We extend the form closure framework for rigid parts to holding a class of deformable parts. In this class, a part is a linearly elastic, frictionless polygon with a finite element mesh and given stiffness matrix. We define the D-space (deformation-space) of a part as the C-space of all its mesh nodes. We define the free space as the intersection of the set of topology preserving mesh configurations with the complement of the union of all D-obstacles that represent collisions of the part with finger bodies. Consider a given set of finger bodies in frictionless contact with a part. When positive work is needed to release the part, we say that it is in deform closure. We present a numerical example and prove two results: (1) if contact set holds a rigid part in form closure, it will hold the equivalent deformable part in deform closure; and (2) deform closure is frame invariant.

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/sup 4/3/log/sup 1/3/n+glogg) 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/sup 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.

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(n/sup 2/) 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(m/sup 3/r) time to arrange r secondary contacts for each vg-grip.

A simple gripper with two vertical cylindrical jaws can make contact with external or internal concavities in polygonal and polyhedral parts to align and grip parts in form closure. This is called a /spl nu/-grip. We begin by defining 2D /spl nu/-grips, where a pair of frictionless point jaws makes contact with a pair of polygonal part concavities to achieve form-closure. We define a /spl nu/-grip quality metric based on the maximum possible change in the part's orientation when jaw position is relaxed infinitesimally. For a polygonal part with polygonal holes, we give an algorithm for computing and ranking 2D /spl nu/-grips. We also extend the definition to jaws with non-zero radii. In 3D, /spl nu/-grips are achieved with a pair of frictionless vertical cylinders. We define 3D /spl nu/-grips and give a numerical algorithm for computing all 3D /spl nu/-grips of a polyhedral part. If n is the number of vertices that describe the part and k is the number of concave vertices, we can compute all 2D /spl nu/-grips in O(n+k/sup 2/) time. Measures of complexity are given for computing offsets for jaws with nonzero radii, a ranked list of 2D /spl nu/-grips based on the quality metric, and all 3D /spl nu/-grips. A Java implementation of the 2D algorithm is available.