A geometric algorithm for symmetric workpiece localization

This paper introduces a geometric algorithm for symmetric workpiece localization. First, we formulate the localization problem as a geometric problem in the configuration space Q=SE(3)timesR³ⁿ, where SE(3) is the Euclidean group, and n is the number of measurement points to be matched by corresponding home surface points of the workpiece. Then, we compute for the critical points of the objective function. So we derive a geometric formula for the optimal Euclidean transformation in terms of the measurement points and the corresponding home surface points. The corresponding home surface point is given by two nonlinear equations. Each measurement point and its corresponding home surface point nearest to the measurement point are solved in every iterations. For symmetric features, their configuration spaces are identified with the quotation space of SE(3). Finally, based on these results an iterative algorithm for obtaining the complete solution is addressed and the proof is given. Experimental results show the algorithm is more efficient than the variation algorithm, which is an algebraic algorithm.