Least-squares Fit of Measured Points for Square Line-profiles

摘要

The pseudoinverse of a rectangular matrix is used to compute the least-squares fit of a set of points that have been measured along a line-profile. Tolerances on line profiles are used to control cross-sectional shapes of parts, such as turbine blades. The specified profile is treated as a moving platform of a hypothetical, redundant, and planar in-parallel-actuated robot, and all the measured points are presumed to be fixed in it. The locations of the linear actuators are represented with screw (torsor) coordinates, and these are arranged in a matrix equation that relates the three small displacements of the platform to the corresponding deviations (treated as small displacements) of the measured points. The Moore-Penrose (pseudoinverse) solution uniquely produces displacements of the platform which correspond to the least-squares minimum for the deviations at all of the measured points.