Surface alignment based on the moment of inertia and improved least-squares methods

As part of dimensional inspection and error analysis of components it is us ually required to place the component in a fixture where its position can b e related to its computer aided design (CAD) nominal coordinate axis and th e 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 ac hieving the required alignment. This method, however, works only if the mis alignment between two data sets is very small. Furthermore, there is no mea sure to establish whether this method is likely to converge or not before p erforming the actual iteration. The requirement for a small angle implies t hat 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 irresp ective of the degree of alignment. The problem with the mass property appro ach is that its accuracy diminishes when the error is small. This paper compares the two methods and demonstrates that both the r.m.s. m inimization and the mass property methods can be expressed as eigenvalue pr oblems, and both approaches produce identical eigenvectors despite having d ifferent eigenvalues (error measurements). A method is proposed to determin e whether convergence is expected in the least-squares minimization at the first step of iteration. The proposed method may be used for accelerating t he convergence operation.