## 5 Summary and Conclusions  
This study presented a novel iterative algorithm for simultaneous estimation of the User Coordinate System (UCS) and Tool Center Point (TCP) and benchmarked its performance against three industrially recognized methods: internal three-point calibration (Method A), absolute orientation with predefined TCP (Method B), and geometrical estimation of the robot's world frame (Method C). In contrast to mathematical optimization-focused techniques such as SE(3)-based minimization [12] or LMI-SDP formulations [23], the presented method is measurement-driven and specifically suited for industrial de  
Figure 5: Comparison of the accuracy of different calibration methods for local bases of an industrial robot in terms of translational errors (\(\Delta\)T). ployment scenarios. It addresses practical limitations highlighted in prior studies, such as rotational instability in geometric methods [24]. By combining the strengths of data fidelity with iterative stability, the method positions itself as a robust, flexible, and implementation-ready alternative for digital twin alignment and offline program commissioning in Industry 4.0 applications. The experimental validation was conducted using two ABB IRB-6640 industrial robots and a high-precision laser tracker, yielding the following key findings:  
* Accuracy improvement: The proposed method (M4) consistently achieved the lowest average translational errors, with mean \(\Delta\)T values of 0.345 mm (R01) and 0.364 mm (R02) when paired with its own TCP estimation. In comparison, the commonly used internal three-point method (M1) resulted in average errors of 2.399 mm (R01) and 2.483 mm (R02), while the absolute orientation method (M2) showed errors of 1.878 mm (R01) and 1.886 mm (R02). Even when those methods were enhanced with the proposed TCP, their errors remained significantly higher than those achieved by the full M4 approach. This represents an accuracy improvement of over 80% in translational error reduction relative to standard industrial techniques, confirming that joint optimization of UCS and TCP yields superior positional precision. * Stability across configurations: The proposed algorithm maintained robust performance across varying robot poses and tool orientations, showing low variance in \(\Delta\)T across all ten test points. It outperformed or matched the absolute orientation method (M2), which showed average \(\Delta\)T values of 1.88 mm (R01) and 1.89 mm (R02). * Limitations of geometric estimation: The geometric estimation approach (M3) demonstrated the poorest results, with translational errors as high as 18.05 mm in some positions and significant instability across measurement series. These errors stem from misalignment in WCS estimation, particularly in rotation, which is not detectable during measurement but emerges only after offline computation. This approach is not recommended as it depends directly on the quality of the robot's structural elements, which may be inconsistent. * Impact of TCP estimation: In all evaluated methods, replacing the robot's internally defined TCP with the one estimated by the proposed algorithm led to substantial improvements. For example, using internal TCP with the three-point method yielded a mean \(\Delta\)T of 2.40 mm (R01) and 2.48 mm (R02), while integrating the proposed TCP reduced errors to 1.70 mm and below. * Industrial applicability: The novel method does not rely on external TCP estimations or measurements and therefore offers a robust, repeatable, and flexible alternative for integrating simulated programs into real robot systems, especially in scenarios where direct robot control access is limited. * Setup efficiency: This approach to UCS determination can significantly shorten the setup time, especially in multi-robot workstations. This efficiency increase is achieved through a single calibration of the measurement system and a one-time alignment with the workstation.