CaltechTHESIS
A Caltech Library Service

# Two and Three Finger Caging of Polygons and Polyhedra

## Citation

Allen, Thomas F. (2016) Two and Three Finger Caging of Polygons and Polyhedra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z93X84KR . https://resolver.caltech.edu/CaltechTHESIS:12062015-164238181

## Abstract

Multi-finger caging offers a rigorous and robust approach to robot grasping. This thesis provides several novel algorithms for caging polygons and polyhedra in two and three dimensions. Caging refers to a robotic grasp that does not necessarily immobilize an object, but prevents it from escaping to infinity. The first algorithm considers caging a polygon in two dimensions using two point fingers. The second algorithm extends the first to three dimensions. The third algorithm considers caging a convex polygon in two dimensions using three point fingers, and considers robustness of this cage to variations in the relative positions of the fingers.

This thesis describes an algorithm for finding all two-finger cage formations of planar polygonal objects based on a contact-space formulation. It shows that two-finger cages have several useful properties in contact space. First, the critical points of the cage representation in the hand’s configuration space appear as critical points of the inter-finger distance function in contact space. Second, these critical points can be graphically characterized directly on the object’s boundary. Third, contact space admits a natural rectangular decomposition such that all critical points lie on the rectangle boundaries, and the sublevel sets of contact space and free space are topologically equivalent. These properties lead to a caging graph that can be readily constructed in contact space. Starting from a desired immobilizing grasp of a polygonal object, the caging graph is searched for the minimal, intermediate, and maximal caging regions surrounding the immobilizing grasp. An example constructed from real-world data illustrates and validates the method.

A second algorithm is developed for finding caging formations of a 3D polyhedron for two point fingers using a lower dimensional contact-space formulation. Results from the two-dimensional algorithm are extended to three dimension. Critical points of the inter-finger distance function are shown to be identical to the critical points of the cage. A decomposition of contact space into 4D regions having useful properties is demonstrated. A geometric analysis of the critical points of the inter-finger distance function results in a catalog of grasps in which the cages change topology, leading to a simple test to classify critical points. With these properties established, the search algorithm from the two-dimensional case may be applied to the three-dimensional problem. An implemented example demonstrates the method.

This thesis also presents a study of cages of convex polygonal objects using three point fingers. It considers a three-parameter model of the relative position of the fingers, which gives complete generality for three point fingers in the plane. It analyzes robustness of caging grasps to variations in the relative position of the fingers without breaking the cage. Using a simple decomposition of free space around the polygon, we present an algorithm which gives all caging placements of the fingers and a characterization of the robustness of these cages.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:caging, grasping, robotics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Mechanical Engineering
Thesis Availability:Public (worldwide access)
• Burdick, Joel Wakeman
Thesis Committee:
• Burdick, Joel Wakeman (chair)
• Murray, Richard M.
• Pellegrino, Sergio
• Perona, Pietro
• Hunt, Melany L.
Defense Date:27 October 2015
Non-Caltech Author Email:tomfallen (AT) yahoo.com
Record Number:CaltechTHESIS:12062015-164238181
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:12062015-164238181
DOI:10.7907/Z93X84KR
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/ICRA.2012.6225281Related ItemConference paper: Two-fingered caging of polygons via contact-space graph search
http://dx.doi.org/10.1109/ICRA.2014.6907125Related ItemConference paper: Two-fingered caging of 3D polyhedra using contact space search
http://dx.doi.org/10.1109/ICRA.2015.7139795Related ItemConference paper: Robust three-finger three-parameter caging of convex polygons
http://dx.doi.org/10.1109/TRO.2015.2463651Related ItemArticle: Two-finger caging of polygonal objects using contact space search
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9307
Collection:CaltechTHESIS
Deposited By: Thomas Allen
Deposited On:18 Dec 2015 18:24

## Thesis Files

 Preview
PDF (Thesis) - Final Version
See Usage Policy.

6MB
 Preview
PDF (Bibliography of published contributions) - Supplemental Material
See Usage Policy.

40kB

Repository Staff Only: item control page