Abstract
Abstract
As part of dimensional inspection and error analysis of components it is usually required to place the component in a fixture where its position can be related to its computer aided design (CAD) nominal coordinate axis and the coordinate frame of the measuring system. The fixturing can be expensive and does not completely eliminate the mathematical matching needed between measured and nominal surfaces.
Least-squares minimization is one of the most common methods employed in achieving the required alignment. This method, however, works only if the misalignment between two data sets is very small. Furthermore, there is no measure to establish whether this method is likely to converge or not before performing the actual iteration. The requirement for a small angle implies that this method is only suitable if fixturing is also used.
The other technique used in obtaining alignment is by consideration of the mass properties of surfaces. This method is more effective and works irrespective of the degree of alignment. The problem with the mass property approach is that its accuracy diminishes when the error is small.
This paper compares the two methods and demonstrates that both the r. m.s. minimization and the mass property methods can be expressed as eigenvalue problems, and both approaches produce identical eigenvectors despite having different eigenvalues (error measurements). A method is proposed to determine whether convergence is expected in the least-squares minimization at the first step of iteration. The proposed method may be used for accelerating the convergence operation.