Qualitative kinematics in mechanisms

Qualitative Kinematics is the qualitative analysis of the possible geometric interactions of physical objects. This thesis investigates the problem of reasoning about the kinematic interactions between parts of a 2-dimensional mechanism as a first step towards a general theory of qualitative kinematics. We introduce the concept of place vocabularies as a generative symbolic description of the possible motion of the parts of a mechanism. Place vocabularies are particularly useful to describe higher kinematic pairs such as ratchets and escapements. We examine the requirements for the representation and introduce a definition of place vocabularies that satisfies them. We show how this representation can be computed from metric data about the shapes of the parts of a mechanism and used as a basis for qualitative envisionments of its behavior. In particular, we give implemented algorithms for computing place vocabularies for 2-dimensional higher kinematic pairs. The objects involved in the pair can have arbitrary shapes, but their boundaries must consist of straight lines and arcs, and each object must have one degree of freedom only. Complete mechanisms are analyzed as compositions of kinematic pairs. The algorithms for computing place vocabularies fall within the qualitative reasoning paradigm. They are based on splitting the computation into 2 parts: a purely symbolic reasoning part and a metric diagram. The metric diagram is an abstract device that gives access to information about quantities by evaluating predicates on them. We propose that this division is a plausible model of human reasoning. When quantitative information is incomplete, the metric diagram defines a set of landmark values for unknown metric parameters. The resulting place vocabulary changes only at these landmark values, and an exhaustive list of all place vocabularies that can be achieved by varying the parameters is found by computation at representative points in each interval. We show how the mechanism design problem of picking suitable values for parameters can be solved by searching the list, and give an implemented example. The exhaustive list forms an ambiguous but complete prediction of the possible behavior. This proves that contrary to common belief, it is in principle possible to reason about kinematics with limited or no knowledge of the metric dimensions of the objects.