Constrained Least-Squares Fitting for Tolerancing and Metrology

Recent years have seen a remarkable emergence of a particular type of least-squares fitting, called constrained least-squares fitting, in national and international standards on tolerancing and metrology. Fitting, which is called an association operation in the international standards on tolerancing and related metrological practices, is an optimization process that associates an ideal-form feature to a non-ideal-form feature. Unconstrained least-squares fitting has been a very well-known practice in general metrology for a long time, but it has not been the standardized operation in tolerancing and related metrology standards till recently. Instead, other fitting methods, such as Chebyshev fitting, have dominated the standards thus far. Now constrained least-squares is emerging as an attractive alternative in these standards, especially for the establishment of datums. This paper describes these recent developments, with particular attention to the mathematical and computational aspects of the optimization problem, and their impact on the digitization of industrial metrology.