Fundamental Development of Mathematical Techniques and Computational Metrology for Coordinate Metrology

该研究项目的目标是推进数学和计算几何工具，以有效和高效地确定工件的几何形状是否满足指定的目标几何形状。为完成这项任务，将追求两个主要目标。首先是使用改进的Voronoi图技术开发基于线性面几何模型的点到面分配。数学证明表明，面模型与本研究中开发的计算计量技术相结合，将被开发出来，以证明点到面分配对于真实（非面）模型是正确的。第二个主要目标是推导任意阶样条曲面的解析偏导数。这些偏导数将针对数值导数以及目前使用的近似技术进行速度和准确性测试。最后，为了证明数学和计算计量学的基本发展是现实的和有用的，它们将在各种工业应用中实施和测试。如果成功，这项研究的结果将为制造零件提供更快、更准确的分析。在这项研究中开发的数学精确对准点复杂的几何表面模型将允许改进的零件分析，如复杂的注塑模具和翼型几何形状。此外，待开发的技术还将大大减少检查复杂几何形状所需的计算时间，这些几何形状已经通过新的制造技术（如快速原型制造）变得可行。预计这项研究的准确性和速度结果将提供使能技术，直接和经济地将光学和触觉扫描技术集成到生产设施中，如汽车和航空航天设施，用于质量控制的实时反馈。

The goal of this research project is to advance mathematical and computational geometry tools to effectively and efficiently determine if the geometry of an artifact meets the target geometry that is specified. To achieve this task two primary objectives will be pursued. The first is the development of point-to-surface assignment based on a linearly faceted geometric model using a modified Voronoi diagram technique. A mathematical proof demonstrating that the faceted model in conjunction with the computational metrology techniques developed in this research will be developed to prove that the point-to-surface assignment is correct for the true (non-faceted) model. The second major objective is to derive the analytic partial derivatives for spline surfaces of any order. These partial derivatives will be tested against both numerical derivatives as well as currently used approximation techniques for both speed and accuracy. Finally, to demonstrate that the fundamental developments in mathematics and computational metrology are realistic and of use, they will be implemented and tested on a variety of industrial applications. If successful, the results of this research will provide faster and more accurate analyses of manufactured parts. The mathematics developed in this research to precisely align points to complex geometric surface models will permit improved analysis of parts such as complex injection molds and airfoil geometries. Furthermore, the techniques to be developed will also substantially reduce the computational time necessary to inspect complex geometries that have been made feasible via new manufacturing techniques, such as rapid prototyping. It is anticipated that the accuracy and speed results of this research will provide the enabling technology to directly and economically integrate optical and tactile scanning technology into production facilities such as automotive and aerospace facilities for use in real-time feedback for quality control.