The Synthesis of Stable Force-Closure Grasps

This thesis addresses the problem of synthesizing grasps that are force-closure and stable. The synthesis of force-closure grasps constructs independent regions of contact for the fingertips, such that the motion of the grasped object is totally constrained. The synthesis of stable grasps constructs virtual springs at the contacts, such that the grasped object is stable, and has a desired stiffness matrix about its stable equilibrium. A grasp on an object is force-closure if and only if we can exert, through the set of contacts, arbitrary forces and moments on the object. So force-closure implies equilibrium exists because zero forces and moment is spanned. In the reverse direction, we prove that a non-marginal equilibrium grasp is also a force-closure grasp, if it has at least two point contacts with friction in 2D, or two soft-finger contacts or three hard-finger contacts in 3D. Next, we prove that all force-closure grasps can be made stable, by using either active or passive springs at the contacts. The thesis develops a simple relation between the stability and stiffness of the grasp and the spatial configuration of the virtual springs at the contacts. The stiffness of the grasp depends also on whether the points of contact stick, or slide without friction on straight or curved surfaces of the object. The thesis presents fast and simple algorithms for directly constructing stable fore-closure grasps based on the shape of the grasped object. The formal framework of force-closure and stable grasps provides a partial explanation to why we stably grasp objects to easily, and to why our fingers are better soft than hard.

Automatic Recognition of Tractability in Inference Relations

A procedure is given for recognizing sets of inference rules that generate polynomial time decidable inference relations. The procedure can automatically recognize the tractability of the inference rules underlying congruence closure. The recognition of tractability for that particular rule set constitutes mechanical verification of a theorem originally proved independently by Kozen and Shostak. The procedure is algorithmic, rather than heuristic, and the class of automatically recognizable tractable rule sets can be precisely characterized. A series of examples of rule sets whose tractability is non-trivial, yet machine recognizable, is also given. The technical framework developed here is viewed as a first step toward a general theory of tractable inference relations.

The Synthesis of Stable Grasps in the Plane

This paper addresses the problem of synthesizing stable grasps on arbitrary planar polygons. Each finger is a virtual spring whose stiffnes and compression can be programmed. The contacts between the finger tips and the object are point contacts without friction. We prove that all force-closure grasps can be made stable, and it costs 0(n) time to synthesize a set of n virtual springs such that a given force closure grasp is stable. We can also choose the compliance center and the stiffness matrix of the grasp, and so choose the compliant behavior of the grasped object about its equilibrium. The planning and execution of grasps and assembly operations become easier and less sensitive to errors.

Motion Planning with Six Degrees of Freedom

The motion planning problem is of central importance to the fields of robotics, spatial planning, and automated design. In robotics we are interested in the automatic synthesis of robot motions, given high-level specifications of tasks and geometric models of the robot and obstacles. The Mover's problem is to find a continuous, collision-free path for a moving object through an environment containing obstacles. We present an implemented algorithm for the classical formulation of the three-dimensional Mover's problem: given an arbitrary rigid polyhedral moving object P with three translational and three rotational degrees of freedom, find a continuous, collision-free path taking P from some initial configuration to a desired goal configuration. This thesis describes the first known implementation of a complete algorithm (at a given resolution) for the full six degree of freedom Movers' problem. The algorithm transforms the six degree of freedom planning problem into a point navigation problem in a six-dimensional configuration space (called C-Space). The C-Space obstacles, which characterize the physically unachievable configurations, are directly represented by six-dimensional manifolds whose boundaries are five dimensional C-surfaces. By characterizing these surfaces and their intersections, collision-free paths may be found by the closure of three operators which (i) slide along 5-dimensional intersections of level C-Space obstacles; (ii) slide along 1- to 4-dimensional intersections of level C-surfaces; and (iii) jump between 6 dimensional obstacles. Implementing the point navigation operators requires solving fundamental representational and algorithmic questions: we will derive new structural properties of the C-Space constraints and shoe how to construct and represent C-Surfaces and their intersection manifolds. A definition and new theoretical results are presented for a six-dimensional C-Space extension of the generalized Voronoi diagram, called the C-Voronoi diagram, whose structure we relate to the C-surface intersection manifolds. The representations and algorithms we develop impact many geometric planning problems, and extend to Cartesian manipulators with six degrees of freedom.

Passive and Active Grasping with a Prehensile Robot End-Effector

This report presents a design of a new type of robot end-effector with inherent mechanical grasping capabilities. Concentrating on designing an end-effector to grasp a simple class of objects, cylindrical, allowed a design with only one degree of actuation. The key features of this design are high bandwidth response to forces, passive grasping capabilities, ease of control, and ability to wrap around objects with simple geometries providing form closure. A prototype of this mechanism was built to evaluate these features.

Analysis and Implementation of Robust Grasping Behaviors

This thesis addresses the problem of developing automatic grasping capabilities for robotic hands. Using a 2-jointed and a 4-jointed nmodel of the hand, we establish the geometric conditions necessary for achieving form closure grasps of cylindrical objects. We then define and show how to construct the grasping pre-image for quasi-static (friction dominated) and zero-G (inertia dominated) motions for sensorless and sensor-driven grasps with and without arm motions. While the approach does not rely on detailed modeling, it is computationally inexpensive, reliable, and easy to implement. Example behaviors were successfully implemented on the Salisbury hand and on a planar 2-fingered, 4 degree-of-freedom hand.