Reasoning about Kinematic Topology

Reasoning about kinematics is an important aspect of common sens e physics. In earlier work, we have developed the place vocabulary theory of qualitative kinematics in mechanisms, a formal theory for representin g the kinematic behavior of two-dimensional mechanisms . The computation of a place vocabulary is very complex because it takes into accoun t the details of object shapes . In this paper, we present a representatio n 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 severa l doubly connected regions, whose shape in configuration space indicate s the relative speeds of the two gears . For many applications, reasonin g about kinematics at this level is sufficient . Kinematic topology can be computed in a qualitative manner an d thus gives an existance proof that a purely qualitative kinematics is possible . Like in other qualitative reasoning applications, the qualitativ e computation has the effect that the result is almost always ambiguous. On the other hand, a kinematic topology can be given even fo r mechanisms whose designs are only imprecise sketches, and can be gen eralized to arbitrary object shapes, several degrees of freedom, and thre e dimensions. We hope that such generalizations of kinematic topolog y can provide the basis for efficiently computing place vocabularies, an d reasoning about general kinematic interactions . Figure 1 : A pair of gearwheels . The drawing on the left shows an actuall y working device, while the one on the right is only a sketch that will not wor k