Reasoning about Kinematic Topology

Abstract

Reasoning about kinematics is an important aspect of common sense physics. In earlier work, we have developed the place vocabulary theory of qualitative kinematics in mechanisms, a formal theory for representing the kinematic behavior of two-dimensional mechanisms. The computation of a place vocabulary is very complex because it takes into account the details of object shapes. In this paper, we present a representation which is much more abstract than a place vocabulary, the kinematic topology. Kinematic topology does not define qualitative inference rules, but provides a characterization of the topology of legal configurations. For example, the kinematic topology of a pair of gears is one or several doubly connected regions, whose shape in configuration space indicates the relative speeds of the two gears. For many applications, reasoning about kinematics at this level is sufficient. Kinematic topology can be computed in a purely qualitative manner and thus gives an existance proof that a purely qualitative kinematics is possible. Like in other qualitative reasoning applications, the qualitative computation has the effect that the result is almost always ambiguous. On the other hand, a kinematic topology can be given even for mechanisms whose designs are only imprecise sketches, and can be generalized to arbitrary object shapes, several degrees of freedom, and three dimensions. We hope that such generalizations of kinematic topology can provide the basis for efficiently computing place vocabularies, and reasoning about general kinematic interactions.