## 3 ISO 230-2 and ISO 230-6 Tests Using LaserTRACER
### LaserTRACER Coordinate Determination
Combined with Equations (17) and (18), Equation (19) can be rewritten as:  
\[\Delta d_{x}\left[k\right]=\hat{d}_{x}\left[n\right]\left[\frac{\Delta x+d_{x }\left[n\right]}{\sqrt{\left(\Delta x+d_{x}\left[n\right]\right)^{2}+\Delta y ^{2}+\Delta z^{2}}}-1\right]-\hat{d}_{x}\left[k\right]\left[1-\frac{\Delta x +d_{x}\left[k\right]}{\sqrt{\left(\Delta x+d_{x}\left[k\right]\right)^{2}+ \Delta y^{2}+\Delta z^{2}}}\right] \tag{20}\]  
The simulation results for length difference in different measuring lengths are shown in Figures 12 and 13. In this simulation, we let \(\Delta x\), \(\Delta y\) and \(\Delta z\), respectively, be 0.7978, 0.2715 and 0.1965 mm (these values have no any meaning and are not very important, we just use these values to perform the evaluation), in which the measurement length differences in \(y\)- and \(z\)-axis are computed by:  
\[\Delta d_{y}\left[k\right]=\hat{d}_{y}\left[n\right]\left[\frac{\Delta y+d_{y }\left[n\right]}{\sqrt{\Delta x^{2}+\left(\Delta y+d_{y}\left[n\right]\right)^ {2}+\Delta z^{2}}}-1\right]-\hat{d}_{y}\left[k\right]\left[1-\frac{\Delta y+d _{y}\left[k\right]}{\sqrt{\Delta x^{2}+\left(\Delta y+d_{y}\left[k\right] \right)^{2}+\Delta z^{2}}}\right] \tag{21}\]  
Figure 11: The deviation of the \(\alpha\) angle following measurement distance changes when \(\Delta x\), \(\Delta y\) and \(\Delta z\) are 0.7978, 0.2715 and 0.1965 mm, respectively. Thus, the total measurement length difference is:  
\[\Delta d_{z}\left[k\right]=\hat{d}_{z}\left[n\right]\left[\frac{\Delta z+d_{z} \left[n\right]}{\sqrt{\Delta z^{2}+\Delta y^{2}+\left(\Delta z+d_{z}\left[n \right]\right)^{2}}}-1\right]-\hat{d}_{z}\left[k\right]\left[1-\frac{\Delta z+d_ {z}\left[k\right]}{\sqrt{\Delta x^{2}+\Delta y^{2}+\left(\Delta z+d_{z}\left[k \right]\right)^{2}}}\right] \tag{22}\]  
\[\Delta d\left[k\right]=\sqrt{\Delta d_{x}^{2}\left[k\right]+\Delta d_{y}^{2} \left[k\right]+\Delta d_{z}^{2}\left[k\right]} \tag{23}\]  
In Figures 12 and 13 we can see that the measurement length difference could be ignored when the measurement length is farther than 300 mm. In this simulation, the distances between the target points and the reference sphere of LaserTRACER are 10, 20, \(\dots\), 90, 100, 200, 300, \(\dots\), 1100 mm (these lengths are absolute distance from the reference sphere of LaserTRACER to the target points). After setting the lengths and \(\Delta x\), \(\Delta y\) and \(\Delta z\), the measurement line angle \(\alpha\) can be computed. Therefore, according to Equations (20)-(22), the length difference can be computed. The simulation result is similar to the result that descripted in ISO 230-2. The total measurement length difference becomes smaller with decreasing \(\Delta x\), \(\Delta y\) and \(\Delta z\). Thus, our inference in this section is that: (1) the \(\alpha\) error due to misalignment can be ignored for testing the positioning performance of a high accuracy machine tool because the coordinate determined error of the LaserTRACER is very small; and (2) measurement length difference due to misalignment can be ignored when the measurement length is farther than 300 mm for a larger coordinate determined error.