Friction Cones and Force Closure

A finger can only push, never pull, and friction lets it push a little sideways too, inside a cone around the surface normal. A grasp is in force closure when the fingers together can resist any force and any twist on the object. That is the classic test for a secure grasp.

Friction coefficient μ Number of fingers

Drag the fingers along the object.

The friction cone

With Coulomb friction coefficient $\mu$, a contact force $\mathbf{f}$ with normal part $f_n$ and tangential part $f_t$ is allowed only inside a cone of half-angle $\arctan\mu$:

$$\lvert f_t\rvert \le \mu\, f_n \qquad \text{half-angle } \alpha = \arctan\mu$$

Force closure

In 2D, each cone edge gives a wrench $(f_x, f_y, \tau)$. The grasp is in force closure when these wrenches together can make any wrench, which happens exactly when the origin lies strictly inside their convex hull. This page checks that numerically over many directions.

What to notice

Two fingers on opposite faces achieve force closure only if the line between them lies inside both cones. With low friction the cones are thin, so the fingers must line up almost exactly, and with $\mu = 0$ two fingers can never succeed. Add a third finger and a frictionless grasp becomes possible on many shapes. Grasp planners use this test, or quality scores built from it, to rank candidate grasps.