9500808 Blackmore The objectives of the project are: 1) to further develop the sweep differential equation (SDE) approach so as to enhance its applicability to the analysis and representation of swept volumes; and 2) to translate the improvements of the SDE method into algorithms and software tools for solving motion verification and planning problems in automated manufacturing. In the SDE method, the surface of the region traversed by an object moving through space (the swept volume of the object) is characterized in terms of a differential equation (the SDE). A major goal of the project is to obtain a generalization of the SDE - incorporating the geometry of the object - that has the property that its trajectories foliate most of the boundary of the swept volume. This leads to a reduction of one dimension in the set that generates the swept volume, and it promises to yield an order of magnitude decrease in the computational complexity of algorithms based on this new method. Algorithms based on this new method shall be developed, and the unique features of the approach shall be exploited to obtain approximate numerical and graphical representations of swept volumes. In particular, the foliation of the boundary surface shall be incorporated into new interpolation procedures, and a natural functional criterion for trimming (or excluding) points not belonging to the swept volume boundary shall be employed. The algorithms shall be used to develop computer programs for automatic verification and generation of numerically controlled (NC) machining programs and robot motion planning problems. The project represents an interdisciplinary collaboration between an applied mathematician and a mechanical engineer aimed at developing new theory that is to be implemented for practical manufacturing related problems, thereby advancing the state-of-the-art in advanced manufacturing. A primary objective is to develop new tech niques that enable one to characterize swept volumes essentially by a curve on the boundary surface of an object in its initial position - with the swept volume being generated by solutions of a differential equation. This innovation is to be used to develop accurate computer-aided methods for obtaining numerical and graphical representations of configurations swept out by objects moving through space. The algorithms based upon this theoretical framework have the potential to reduce the computations of existing algorithms by at least an order of magnitude. As swept volume plays an important role in such diverse areas of automated manufacturing as NC machining, robot motion planning and computer-aided design, the applications of the research in this project should have a significant impact on modern manufacturing theory and practice.
