On the Hybrid Workpiece Localization/Envelopment Problems

Hybrid workpiece localization/envelopment plays an important role in automatic setup of workpieces. In this paper, we de ne hybrid workpiece localization/envelopment problems to partially nished workpieces. We rst analyze symmetric properties of nished surfaces of a workpiece and con guration space. We localize the workpiece with measurement points from the nished surfaces so as to obatin a representative transformation of the CAD frame of the workpiece relative to a reference frame. We then form a parameterized family of Euclidean transformations based on the representative transformation and the symmetry group of the nished surfaces. With this family of Euclidean transformations, we formulate the envelopment problem as a nonlinear optimization problem subject to nonlinear inequality constraints. Based on these formulations, we derive a uni ed and explicit solution for the envelopment problems with the duality problem in nonlinear programming. Finally we present simulation results. The results show that the hybrid workpiece localization/envelopment method can determine a satisfactory transformation which satis es not only the prole tolerance zones required by the nished surfaces, but also envelopment requirements of the un nished surfaces.