## 2 Experiments
### Proportional Error  
To further check is the positioning accuracy cab be fully observed from the ISO 230-2 and -6 tests, the second experiment was performed. The second experiment was to give a proportional error for the target points, as seen in Figure 4b. Note that the proportional error is to simulate, as an example, the lead error of ball screw (assuming the ball screw has no backlash and hysteresis) in this study. The proportional error means that the actual position equals target position multiplying a constant (e.g., \(t^{\prime}=k\times\ t\)). There are two examples that we give:  
1. Assuming the target point is (100, 50, 30.2) mm and the given proportional error for \(z\)-axis is 1.01, thus, the command position will be (100, 50, 30.804) mm. 2. Assuming the target point is (82.55, 50, 30.2) mm and the given proportional error for \(z\)-axis is 0.99 for \(x\)-axis, the command position will be on (81.725, 50, 30.2) mm. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Offset Value**} & \multicolumn{3}{c}{**Coordinate Difference**} \\ \cline{2-4}  & \(x\) & \(y\) & \(z\) \\ \hline \(x\)-axis, +1 mm & \(-1.00\) & 0.00 & 0.00 \\ \(x\)-axis, \(-1\) mm & 1.00 & 0.00 & 0.00 \\ \(y\)-axis, +1 mm & 0.00 & \(-1.00\) & 0.00 \\ \(y\)-axis, \(-1\) mm & 0.00 & 1.01 & 0.01 \\ \(z\)-axis, +1 mm & 0.00 & 0.00 & \(-0.99\) \\ \(z\)-axis, \(-1\) mm & 0.01 & 0.00 & 1.00 \\ \(x\)- and \(y\)-axis, +1 mm & \(-1.00\) & \(-1.00\) & 0.01 \\ \(x\)- and \(y\)-axis, \(-1\) mm & 1.00 & 1.00 & 0.01 \\ \(x\)- and \(z\)-axis, +1 mm & \(-1.00\) & 0.00 & \(-0.99\) \\ \(x\)- and \(z\)-axis, -1 mm & 1.00 & 0.00 & 1.01 \\ \(y\)- and \(z\)-axis, +1 mm & 0.01 & \(-1.00\) & \(-1.00\) \\ \(y\)- and \(z\)-axis, \(-1\) mm & 0.00 & 1.00 & 1.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experiment results of the coordinate offset simulation. (Unit: mm). The experimental results are shown in Table 3. We can see that the coordinate determined error not only occurs on the axis that was giving the proportional error, but also appears in other axes. For instance, the actual and the determined coordinates of the LaserTRACER, respectively, locate at \((100,50,30)\) and \((100.60,49.90,29.99)\) mm of the experiment \"\(x\)-axis, \(1.01\)\". The measurement difference of \(x\)-axis test for each point is:  
\[ \begin{array}{l}\left(\begin{array}{cccc}100.60+k\times\Delta\mathrm{s} \times\mathrm{p}_{\mathrm{x}}+\mathrm{s}_{\mathrm{x}}+\sigma_{\mathrm{x}},&49. 9,&29.99\\ \end{array}\right)-\left(\begin{array}{cccc}100+k\times\Delta\mathrm{s}+ \mathrm{s}_{\mathrm{x}},&50,&30\\ \end{array}\right)\\ =\left(\begin{array}{cccc}0.60+\sigma_{\mathrm{x}}+k\times\Delta\mathrm{s}_ {\mathrm{x}}\times\left(\mathrm{p}_{\mathrm{x}}-1\right),&0.10,&0.01\\ \end{array}\right),\end{array}  \tag{9}\]  
where \(p_{\mathrm{x}}\) represents the given proportional error. If the testing axis is \(y\)-axis, the measurement difference for \(k\)-th measurement point will be:  
\[ \begin{array}{l}\left(\begin{array}{cccc}100.60,&49.9+k\times\Delta \mathrm{s}\times\mathrm{p}_{\mathrm{y}}+\mathrm{s}_{\mathrm{y}}+\sigma_{ \mathrm{y}},&29.99\\ \end{array}\right)-\left(\begin{array}{cccc}100,&50+k\times\Delta\mathrm{s}+ \mathrm{s}_{\mathrm{y}},&30\\ \end{array}\right)\\ =\left(\begin{array}{cccc}0.60,&0.10+\sigma_{\mathrm{y}}+k\times\Delta \mathrm{s}\times\left(\mathrm{p}_{\mathrm{y}}-1\right),&0.01\\ \end{array}\right)\end{array}  \tag{10}\]  
From the experimental results, we can see that the measurement results included an error, as shown in Figure 5, and the proportional error could be observed. Figure 5: If the actual and determined positions of the ATLI are different, the measurement difference becomes smaller over the measurement distance. In the figure, \(t_{0}\) to \(t_{n}\) represent the number of target point to be tested.