 1 degree) following measurement distance becoming far away from the origin point (as seen in Figure 11), which means the measurement length difference also becomes small, measurement length difference can be computed by Equation (19). Here, we can see that the effect of \(\Delta x\) has been eliminated.