Computation and Analysis of Natural Compliance in Fixturing and Grasping Arrangements

Abstract

Abstract-This paper computes and analyzes the natural compliance of fixturing and grasping arrangements. Traditionally, linear-spring contact models have been used to determine the natural compliance of multiple contact arrangements. However, these models are not supported by experiments or elasticity theory. We derive a closed-form formula for the stiffness matrix of multiple contact arrangements that admits a variety of nonlinear contact models, including the well-justified Hertz model. The stiffness matrix formula depends on the geometrical and material properties of the contacting bodies and on the initial loading at the contacts. We use the formula to analyze the relative influence of first-and second-order geometrical effects on the stability of multiple contact arrangements. Second-order effects, i.e., curvature effects, are often practically beneficial and sometimes lead to significant grasp stabilization. However, in some contact arrangements, curvature has a dominant destabilizing influence. Such contact arrangements are deemed stable under an all-rigid body model but, in fact, are unstable when the natural compliance of the contacting bodies is taken into account. We also consider the combined influence of curvature and contact preloading on stability. Contrary to conventional wisdom, under certain curvature conditions, higher preloading can increase rather than decrease grasp stability. Finally, we use the stiffness matrix formula to investigate the impact of different choices of contact model on the assessment of the stability of multiple contact arrangements. While the linear-spring model and the more realistic Hertz model usually lead to the same stability conclusions, in some cases, the two models lead to different stability results.

On the mechanics of natural compliance in frictional contacts and its effect on grasp stiffness and stability

“…Let us first describe a generic normal compliance modeling approach which ignores the details of compliant surface deformation and models the resultant contact force as a function of B's configuration [23]. Consider a single contact between B and A i .…”

“…When the preload level is too high this term can possibly destabilize the grasp, a phenomena known as "coin snapping" in the grasping literature [6,23,28]. The third term represents the overlapsegment change of direction due to small displacements of B.…”

“…Moreover, it is intuitively clear (and rigorously justified below) that the contacts experience tangential stiffening as the bodies are pressed harder together. Subsequent papers focused on the inclusion of the Hertz normal compliance model into the analysis and synthesis of frictionless grasps and fixtures [18,23,24,50]. Systematic efforts at incorporating frictional compliance effects were made by Sinha and Abel [37], Li and Melkote [22], and Wang and Kumar [46].…”

Stability analysis of multiple objects grasped by multifingered hands with revolute joints in 2D

“…He investigated planar and spatial frictional grasps by replacing each finger with a two-or threedimensional (2D or 3D) orthogonal spring model and derived a grasp stiffness matrix. Funahashi et al [4], Howard and Kumar [5], Lin et al [6], Yamada et al [7] explored curvature effect at each contact point. Yamada et al [9] investigated the effect of contact surface geometry (metric tensor, curvature, torsion).…”

Stability analysis of a single object grasped by a multifingered hand with angular joints in 2D

“…He investigated planar and spatial frictional grasps by replacing each finger with a two-or threedimensional (2D or 3D) orthogonal spring model and derived a grasp stiffness matrix. Funahashi et al [4], Howard and Kumar [5], and Lin et al [6] explored curvature effect at each contact point.…”