CAREER: Geometric Singularities in Engineering Design and Manufacturing: A Generic Spacetime Approach

这一早期教师职业发展(CAEREAR)奖的研究目标是开发一个通用的理论框架和计算算法，用于预测、量化和纠正由几何奇点引起的潜在故障，例如在存在不确定性的情况下，意外失去联系或移动物体的规定运动的变化。该研究采用了一种新的方法来刻画刚体和非刚体运动形状族的包络中的几何奇点，将问题重新定义为这些奇点邻域中的“折叠点”和“折叠区域”。基于这种对偶性，几何奇异性的检测问题被重塑为原始d维欧氏空间中的点成员分类测试，从而消除了包络计算的需要。反过来，这将提高对此类潜在故障的检测和纠正能力。如果成功，这项研究的结果将通过提供算法来推动计算机辅助制造、路径规划和几何建模的技术水平，例如，将显著改进针对任意复杂形状和运动的刀具路径和数控代码的在线测试，这将减少机械加工中的欠切或过切，改进的扫掠体积计算和改进的碰撞检测。将开发新的课程并将其整合到机械工程课程中。作为该项目的一部分，将开展与一所女子学院的外联活动，以鼓励和积极招收女学生进入机械工程专业，并与现有的成功的K-12外联方案进行合作。

The research objective of this Early Faculty Career Development (CAREER) award is to develop a generic theoretical framework and computational algorithms for predicting, quantifying, and correcting potential malfunctions induced by geometric singularities, such as undesired loss of contact or changes in the prescribed motion of moving bodies in the presence of uncertainty. The research takes a new approach to the characterization of geometric singularities in the envelopes of families of both rigid and non-rigid moving shapes by reframing the problem in terms of the so-called "fold points" and "fold regions" in the neighborhood of these singularities. Based on this duality, the problem of detecting geometric singularities is recast into Point Membership Classification tests in the original d-dimensional Euclidean space, eliminating the need for envelope computations. In turn, this will result in improved detection and corrective capabilities of such potential failures. If successful, the results of this research will advance the state of the art in computer aided manufacturing, path planning, and geometric modeling by providing algorithms which will, for example, significantly improve on-line testing of tool paths and CNC codes for arbitrarily complex shapes and motions that will reduce under- or over-cutting in machining, improved swept volume calculations and improved collision detection. New curriculum will be developed and integrated into the mechanical engineering program. Outreach to an all women's college to encourage and actively recruit female students into mechanical engineering and collaboration with a successful existing K-12 outreach program will be performed as part of this project.