# A Self-Calibration Method for Robot End-Effector Using Spherical Constraints
## 6 Conclusions  
This paper introduces a method for calibration of the robot end-effector coordinate system using spherical constraints. Initially, a mathematical model incorporating both robot geometric errors and end-effector deformation errors was established. Various errors were propagated to the base coordinate system via an error propagation matrix. Utilizing the spherical constraints of the calibration sphere, an error parameter identification model was developed, and the Levenberg-Marquardt algorithm was applied to solve the nonlinear least squares problem. Results from MATLAB simulations and actual robot calibration experiments demonstrate that the proposed self-calibration method converges rapidly and significantly enhances the positioning accuracy of the robot end-effector.  
Methodology, X.W. and W.R.; Software, W.R.; Validation, X.W. and J.L. ([PERSON]); Formal analysis, K.W. and J.Z.; Investigation, X.W.; Resources, J.L. ([PERSON]) and F.L.; Data curation, W.R. and K.W.; Writing--original draft, W.R.; Writing--review & editing, X.W.; Visualization, K.W.; Supervision, J.L. ([PERSON]), J.L. ([PERSON]), J.F. and J.Z.; Project administration, [PERSON] ([PERSON]), J.F. and J.Z.; Funding acquisition, [PERSON] ([PERSON]), J.F. and F.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the infrastructure technology and equipment engineering research center of China State Construction Engineering Corporation (CSCEC-PT-017).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
This research was supported by the infrastructure technology and equipment engineering research center of China State Construction Engineering Corporation (CSCEC-PT-017). Authors [PERSON], [PERSON], [PERSON], [PERSON] were employed by China Construction Second Engineering Bureau Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.  
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