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.