BIB-VERSION:: CS-TR-v2.0
ID:: CORNELLCS//TR92-1296
ENTRY:: 1993-10-14
ORGANIZATION:: Cornell University, Computer Science Department
LANGUAGE:: English
TITLE:: The Motion of Planar Compliantly-Connected Rigid Bodies in Contact
        With Applications to Automatic Fastening
AUTHOR:: Donald, Bruce Randall
AUTHOR:: Pai, Dinesh K.  
DATE:: July 1992
PAGES:: 50
NOTES:: Replaces 89-1047, 89-1048, 90-1177
ABSTRACT::
We consider the problem of planning and predicting the motion of a flexible 
object amidst obstacles in the plane. We model the flexible object as a rigid 
``root'' body, attached to compliant members by torsional springs. The root's 
position may be controlled, but the compliant members move in response to 
forces from contact with the environment. Such a model encompasses several 
important and complicated mechanisms in mechanical design and automated 
assembly: snap-fasteners, latches, ratchet and pawl mechanisms, and 
escapements. The problem is to predict the motion of such a mechanism amidst 
fixed obstacles. For example, our algorithm could be used to determine 
whether a snap-fastener design can be assembled with a certain plan.

In this paper, we analyze the physics of these flexible devices, and develop 
combinatorially precise algorithms for predicting their movement under a 
motion plan. Our algorithms determine when and where the motion will 
terminate, and also computes the time-history of contacts and mating forces. 
In addition to providing the first known exact algorithm that addresses 
flexibility in motion planning, we also note that our approach to compliance 
permits an exact algorithm for predicting motions under rotational 
compliance, which was not possible in earlier work.

We discuss the following issues: the relevance of our approach to engineering 
(which we illustrate through the examples we ran using our system), the 
computational methods employed, the algebraic techniques for predicting 
motions in contact with rotational compliance, and issues of robustness and 
stability of our geometric and algebraic algorithms. Our computational 
viewpoint lies in the interface between differential theories of mechanics, 
and combinatorial collision detection algorithms. From this viewpoint, subtle 
mathematical difficulties arise in predicting motions under rotational 
compliance, such as the forced non-genericity of the intersection problems 
encountered in configuration space. We discuss these problems and their 
solutions. Finally, we extend our work to predict the forces on the 
manipulated objects as a function of time, and show how our algorithm can 
easily be extended to include uncertainty in control and initial conditions. 
With these extensions, we hope that our system could be used to analyze and 
design objects that are easy to assemble, even given control and sensing 
errors, and that require more force to disassemble than to mate.
END:: CORNELLCS//TR92-1296
BODY::
The Motion of Planar Compliantly-Connected
Rigid Bodies in Contact
With Applications to Automatic Fastening
Bruce R. Donald*
Dinesh K. Pal**
TR 92-1296
July1992
Department of Computer Science
Cornell University
Ithaca, NY 14853-7501
*This paper describes research done in the Robotics and Vision Laboratory and the
Computer Science Department at Cornell University. Support for our robotics
research is provided in part by the National Science Foundation under grants No. RI-
8802390, lRI-9000532 and by a Presidential Young Investigator award to Bruce
Donald, and in part by the Air Force Office of Sponsored Research, the Mathematical
Sciences Institute, Intel Corporation and AT&T Bell Laboratories.
**Supported in part by ONR Grant N0001 4-88K-0591, ONR Grant N0001 4-89J-1 946,
NSF Grant DMC-86-17355 at Cornell University and by NSERC Operating Grant
OGP0122128 at UBC.
To Appear in The International Joarnal of Robotics Research
The Motion of Planar Compliantly-Connected
R?igid Bodies in Contact
with Applications to Automatic Fastening
Bruce R. Donald*
Computer Science Department
Cornell University
Dinesh K. Pait
Computer Science Department
University of British Columbia
July 22,1992
Abstract
We consider the problem of planning and predicting the motion of a flexible object
amidst obstacles in the plane. We model the flexible object as a rigid "root" body,
attached to compliant members by torsional springs. The root's position may be con-
trolled, but the compliant members move in response to forces from contact with the
environment. Such a model encompasses several important and complicated mecha-
nisms in mechanical design and automated assembly: snap-fasteners, latches, ratchet
and pawi mechanisms, and escapements. The problem is to predict the motion of such
a mechanism amidst fixed obstacles. For example, our algorithm could be used to
determine whether a snap-fastener design can be assembled with a certain plan.
In this paper we analyze the physics of these flexible devices, and develop com-
binatorially precise algorithms for predicting their movement under a motion plan.
0ur algorithms determine when and where the motion wilt terminate, and also com-
putes the time-history of contacts and mating forces. In addition to providing the first
known exact algorithm that addresses flexibility in motion planning, we also note that
our approach to compliance permits an exact algorithm for predicting motions under
rotational compliance, which was not possible in earlier work.
We discuss the following issues: the relevance of our approach to engineering (which
we illustrate through the examples we ran using our system), the computational meth-
ods employed, the algebraic techniques for predicting motions in contact with rotational
compliance, and issues of robustness and stability of our geometric and algebraic algo-
rithms. Our computational viewpoint lies in the interface between differential theories
*This paper describes research done in the Robotics and Vision Laboratory and the computer Science
Department at Cornell University. Support for our robotics research is provided in part by the National
Science Foundation under grants No. IRI-8802390, IRI-9000532 and by a Presidential Young Investigator
award to Bruce Donald, and in part by the Air Force 0ffice of Sponsored Research, the Mathematical
Sciences Institute, Intel corporation, and AT&T Bell Laboratories.
tsupported in part by 0NR Grant N00014-88K-O591, ONR Grant NOOOl4-89J-1946, NSF Grant DMc-
86-17355 at cornell University, and by NSERC Operating Grant 0GPO122128 at UBC
0
of mechanics, and combinatorial collision detection algorithms. From this viewpoint,
subtle mathematical difficulties arise in predicting motions under rotational compli-
ance, such as the forced non-genericity of the intersection problems encountered in
configuration space. We discuss these problems and their solutions. Finally, we extend
our work to predict the forces on the manipulated objects as a function of time, and
show how our algorithm can easily be extended to include uncertainty in control and
initial conditions. With these extensions, we hope that our system could be used to
analyze and design objects that are easy to assemble, even given control and sensing
errors, and that require more force to disassemble than to mate.
Contents
1 Introduction
2 Problem Statement
3
4
Examples
3.1 The Two-Pawi Example
3.2 "Motion Diode" Example
2
5
Overview of Results
4.1 Differential Theories of Mechanics: An Algebraic Approach
4.2 Computing Motions and Intersection Problems
4.3 Exact solutions for mechanical simulations
4.4 A Classification of Motion Diodes
5 Simulation and Algebraic Intersection Problems
6 Sticking due to Eriction
7 The Sliding Direction
8
9
Collision Detection in Configuration Space
8.1 Computing the "Snap:" Pure Rotational Collision Detection
8.2 Collision Detection Subject to a Holonomic Constraint . . .
8.3			Choosing the Correct Branch.
8.3.1			Application: Snapping Off and Jamming.
8.3.2			The Branch Problem
8.4			The Genericity of Intersection Problems			. .
8.4.1 The Inherent Genericity of C-functions in Trigonometric Quadratic Form
8.4.2 On the Forced Non-Genericity of Intersection Problems 