Comparison of the closed analytical solution on the parameters of the alignment and surveying of the elements of accelerator under random errors

Purpose Compared with the iterative approximation solution, the analytically rigorous solution of the parameters of the accelerator alignment and surveying is the closed theoretical solution. To make the calculated error meet the negligible principle, in this paper, such examples of the zero, small angle and small translation, only large translation and no angle, only large angle without translation and large translation and large angle parameters under random errors are constructed. Methods Furthermore, Singular Value Decomposition, Eigen Value Decomposition, Polar Decomposition, Orthogonal Procrustes Analysis, and Successive Equivalent Differences Transformation are compared. Results In all the above examples, the mutual calculation deviation of Singular Value Decomposition, Eigen Value Decomposition, Polar Decomposition, and Orthogonal Procrustes analysis methods is less than 10 um. Its calculation accuracy is the highest, and Successive Equivalent Differences Transformation-3 has the largest calculation error, and Successive Equivalent Differences Transformation-N is in the middle. The results show that the coordinate residuals, parameters offsets and point position errors of the above solutions with random errors are all nearly zero. Conclusion It shows that the closed analytical solution is a rigorous solution. Moreover, it also shows that the constructed method of the coordinate point pairs is correct.