## 4 Comparative Validation Based on Measurement Data  
A verification process has been conducted with all possible combinations of TCP and UCS, giving 80 measurement pairs with fixed coordinates in batches of 10 points. A representation of this process is presented in Figure 4 together with a block diagram. The visible difference in measured UCS highlights the discrepancy between simulation assumptions and the actual mounting positions of these two robots, reflecting deviations from real-world conditions. Measurements were collected, analyzed, and organized into four main series of 20 samples each. Each series corresponds to a specific UCS estimation method, and within each series, two subsets of 10 samples were recorded using the TCP from internal measurements and from the algorithm-estimated solution. Using the robot's internal TCP measurement method is inherently dependent on manual input, reference point positioning within the workspace, and operator skill. In this study, the accuracy of the internal TCP procedure was validated by ensuring the mean measurement error did not exceed 0.5 mm--an accuracy level typically sufficient for most industrial applications. This internal method is a commonly used procedure for calibrating the Tool Center Point (TCP) of an industrial robot. It involves reaching a fixed reference point with the tool from at least four different orientations, allowing the robot to calculate the exact position of the TCP relative to the robot flange. A threshold of 0.5 mm for the allowable average measurement error using the internal method is sufficient, as such limits are typically defined by individual facilities during manual measurements with the robot. When working with high-volume tooling, even achieving this level of accuracy can be  
Figure 4: Verification process of two ABB industrial robots in the RobotStudio 6.08 environment, showing robot motion trajectories based on different UCS configurations. The visualization demonstrates the process of collecting measurement data using a laser tracker. challenging in manual processes. Moreover, the issue of absolute accuracy in industrial robots can further worsen the measurement reliability depending on the location within the robot workspace [2; 21]. In the first subset, the TCP was introduced using the robot's internal measurement procedure. In the second subset, the introduced TCP was a result of the proposed algorithm in this paper. The four UCS estimation approaches evaluated in this study were as follows:  
* Robot internal method (3 Point)--Method A: UCS defined using the robot's internal three-point procedure. * Absolute orientation solution (Absor)--Method B: UCS estimated using the absolute-orientation-based algorithm, with a predefined TCP. * Geometrical estimation--Method C: UCS defined from geometrical referencing of the robot base and surrounding fixture. * Proposed algorithm (AbsIter)--Novel Method D: UCS and TCP jointly estimated using the proposed method with additional robot orientations. For each sample, the tool position was measured and compared to the reference position computed from the robot's transformation data. Absolute translational differences along the X, Y, and Z axes were calculated, and both per-axis and overall positional errors were evaluated. Each selected result has been gathered in Table 1.  
The results are presented in Plots 1-4, showing the absolute average translational difference in all three directions for each individual measurement point. The error is calculated using Formula (15), which compares the position measured by the external system with the corresponding robot read values. For each robot, two diagrams are provided: one using the TCP obtained from the internal robot measurement routine and the other using the TCP estimated with the proposed method.