Gripping Parts at Concave Vertices

Ken Goldberg, and K.Gopalakrishnan
<Initiated: August, 2000> 
ALPHA Lab, University of California, Berkeley. 
 

Examples of v-grips.

Abstract

A simple gripper with two vertical cylindrical jaws can make contact with external or internal concavities in polygonal and polyhedral parts to align and grip parts in form closure. Such grippers are inexpensive, lightweight, and their small footprint facilitates access and insertion for industrial applications.

We refer to this as a v-grip. We begin by defining 2D v-grips, where a pair of frictionless point jaws makes contact with a pair of polygonal part concavities to achieve form-closure. We define a v-grip quality metric based on the maximum possible change in the part’s orientation when jaw position is relaxed infinitesimally. For a polygonal part with polygonal holes, we give an algorithm for computing and ranking 2D v-grips. We also extend the definition to jaws with non-zero radii. In 3D, v-grips are achieved with a pair of frictionless vertical cylinders. We define 3D v-grips and give a numerical algorithm for computing all 3D v-grips of a polyhedral part.

If n is the number of vertices that describe the part and k is the number of concave vertices, we can compute all 2D v-grips in O(n+k2) time. Computing offsets for jaws with non-zero radii takes O(n log n) time. A ranked list of 2D v-grips based on the quality metric can be computed in an additional O(k2 log k) time. We find all 3D v-grips in O(n3 k2) time. A Java implementation of the 2D v-grip algorithm is available online for testing.

Introduction

As illustrated in the figure, parts can be gripped with two cylindrical jaws by contracting or expanding the jaws toward a pair of concavities. One frictionless contact at a concave vertex can generate forces equivalent to two contacts at part edges. In this paper, we formalize conditions for, and study the properties of v-grips for rigid polygonal parts with jaws of zero radii. A v-grip is contracting if the jaws move towards each other and expanding if the jaws move away from each other. We analyze v-grips using a distance function. We define a new quality metric based on the maximum possible change in the part’s orientation when jaw position is relaxed infinitesimally. This can be computed efficiently to rank v-grips and is consistent in most cases with physical intuition.

We then consider v-grips in 3D where a given polyhedral part rests on a horizontal planar work-surface under the influence of gravity. We generate all possible 3D v-grips and the part trajectory leading to the v-grip. Using the 2D algorithm, the finding of points on the trajectory is reduced from a 6 dimensional search to a 1 dimensional search.

We have implemented the algorithm in Visual BASIC for zero jaw radius for one polygon. A screenshot of this implementation is shown below. A Java implementation can be found here.

Papers

  • K. Gopalakrishnan, Ken Goldberg, "Gripping Parts at Concave Vertices", to appear in IEEE ICRA 2002.[pdf]

Acknowledge:

    This work was supported in part by the National Science Foundation under DMI-0010069. Research funding was also provided by Adept Technology, and California State MICRO Grant 00-032.