## Overview

We shall introduce three types of effectors (Fig. 1), that is, robotic hands, which can grasp *every* object modeled as a polyhedron, in a *stable* manner. The effectors simply have a planar, cylindrical, or spherical surface for contacting objects; in fact, a wider range of shapes can be used (this result will be published in the near future).

## Immobilizing Objects

The effectors can *immobilize* every object modeled as a polyhedron: an immobilized object can neither translate nor rotate. Fig. 2 shows three types of immobilizing grasps with at most three such effectors; the collection of the grasps is complete in the sense that every polyhedron can be immobilized by at least one of the grasps. For more details, see our paper.

## Caging Objects

The effectors can *cage* every object modeled as a polyhedron: a caged object cannot escape from the grip of the effectors. Fig. 3 shows three types of cages with two such effectors; see how they are related to the immobilizing grasps in Fig. 2.

In the cages of Fig. 3, the effectors are contacting the objects; however, the effectors do not have to make contacts. Suppose that the two effectors, in each panel of Fig. 3, are allowed to move in such a way that they relatively translate along the axis shown in each panel. Then the upper bound of the distance between the two effectors that does not break the cage can be obtained in an analytical (Fig. 4) or empirical (Fig. 5) manner.

Fig 4, 5 suggest that we can quantify the capability of an effector pair in terms of caging objects in a standardized manner, which can also be thought of as a grasp quality measure.

## Stable Grasping

The effectors were used as the modular end-effectors of the manipulator shown in Fig. 6. The manipulator grasps an object by executing a two-stage algorithm: preshaping (caging the object) followed by squeezing (decreasing the distance between the end-effectors). Click here for a video. The distance between the end-effectors can be thought of as a Lyapunov function. We thus physically guarantee Lyapunov stability in grasping by the physical contour of the effectors.