The Motion of Planar, Compliantly Connected Rigid Bodies in Contact, With Applications to Automatic Fastening

Abstract

We consider the problem of planning and predicting the mo tion of a flexible object amid obstacles in the plane. We model the flexible object as a rigid "root" body attached to compli ant 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 en compasses 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 amid 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 article we analyze the physics of these flexible de vices and develop combinatorially precise algorithms for predicting their movement under a motion plan. Our algo rithms determine when and where the motion will terminate and also compute 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 ap proach to engineering (which we illustrate through examples we ran using our system), the computational methods em ployed, 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 differen tial theories of mechanics and combinatorial collision detection algorithms. From this viewpoint, subtle mathematical difficul ties arise in predicting motions under rotational compliance, such as the forced nongenericity 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.