Mathematical Sciences: Applications of Sweep Differential Equations to Automated Manufacturing

该项目的目标是：1)进一步发展扫描微分方程（SDE）方法，以提高其对扫描体积分析和表示的适用性；2)将SDE方法的改进转化为算法和软件工具，以解决自动化制造中的运动验证和规划问题。在SDE方法中，通过空间移动的物体所遍历的区域表面（物体的扫描体积）用微分方程（SDE）来表征。该项目的一个主要目标是获得SDE的泛化-结合物体的几何形状-具有其轨迹在扫描体的大部分边界处形成叶状的特性。这导致生成扫描体积的集合减少了一个维度，并且它有望使基于这种新方法的算法的计算复杂性降低一个数量级。基于这种新方法的算法将被开发，并利用该方法的独特特征来获得扫描体积的近似数值和图形表示。特别是，边界表面的叶理应纳入新的插值程序，并应采用自然函数准则来修剪（或排除）不属于扫描体边界的点。该算法用于开发计算机程序，用于自动验证和生成数控加工程序和机器人运动规划问题。该项目代表了应用数学家和机械工程师之间的跨学科合作，旨在为实际制造相关问题开发新的理论，从而推进先进制造的最新技术。主要目标是开发新技术，使人们能够基本上通过物体初始位置的边界表面上的曲线来表征扫描体积-扫描体积由微分方程的解生成。这项创新将用于开发精确的计算机辅助方法，以获得在空间中移动的物体扫出的构型的数值和图形表示。基于该理论框架的算法具有将现有算法的计算量至少减少一个数量级的潜力。由于扫描体积在数控加工、机器人运动规划和计算机辅助设计等自动化制造的各个领域中发挥着重要作用，因此本课题的研究应用将对现代制造理论和实践产生重大影响。

9500808 Blackmore The objectives of the project are: 1) to further develop the sweep differential equation (SDE) approach so as to enhance its applicability to the analysis and representation of swept volumes; and 2) to translate the improvements of the SDE method into algorithms and software tools for solving motion verification and planning problems in automated manufacturing. In the SDE method, the surface of the region traversed by an object moving through space (the swept volume of the object) is characterized in terms of a differential equation (the SDE). A major goal of the project is to obtain a generalization of the SDE - incorporating the geometry of the object - that has the property that its trajectories foliate most of the boundary of the swept volume. This leads to a reduction of one dimension in the set that generates the swept volume, and it promises to yield an order of magnitude decrease in the computational complexity of algorithms based on this new method. Algorithms based on this new method shall be developed, and the unique features of the approach shall be exploited to obtain approximate numerical and graphical representations of swept volumes. In particular, the foliation of the boundary surface shall be incorporated into new interpolation procedures, and a natural functional criterion for trimming (or excluding) points not belonging to the swept volume boundary shall be employed. The algorithms shall be used to develop computer programs for automatic verification and generation of numerically controlled (NC) machining programs and robot motion planning problems. The project represents an interdisciplinary collaboration between an applied mathematician and a mechanical engineer aimed at developing new theory that is to be implemented for practical manufacturing related problems, thereby advancing the state-of-the-art in advanced manufacturing. A primary objective is to develop new tech niques that enable one to characterize swept volumes essentially by a curve on the boundary surface of an object in its initial position - with the swept volume being generated by solutions of a differential equation. This innovation is to be used to develop accurate computer-aided methods for obtaining numerical and graphical representations of configurations swept out by objects moving through space. The algorithms based upon this theoretical framework have the potential to reduce the computations of existing algorithms by at least an order of magnitude. As swept volume plays an important role in such diverse areas of automated manufacturing as NC machining, robot motion planning and computer-aided design, the applications of the research in this project should have a significant impact on modern manufacturing theory and practice.