A procedure for the evaluation and compensation of form errors by means of global isometric registration with subsequent local reoptimization

Stresses remaining in the component after sheet metal forming processes can result in complex form errors, such as springback and torsions. In order to compensate these process-induced deformations, the local and global deformations have to be analyzed. Hence, an appropriate comparison between the actually manufactured and the target design is required. For this purpose, the surface of the actual workpiece is scanned and the so-obtained scan points have to be assigned to corresponding points of the target shape defined by the workpiece model. From these correspondences, a field of deformation vectors can be computed which represents the basis for the compensation strategy. The task of finding appropriate correspondences is called registration. It is usually solved using rigid transformations, i.e., translation and rotation. Due to the locality, strength and complexity of the deformations, rigid transformations are usually not sufficient. As a more flexible alternative, a procedure for non-rigid registration is presented in this paper. Therein, isometry, i.e., the conservation of distances between corresponding points within an appropriate neighborhood structure, is defined as the objective function. The procedure consists of three steps: definition of the neighborhood structure, global registration, and local reoptimization. The main focus of the paper is set to the latter, where an adapted gradient descent method also allowing projections into the triangles of the target shape is presented and experimentally validated. With these three steps, an assignment between both shapes can be calculated, even for strong local deformations and coarse triangular meshes representing the workpiece model.