Coordinate metrology is an important branch of modern manufacturing science. Workpiece localization and mathematization of the tolerancing notations contained in ANSI Y14.5M are two fundamental problems in coordinate metrology. This dissertation is concerned with theory and algorithms for coordinate metrology. The contributions of the thesis are two-fold. First, by making use of tools from differential geometry and Lie group theory, we provide a unified treatment for localization of symmetric features of all types; second, we extend this powerful geometric tool to give a precise formulation of the tolerancing notations contained in ANSI Y14.5M standard and develop efficient algorithms for tolerance evaluation. The theory is based on an important observation that a symmetric feature has...[ Read more ]

Coordinate metrology is an important branch of modern manufacturing science. Workpiece localization and mathematization of the tolerancing notations contained in ANSI Y14.5M are two fundamental problems in coordinate metrology. This dissertation is concerned with theory and algorithms for coordinate metrology. The contributions of the thesis are two-fold. First, by making use of tools from differential geometry and Lie group theory, we provide a unified treatment for localization of symmetric features of all types; second, we extend this powerful geometric tool to give a precise formulation of the tolerancing notations contained in ANSI Y14.5M standard and develop efficient algorithms for tolerance evaluation. The theory is based on an important observation that a symmetric feature has a symmetry subgroup G₀ under the action by the group SE(3) of Euclidean motions in [double-struck R]³. Thus, the configuration space of a symmetric feature can be identified with the homogeneous space SE(3)/G₀. By exploring the geometric structure of SE(S)/G₀, we develop a simple and efficient algorithm, called the Fast Symmetric Localization (FSL) algorithm, for localization of symmetric features. Second, we examine tolerancing notations contained in the ANSI Y14.5M (also ANSI Y14.5.1M) standard. By extending the geometric tools developed for symmetric localization, we give precise formulations for all cases of form, profile and orientation tolerances as appropriate minimization problems in the homogeneous spaces of SE(3). For orientation tolerances, suitable defined constraints can be incorporated. We present a simple and unified algorithm, called the Symmetric Minimum Zone (SMZ) algorithm for verification of form, profile and orientation tolerances. Finally, we address the problem of establishing a datum reference frame which is a coordinate system used to locate and orient part features. The problem can be posed as a minimization problem in SE(3)IG₀. We give conditions under which a datum feature qualifies to be a secondary or a tertiary datum. We present a sequential procedure that transforms the primary, secondary and tertiary datum problems as a minimization or a constrained minimization problem in the homogeneous spaces of SE(3). We develop simple algorithms to solve these problems and thus the problem of establishing datum reference frames from actual datum features.