## 4 Comparative Validation Based on Measurement Data
\[\overline{\Delta T}=\left(\frac{|X_{M}-X_{R}|+|Y_{M}-Y_{R}|+|Z_{M}-Z_{R}|}{3}\right) \tag{15}\]  
The outcome from the geometrical estimation method differs significantly from the other results due to high translational errors, with deviations \(\Delta\)T reaching over 5 mm, as shown in Tables 2 and 3. The error in the geometric method indicates a very poor fit of \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**IRB 6640-R01** & **X [mm]** & **Y [mm]** & **Z [mm]** & **Q0** & **Q1** & **Q2** & **Q3** \\ \hline UCS\_3 point & 1629.190 & 1071.970 & 915.263 & 0.974142 & 0.001790 & 0.001257 & \(-\)0.225926 \\ \hline UCS\_AbsOri\_withTCP & 1632.170 & 1070.160 & 917.030 & 0.974113 & 0.000780 & 0.001124 & \(-\)0.226058 \\ \hline UCS\_Geometrical & 1632.265 & 1052.513 & 918.997 & 0.973929 & 0.004281 & 0.003842 & \(-\)0.226782 \\ \hline UCS\_AbsIter & 1629.342 & 1074.242 & 915.302 & 0.974272 & 0.000858 & 0.001591 & \(-\)0.225369 \\ \hline TCP\_AbsIter & \(-\)178.748 & 7.301 & 76.210 & 1.000000 & 0.000000 & 0.000000 & 0.000000 \\ \hline TCP\_Internal & \(-\)179.206 & 7.533 & 71.334 & 1.000000 & 0.000000 & 0.000000 & 0.000000 \\ \hline
**IRB 6640-R02** & & & & & & & \\ \hline UCS\_3 point & 1616.480 & \(-\)2494.640 & 914.222 & 0.974081 & 0.001109 & \(-\)0.000078 & \(-\)0.226198 \\ \hline UCS\_AbsOri\_withTCP & 1618.838 & \(-\)2492.048 & 914.041 & 0.974106 & 0.000684 & 0.000748 & \(-\)0.226091 \\ \hline UCS\_Geometrical & 1612.251 & \(-\)2486.860 & 22.398 & 0.973748 & \(-\)0.000803 & 0.002966 & \(-\)0.227607 \\ \hline UCS\_AbsIter & 1616.328 & \(-\)2495.413 & 913.986 & 0.974088 & 0.000633 & 0.000703 & \(-\)0.226167 \\ \hline TCP\_AbsIter & \(-\)349.873 & 75.604 & 23.609 & 1.000000 & 0.000000 & 0.000000 & 0.000000 \\ \hline TCP\_Internal & \(-\)345.739 & 75.405 & 23.877 & 1.000000 & 0.000000 & 0.000000 & 0.000000 \\ \hline \end{tabular}
\end{table} 
Table 1: Gathered values of UCS translation and rotation and TCP translation. the WCS, which cannot be detected during the measurement stage itself but only becomes evident after performing the calculations, ultimately making this method unsuitable for effective deployment of simulations into real-world environments. However, it is worth noting that geometric estimation methods are still widely used in industrial settings due to their non-invasive nature and independence from robot controller access, similar to the motivations discussed in [22; 23], where external measurements were leveraged to reduce reliance on internal robot interfaces. Compared to the method proposed in this paper, the geometric approach lacks robustness, especially under varying orientations. In contrast, the new iterative method resembles techniques such as those in [PERSON] et al. [18] and [PERSON] [12], which also tackled the AX = YB formulation through iterative or separable solutions, but without integrating measurement-driven feedback for simultaneous estimation. The measurement-based refinement strategy adopted here is akin to that in [PERSON] et al. [24], where pose and reprojection errors were minimized iteratively, but differs by grounding the calibration in physically measured transformations rather than simulation-generated or vision-based estimates. Furthermore, while [PERSON] and [PERSON] [12] introduced a metric on SE(3) to optimize the AX = ZB solution based on system stochasticity, and [PERSON] et al. [23] applied convex optimization through LMI-SDP for robustness, these methods primarily focused on mathematical optimality under noise rather than the specific industrial deployment scenario addressed here. The proposed method uniquely aligns with real-world deployment constraints by combining accurate, external laser measurements with iterative optimization of both the TCP and UCS, demonstrating an average translational error below 0.4 mm--surpassing the performance of most state-of-the-art techniques.