On the symmetric location problem

Accurate and efficient localization of symmetric features plays an important role in dimensional inspection of machined parts and machining of partially finished workpieces. We present a geometric theory for efficient and accurate localization of symmetric features. First, we show that the configuration space of a symmetric feature can be naturally identified with the homogeneous space SE_(3/)G_o of the Euclidean group SE_{(3/), where G}sub o/ is the symmetry group of the feature. Then, we explore the geometric structure of the homogeneous space and present a simple and unifying algorithm for symmetric localization. Finally, we give simulation results illustrating several unique features of the algorithm: 1) implementational simplicity; 2) robustness with respect to initial conditions; 3) high accuracy in computed results; and 4) computational efficiency