Analysis of Swept Volume via Lie Groups and Differential Equations

The development of useful mathematical techniques for an alyzing swept volumes, together with efficient means of im plementing these methods to produce serviceable models, has important applications to numerically controlled (NC) machin ing, robotics, and motion planning, as well as other areas of automation. In this article a novel approach to swept volumes is delineated—one that fully exploits the intrinsic geometric and group theoretical structure of Euclidean motions in or der to formulate the problem in the context of Lie groups and differential equations. Precise definitions of sweep and swept volume are given that lead naturally to an associated ordinary differential equation. This sweep differential equation is then shown to be related to the Lie group structure of Euclidean motions and to generate trajectories that completely determine the geometry of swept volumes. It is demonstrated that the notion of a sweep differential equation leads to criteria that provide useful insights concern ing the geometric and topologic features of swept volumes. Several new results characterizing swept volumes are obtained. For example, a number of simple properties that guarantee that the swept volume is a Cartesian product of elementary mani folds are identified. The criteria obtained may be readily tested with the aid of a computer.