Unilateral Fixtures for Sheet-Metal Parts With Holes

In this paper, we introduce unilateral fixtures, a new class of fixtures for sheet-metal parts with holes. These fixtures use cylindrical jaws with conical grooves that facilitate part alignment; each jaw provides the equivalent of four point contacts. The fixtures are unilateral in the sense that their actuating mechanisms are restricted to one side/surface of the part, facilitating access to the other side/surface for assembly or inspection. We present a two-phase algorithm for computing unilateral fixtures. Phase I is a geometric algorithm that assumes the part is rigid and applies two-dimensional (2-D) and three-dimensional (3-D) kinematic analysis of form closure to identify all candidate locations for pairs of primary jaws. We prove three new grasp properties for 2-D and 3-D grips at concave vertices and define a scale-invariant quality metric based on the sensitivity of part orientation to infinitesimal relaxation of jaw position. Phase II uses a finite element method to compute part deformation and to arrange secondary contacts at part edges and interior surfaces. For a given sheet-metal part, given as a 2-D surface embedded in 3-D with $e$ edges, $n$ concavities and $m$ mesh nodes, Phase I takes $O(e + n^{4/3} \log^{1/3}n + g \log g)$ time to compute a list of $g$ pairs of primary jaws ranked by quality. Phase II computes the location of $r$ secondary contacts in $O(grm^{3})$ time. Note to Practitioners—This paper was motivated by the problem of holding sheet-metal parts for automobile bodies but it also applies to other sheet-metal components that have cut or stamped holes. Existing approaches to fixturing such parts generally have contacting mechanisms on both sides of the sheet that restrict access for welding or inspection. This paper suggests a new approach using pairs of grooved cylinders, activated from only one side of the part (hence “unilateral”). These cylinders mate with opposing corners of holes in the sheet and push apart to hold the sheet in tension, thus acting as both locators and clamps. In this paper, we mathematically characterize the mechanics and conditions for a unilateral fixture to hold a given part. We then show how such fixtures can be efficiently computed; this can allow a computer-aided design (CAD) system (with finite element capability) to automatically generate and propose unilateral fixtures for a given part. Preliminary physical experiments suggest that this approach is feasible but it has not yet been incorporated into a CAD system nor tested in production. In future research, we will address the design of unilateral fixtures that hold two or more parts simultaneously for welding.