We present a unified geometric theory for localization of three types of workpieces: 1) general three-dimensional (3D) workpieces where points from the finished surfaces fully constrain the rigid motions of the workpieces; 2) symmetric workpieces; 3) partially machined workpieces where points from the finished surfaces are inadequate to fully constrain the rigid motions of the workpieces. Applications of the study include workpiece setup, refixturing and dimensional inspections in a flexible manufacturing environment. First, we formulate the localization problem for a general 3D workpiece and study the mathematical properties of the underlying problem. We discuss an iterative approach for solving the general localization problem and show how different considerations in updating the Euclidean transformation lead to various geometric algorithms. Then, we extend the localization techniques to symmetric workpieces and partially machined workpieces and present a simple algorithm for each of the problems. Finally, we present simulation results showing convergence and robustness properties of the various geometric algorithms.
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