Mobility of bodies in contact. I. A 2nd-order mobility index for multiple-finger grasps

Abstract

Using a configuration-space approach, this paper develops a novel 2nd-order mobility theory for rigid bodies in contact. A major component of this theory is a coordinate invariant 2nd-order mobility index for a body, B, in frictionless contact with finger bodies A 1 ; 1 1 1 ; A k . The index is an integer that captures the inherent mobility of B in an equilibrium grasp due to second order, or surface curvature, effects. It differentiates between grasps which are deemed equivalent by classical 1storder theories, but are physically different. We further show that 2nd-order effects can be used to lower the effective mobility of a grasped object, and discuss implications of this result for achieving new lower bounds on the number of contacting finger bodies needed to immobilize an object. Physical interpretation and stability analysis of 2nd-order effects are taken up in the companion paper.

Computation and Analysis of Natural Compliance in Fixturing and Grasping Arrangements

“…An essential grasp is defined as a grasp where all fingers must apply a nonzero force in order to positively span the origin. Essential grasps constitute a large class of practical grasps; all generic 2-D grasps with up to four fingers and all generic 3-D grasps with up to seven fingers are essential [30]. In nonessential grasps, the finger-force magnitudes must be determined by direct measurements, rather than by the geometrical procedure described below.…”

On the complexity of the set of three-finger caging grasps of convex polygons

“…Mason's text book [8]), and immobilizing and equilibrium grasps [13]. A part is immobilized by a number of fixed fingers (forming an immobilizing grasp) when any motion of the part violates the rigidity of the part or the fingers.…”

The stability of heavy objects with multiple contacts

“…For an equilibrium grasp, the domain of the gravity relative curvature form, D2dG, is We define the 2 n d -~r d e r gravity mobility index, m:hG), t o be the dimension of the largest possible subspace in on which the gravity relative curvature form KG is non-negative (positive semi-definite). Note that by proposition 4.2, a grasp is stable if An analogy may be made t o the lSt-and 2 n d -~r d e r mobility indices introduced in [8] and [9]. When considering whether an equilibrium y p permits any unstable motions, one can imagine t e gravity equipotential passing through 40 t o be the impenetrable surface of a ( k + ,)st finger with distance function -U ( q ) .…”

“…NC(Q0) -4 1-# 0 (9) Grasps which satisfy equation 7 are equilibrium grasps and qo is the equilibrium configuration. Note that e ui librium grasps implicitly satisfy the condition that 06 span(n1,.…”