"Unilateral" fixturing of sheet metal parts using modular jaws with plane-cone contacts

To fixture sheet metal parts for welding, we propose "unilateral fixtures" consisting of modular fixturing elements that lie almost completely on one side of the part. These are based on cylindrical jaws with conical grooves which provide the equivalent of 4 point contacts. We propose a two-phase procedure for designing unilateral fixtures. The first phase is a geometric algorithm that assumes the part is rigid and computes vg-grips (vertex-groove grips). The vg-grip algorithm uses a fast sufficient test for immobility to generate a list of vg-grips and find bounds on jaw cone angles for each. The second phase is a fast heuristic procedure that uses FEM to arrange secondary contacts to reduce part deformation. For a part described by n concave "virtual vertices", a list of vg-grips and minimum half cone angles for each vg-grip can be generated in O(n²) time. We also propose a quality metric based on the sensitivity of the part\´s orientation to an infinitesimal relaxation of the jaws that can be evaluated in constant time for a given fixture. For an FEM model with m nodes, the second phase takes O(m³r) time to arrange r secondary contacts for each vg-grip.