The Geometry of Grasping

A form closure (or complete restraint) of a solid object is a finite set of wrenches (force-moment combinations) applied on the object, with the property that any other wrench acting on the object can be balanced by a positive combination of the original ones. Intuitively, form closure is a way of defining the notion of a "firm grip" of the object (when friction is not taken into account). It has been pointed out by Reuleaux (1875) and Somoff (1897), and more recently by Lakshmin arayana (1978), that the form closure of a two-dimensional object requires at least four wrenches, and that form closure of a three-dimensional object requires at least seven wrenches. It was also conjectured that these numbers can be achieved by wrenches realizable as forces normal to the surface of the object (such wrenches are called fingers). In this paper we prove this conjecture. In particular we show that form closure of any two-dimensional bounded object with piecewise smooth boundary (except a circle) can be achieved by four fingers. For three dimensions, we prove that form closure of any bounded object with piecewise smooth boundary can be achieved with 12 fingers if and only if the object does not have a rotational symmetry (in which case, of course, form closure is not achievable, since no moment along the axis of symmetry can be opposed). We also show that, under very general conditions, form closure of three-dimensional objects without rotational symmetries can be achieved with seven fingers. Finally, we show that when Coulomb friction is taken into account, under the most relaxed assumptions three fingers are necessary and sufficient in two dimensions, and four fingers in three dimensions.