## 4 Comparative Validation Based on Measurement Data
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & \(\widetilde{\Delta T}\) [mm] \\ \hline
**TCP/UCS** & \(\widetilde{\Delta T}\) **(P01)** & \(\widetilde{\Delta T}\) **(P02)** & \(\widetilde{\Delta T}\) **(P03)** & \(\widetilde{\Delta T}\) **(P04)** & \(\widetilde{\Delta T}\) **(P05)** & \(\widetilde{\Delta T}\) **(P06)** & \(\widetilde{\Delta T}\) **(P07)** & \(\widetilde{\Delta T}\) **(P08)** & \(\widetilde{\Delta T}\) **(P09)** & \(\widetilde{\Delta T}\) **(P10)** & \(\widetilde{\Delta T}\) **(P10)** \\ \hline Internal/3 Point & 1.904 & 1.925 & 3.379 & 2.456 & 1.910 & 1.510 & 3.549 & 2.515 & 3.477 & 1.370 & 2.399 \\ \hline Internal/Absor & 1.184 & 0.878 & 2.785 & 3.086 & 1.257 & 2.695 & 1.849 & 0.977 & 3.134 & 0.936 & 1.878 \\ \hline Internal/Geo & 11.694 & 9.458 & 11.960 & 15.701 & 16.339 & 12.457 & 15.698 & 11.437 & 13.257 & 12.947 & 13.095 \\ \hline Internal/Abslter & 2.364 & 2.202 & 2.114 & 2.001 & 2.545 & 2.991 & 2.583 & 2.756 & 2.091 & 2.192 & 2.384 \\ \hline Abslter/3 Point & 0.796 & 0.944 & 1.598 & 1.597 & 2.132 & 1.177 & 2.353 & 2.584 & 1.847 & 1.921 & 1.695 \\ \hline Abslter/Abslter & 2.154 & 2.024 & 2.157 & 2.010 & 2.417 & 2.443 & 2.115 & 1.641 & 1.778 & 2.993 & 2.173 \\ \hline Abslter/Geo & 13.376 & 11.747 & 13.789 & 10.191 & 18.052 & 13.071 & 14.792 & 13.536 & 14.355 & 15.180 & 13.809 \\ \hline Abslter/Abslter & 0.258 & 0.118 & 0.341 & 0.150 & 0.459 & 0.276 & 0.399 & 0.489 & 0.481 & 0.475 & 0.345 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of positional results for each measured point using different UCS and TCP estimation methods on the IRB 6640-R01 robot. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & \(\widetilde{\Delta T}\) [mm] \\ \hline
**TCP/UCS** & \(\widetilde{\Delta T}\) **(P01)** & \(\widetilde{\Delta T}\) **(P02)** & \(\widetilde{\Delta T}\) **(P03)** & \(\widetilde{\Delta T}\) **(P04)** & \(\widetilde{\Delta T}\) **(P05)** & \(\widetilde{\Delta T}\) **(P06)** & \(\widetilde{\Delta T}\) **(P07)** & \(\widetilde{\Delta T}\) **(P08)** & \(\widetilde{\Delta T}\) **(P09)** & \(\widetilde{\Delta T}\) **(P10)** & \(\widetilde{\Delta T}  
Figure 5 presents a comparison of translational errors (\(\Delta\)T) obtained through different calibration methods used to define local coordinate systems (UCS) for an industrial robot. A noticeable fluctuation in error values, particularly in the geometrical method, highlights the inherent instability in estimating rotational components of the coordinate system, which directly affects translational consistency. This apparent instability stems from the fact that during the reference-point-based calibration process, the tool approaches the target from various angles. As the orientation of the tool (i.e., angle of incidence) changes, small angular misalignments in the UCS lead to amplified deviations in the calculated TCP position, especially when the tool is long or offset from the flange. These changes introduce variations in the resulting translational components, as clearly visualized in the plotted curves. If one were to measure rotational discrepancies directly between UCS pairs, it would require collecting laser-based 3D coordinates of three non-collinear points at each robot pose to accurately compute the orientation of the local frame. While this approach was not applied here, similar angular effects can already be indirectly inferred from the graph: by examining the number and magnitude of outlier points (peaks or drops) deviating from the general trend line. These spikes suggest momentary inconsistencies in orientation estimation that manifest as fluctuating translational error.