## 5 Conclusions  
This paper proposes a TM-MDH error model and a C-LM algorithm for identifying the actual structural parameters of industrial robots. The simulation results indicate that the proposed C-LM algorithm overcomes the drawback of easily falling into local optima when identifying geometric parameters simultaneously and exhibits a wider convergence range. Furthermore, the proposed TM-MDH model accurately describes the relationship between geometric errors and end-effector pose errors. After calibration by the C-LM algorithm, the positional error of the industrial robot's flange end was reduced by 0.0122 mm from 1.9536 mm, and the orientation error was reduced by 1.31 \(\times\) 10\({}^{-4}\) rad from 1.46 \(\times\) 10\({}^{-2}\) rad.  
This study is an initial exploration of error models for calibrating industrial robots, considering only geometric parameters such as joint angles. Further research can incorporate motor gear ratios, coupling ratios, and link deformation parameters into the robot error model. It is worth noting that, besides geometric errors, there are other non-geometric errors and random errors. Therefore, establishing an end-effector accuracy compensation model for industrial robots using neural networks is an interesting research direction. Lastly, considering intelligent optimization algorithms such as differential evolution and genetic algorithms for identifying the structural parameters of industrial robots is also meaningful.  
Methodology, D.L. and Y.Q.; Software, D.L. and Y.Q.; Validation, D.L., Y.Q. and F.W.; Formal analysis, D.L., Y.Q. and F.W.; Investigation, D.L. and Y.L.; Resources, D.L. and Y.L.; Data curation, D.L. and Y.Q.; Writing--original draft, D.L., Y.Q., F.W. and Y.L.; Writing--review & editing, D.L.; Visualization, D.L. and Y.Q.; Supervision, Y.Q.; Project administration, D.L., Y.Q. and Y.L.; Funding acquisition, D.L. and Y.Q. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the Scientific and Technological Innovation Foundation of the Shunde Innovation School (Grant No. BK20 BE019).  
Not applicable.  
The authors are grateful to the editor and the anonymous referees for their constructive comments and recommendations, which have helped to improve this paper significantly. The authors would also like to express their sincere thanks to [PERSON] for the useful information regarding the MDH model.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Error Model** & **Mean** & **Max** & **Min** \\ \hline Before compensation & 1.9536 & 4.1821 & 0.1847 \\ MDH & 0.9292 & 2.2057 & 0.2764 \\ TM-MDH & 0.0122 & 0.0228 & 0.0045 \\ \hline \end{tabular}
\end{table}
Table 10: The position accuracy for different error models (mm).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Error Model** & **Mean** & **Max** & **Min** \\ \hline Before compensation & 1.46 \(\times\) 10\({}^{-2}\) & 3.16 \(\times\) 10\({}^{-2}\) & 3.61 \(\times\) 10\({}^{-3}\) \\ MDH & 9.50 \(\times\) 10\({}^{-3}\) & 3.70 \(\times\) 10\({}^{-2}\) & 3.47 \(\times\) 10\({}^{-3}\) \\ TM-MDH & 1.31 \(\times\) 10\({}^{-4}\) & 1.36 \(\times\) 10\({}^{-3}\) & 4.74 \(\times\) 10\({}^{-5}\) \\ \hline \end{tabular}
\end{table}
Table 11: The orientation accuracy for different error models (rad).