An algebraic algorithm for workpiece localization

Presents an algebraic algorithm for workpiece localization. First, we formulate the problem as a least-square problem in the configuration space Q=SE(3)×R³ⁿ, 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, the authors use the geometric properties of the Euclidean group to compute for the critical points of the objective function. Doing so the authors derive an algebraic formula for the optimal Euclidean transformation in terms of the measurement points and the corresponding home surface points. The authors also give for each measurement point a system of two nonlinear equations from which the corresponding home surface point nearest to the measurement point can be solved. Finally, based on these analytic results the authors present an iterative algorithm for obtaining the complete solution of the least-square problem