Theory and Practice of Point Line Kinematics

该基金为点线运动学理论的发展及其在点线运动的分析、表征、综合和控制方面的应用提供资金。一个基于几何和Clifford代数的点-线算子将被开发来表示点-线变换。随后，该变换将允许将点线视为具有完全指定运动的独立刚性单元，而无需调用刚体变换。基于类似于点-线算子的原理，该理论将扩展到包括线位移和刚体运动，其中可以根据需要删除或添加导致不完全指定运动的参数。有限和无穷小点线运动的综合问题将被研究。研究结果将用于提高空间机构和机械臂的设计能力。该项目还将首次探索五轴数控加工中刀具轨迹生成的全构型空间，以避免干涉。如果成功，结果将导致第一个可与发达的刚体运动学理论相媲美的点线运动学理论。此外，它将导致线，点线和刚体的统一运动学理论。它允许人们以类似于点或刚体的方式将点线视为一个完整的刚性元素来理解和处理点线运动。它将导致一种数学语言，这样点-线运动的特征就可以被严格地表达出来。它将成为运动学、机器人、计算机辅助几何设计和制造技术进步的催化剂和基础。由于数控加工、钻孔、旋紧、铆接、切削、探测、插入、电火花加工等许多刀具操作基本上都是点线操作，本项目的成果将增强这些操作的运动综合、控制和质量改进的能力。人们还认识到，由于该学科的基本性质，其影响可能会扩展到科学和工程的其他方面。

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.