Theory and Practice of Point Line Kinematics

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This grant provides funding for the development of a theory on point-line kinematics and its applications on the analysis, characterization, synthesis, and control of point-line motion. A point-line operator based on geometry and Clifford algebra will be developed to express a point-line transformation. Subsequently, this transformation will allow a point-line to be considered as an independent rigid element having a completely specified motion without invoking rigid body transformation. Based on a principle similar to the point-line operator, the theory will be extended to include line displacement and rigid body motion, in which parameters responsible for causing incompletely specified motion can be removed or added as needed. Issues on the synthesis for finite and infinitesimal point-line motions will be investigated. The results will be applied to enhance the design capability of spatial mechanisms and manipulators. The project will also explore for the first time the full configuration space concerning the tool path generation in five-axis numerical control machining for interference avoidance.If successful, the results will lead to the first point-line kinematics theory comparable to the well-developed rigid body kinematics theory. In addition, it will lead to a unified kinematics theory for lines, point-lines, and rigid bodies. It allows one to understand and treat a point-line motion by viewing the point-line as an integrated rigid element in a way similar to that of points or rigid bodies. It will lead to a mathematical language such that the characteristics of point-line motions can be rigorously articulated. It will serve as a catalyst and foundation for advancements in kinematics, robotics, computer-aided geometric design, and manufacturing technology. Because many tool operations such as numerical control machining, drilling, screwing, riveting, cutting, probing, insertion, and electric wire discharge machining are basically point-line operations, the project's results will enhance the capability of making improvements on the motion synthesis, control, and quality of these operations. It is also recognized that because of the fundamental nature of the subject, the impact may extend to other aspects of science and engineering.