Robust 3D line fitting and its applications in alignment engineering

Abstract Accurate three-dimensional (3D) line fitting to point data is a fundamental task in alignment engineering, particularly in applications such as quadrupole smoothing and undulator shimming. However, conventional fitting methods often neglect errors in certain coordinate components and exhibit limited robustness in the presence of outliers. To overcome these limitations, we formulate the 3D line fitting problem as an errors-in-variables (EIVs) model, explicitly accounting for the statistical characteristics of all observed variables. The EIV model is solved iteratively using a total least squares approach. To further improve robustness, we introduce an equivalent weighting scheme based on standardized residuals, which effectively suppresses the influence of outliers while preserving the correlation structure among observations. The proposed method is validated through three case studies: a simulated line fitting experiment, an elliptical undulator girder measurement, and a quadrupole smoothing task. Comparative results with the existing algorithms and the spatial analyzer software demonstrate the superior robustness and accuracy of the proposed approach.