Immobilizing polygons against a wall

Abstract

A familiar task in industrial applications is grasping an object to con- strain its motions. When the external forces and torques acting on the object are uncertain or varying, form-closure grasps are preferred; these are grasps that constrain all innitesimal and nite motion of the object. Much of previous work on computing form-closures has involved achieving it with point-contacts; for a planar object, four point-contacts were proven to be necessary and sucient. Inspired by the intuitive habit of supporting an object against something at to immobilize it, in this paper we propose a new class of contacts called edge-contacts; these oer a straight-line sup- port against which the object rests. Our rst result is that almost any polygonal part can be constrained in form-closure with an edge-contact and two point-contacts. A related problem is that of immobilizing an object with modular x- tures. These typically comprise of a regular lattice of holes on which the object is placed and a set of precision contacting elements or xels whose locations are constrained by the grid. Given the input object mod- eled as an n-gon, the grid size, and the shapes of the xels available, we present an algorithm that returns all valid modular xtures, each ac- complished by an edge-xel, a locator-xel and a clamp. The algorithm uses sophisticated data-structures to achieve output-sensitivity; it runs in O(n(n+p) 4 = 3+ +K) time, where K is the number of valid solutions (xture congurations) returned, and p is the object's perimeter in grid units.