Fixturing hinged polygons

We study the problem of fixturing a chain of hinged objects in a given placement with frictionless point contacts. We define the notions of immobility and robust immobility - which are comparable to the second and first order immobility for a single object - to capture the intuitive requirement for the fixture of a chain of hinged objects. Robust immobility differs from immobility in that it additionally requires insensitivity to small perturbations of contacts. We show that (p+2) frictionless point contacts can immobilize any chain of p/spl ne/3 polygons without parallel edges; six contacts can immobilize any chain of three such polygons. Any chain of p arbitrary polygons can be immobilized with at most (p+4) contacts. We also show that /spl lceil/(6/5)(p+2)/spl rceil/ contacts suffice to robustly immobilize p polygons without parallel edges, and that /spl lceil/(5/4)(p+2)/spl rceil/ contacts can robustly immobilize p/spl ne/3 arbitrary polygons, and eight contacts can robustly immobilize three polygons.

1 Introduction

Many manufacturing operations, such as machining and assembly, require the parts that are subjected to these operations to be fixtured, i.e., to be held in such a way that they can resist all external wrenches. Fixturing is a problem that is studied extensively, see e.g. [2], [3], [5], [14], [15], [16]. We consider the planar version of part fixturing (or immobilization), which appears e.g. in preventing all sliding motions of a part resting on a table. The concept of form closure, formulated by Reuleaux [8] in 1876, provides a sufficient condition for constraining, despite the application of possible external wrenches, all finite and infinitesimal motions of a rigid part by a set of contacts along its boundary. Any motion of a part in form closure has to violate the rigidity of the contacts. Markenscoff et a1. [6] and Mishra et al. [7] independently showed that-in the absence of friction-four point contacts are sufficient and often necessary to put any polygonal object in form closure. In fact their result applies to almost any planar rigid part.