Caging planar objects with a three-finger one-parameter gripper

This paper extends the caging theory of Rimon and Blake (1996) for one-parameter two-finger grippers to one-parameter three-finger grippers. The caging theory describes how the fingers of a robot gripper can be placed around an object so that it cannot escape. As the fingers close from such a configuration, the freedom of the object to move is gradually restricted, until, in the absence of friction, it is completely immobilized by the fingers at an immobilizing grasp. The extension of the caging theory to three-fingered grippers is important because convex objects cannot be caged by two-finger grippers. The computation of the set of caging formations requires the identification of both the two- and three-finger frictionless grasps. These grasps correspond to critical points of the opening parameter in the gripper´s configuration space. There are two main problems here. Firstly, a method is needed to compute the critical points of the opening parameter in the three-finger contact space. This is complicated by the fact that this space may have more that one connected component. The second problem is how to associate the immobilizing grasps with punctures. In this paper we solve both these problems, presenting efficient algorithmic solutions