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

这个早期教师职业发展（CAREER）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，例如不希望的接触损失或在存在不确定性的情况下移动物体的规定运动的变化。该研究采取了一种新的方法来表征的几何奇点在家庭的刚性和非刚性的移动形状的包络线，通过重新定义的问题，在所谓的“折叠点”和“折叠区域”在这些奇点附近。基于这种对偶性，检测几何奇异性的问题被重新转换为原始d维欧氏空间中的点隶属分类测试，从而消除了对包络计算的需要。反过来，这将导致这种潜在故障的改进的检测和校正能力。如果成功的话，这项研究的结果将推进最先进的计算机辅助制造，路径规划和几何建模提供的算法，这将，例如，显着提高在线测试的工具路径和数控代码的任意复杂的形状和运动，将减少下或过切削加工，提高扫描体积计算和改进的碰撞检测。新的课程将被开发和整合到机械工程计划。作为该项目的一部分，将与一所女子学院进行外联，鼓励和积极招收女学生进入机械工程，并与现有的一个成功的K-12外联方案合作。

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.