The Application of the Principal Curve Analysis Technique to Smooth Beam Lines

The smoothness of a beam line refers to the quality of the relative positioning of a number of adjacent beam guiding components. The fact that smoothness is of highest priority when positioning magnets can be seen in the local tolerances imposed by the beam optics. In the past, smoothing has been done by separating horizontal and vertical misalignments and then applying some sort of analytical or manual ''feathering'' technique. The Stanford Linear Collider (SLC) did not easily lend itself to this sort of smoothing because of the highly coupled nature of its pitched and rolled beam line. This paper will discuss an attempt to develop a repeatable method which is independent of the inconsistencies of human judgment and can simultaneously smooth in two or more dimensions. Four major goals were defined for the smoothing algorithm used on the SLC alignment. The first, was to simultaneously model errors for both horizontal and vertical directions. Secondly, a smooth curve whose shape was suggested by the data and not by a predetermined model was implied by the fact that unknown systematic errors were being eliminated. Thirdly, this curve must be a reproducibly fit, independent of the inconsistent nature of human judgment. Fourth, themore » result of the procedure was to minimize the number and size of magnet movements to reach the final alignment criteria.« less