From caging to grasping

This paper digs into the relationship between cages and grasps of a rigid body. In particular, it considers the use of cages as waypoints to grasp an object. We introduce the concept of pregrasping cages, caging configurations from which an object can be grasped without first breaking the cage. For two-fingered manipulators, all cages are pregrasping cages and, consequently, useful waypoints to grasp an object. A contribution of this paper is to show that the same does not hold for more than two fingers. A second contribution is to show how to overcome that limitation. We explore the natural generalization of the well-understood squeezing/stretching characterization of two-finger cages to arbitrary workspace dimension, arbitrary object shapes without holes, and arbitrary number of point fingers, and exploit it to give sufficient conditions for a cage to be a pregrasping cage. As a product of that generalization, we introduce grasping functions: scalar functions defined on the finger formation that control the process of going from a cage to a grasp. We finish the paper by establishing an analogy between the role of grasping functions in grasping and that of Lyapunov functions in stability theory.