## 5 Calibration Experiment and Algorithm Comparison
### Verify the Accuracy of the Calibration Results  
To validate the accuracy of TCP calibration results, an ABB robot repositioning method can be employed as a validation technique to assess the calibrated TCP parameters. It's important to note that while this method provides a means of validation, it may not offer highly precise results. Alternatively, TCP calibration can be conducted using a dedicated TCP calibration tool, and the resultant TCP parameters can be compared with the calculated TCP parameters to validate the calibration accuracy. This approach often provides a more accurate and reliable validation of the TCP calibration results.  
In this experiment, a LEONI standard 6D TCP calibration system was utilized to calibrate the measuring head, as depicted in Figure 11. The LM-D algorithm addresses the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **LM-D** & **L-M** & **DIRECT** & **SA** \\ \hline Average Computational & & & & \\ Efficiency & Medium & Fast & Slow & Slow \\ \hline Data 1 F(X\({}_{i}\)) (mm) & 0.0838 & 0.0838 & 4.975 & 4.076 \\ \hline Data 1 Iteration Time (s) & 11.9 & 8.17 & T\({}_{\text{max}}\) & T\({}_{\text{max}}\) \\ \hline Data 2 F(X\({}_{i}\)) (mm) & 0.007 & 5.526 & 4.525 & 0.007 \\ \hline Data 2 Iteration Time (s) & 87.3 & 4.4 & T\({}_{\text{max}}\) & 2710 \\ \hline Data 3 F(X\({}_{i}\)) (mm) & 0.001 & 1.26 & 1.349 & 0.378 \\ \hline Data 3 Iteration Time (s) & 2438 & 5.5 & T\({}_{\text{max}}\) & T\({}_{\text{max}}\) \\ \hline \multirow{2}{*}{Astringency} & Global & \multirow{2}{*}{\begin{tabular}{c} Global \\ convergence \\ \end{tabular} } & Global & \multirow{2}{*}{\
\begin{tabular}{c} Global \\ convergence \\ \end{tabular} } & Global & \multirow{2}{*}{
\begin{tabular}{c} Global \\gence/Local \\ convergence \\ \end{tabular} } \\ \cline{1-1}  & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: The solution result of data 1, 2, 3.  
contact-based TCP calibration problem, where the calibration result identifies the center of the ruby probe. The LEONI 6D TCP calibration system, on the other hand, is a non-contact calibration method based on vertical laser beam sensors. As shown in Figure 11, this calibration method measures the bottom of the tool. The difference between the two calibration results lies in the distance between the bottom of the ruby probe and its center, which is the radius 2.5 mm of the ruby probe. The TCP calibration results obtained through the tool center point calibration method exhibited a difference of 2.5 mm in the direction of the tool axis when compared to the TCP results obtained using the standard ball calibration method.  
Upon applying the transformation, the results obtained from the tool center point calibration of the measuring head were (X: 583.667, Y: \(-\)5.818, Z: 123.815), allowing for a direct comparison with the previously mentioned results. Notably, the data collected in the first set of measurements are the most extensive, and theoretically, the results (X: 583.56, Y: \(-\)6.07, Z: 123.75) obtained through the least squares method are considered more accurate. The close alignment between the two sets of data signifies that the TCP data obtained using the LM-D method is accurate, affirming the suitability of this combined algorithm for TCP calibration.