Minimal Fixturing of Frictionless Assemblies: Complexity and Algorithms

Abstract

In many assembly tasks it is necessary to ensure the stability of a subcollection of contacting objects. To achieve stability, it is often necessary to introduce fixture elements (also called ``fingers'' in some work) to help hold objects in place. In this paper the complexity of stabilizing multiple contacting bodies with the fewest number of fixture elements possible is considered. Standard fixture elements of the type explored in previous single-object grasping work are considered, along with a generalized variant of fixture elements. Both form-closure (complete immobility of the assembly), and first-order stability (stability of an assembly in the neighborhood of a specific external force and torque on each body) are considered. The major result is that for three of the four combinations of fixture element varieties and stability considered, achieving an optimal solution (that is, finding a smallest set of fixture elements yielding stability) is NP-hard. However, for many fixturing problems it seems likely that suboptimal, yet acceptably small solutions can be found in polynomial time, and some candidate algorithms are presented.