Mathematical Sciences: Applications of Sweep Differential Equations to Automated Manufacturing

9500808布莱克莫尔项目的目标是：1)进一步发展扫描微分方程法，以增强其在分析和表示扫描体方面的适用性；以及2)将扫描微分方程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，通过空间移动的对象所经过的区域的表面(对象的扫掠体积)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-包括对象的几何-具有其轨迹覆盖扫掠体积的大部分边界的特性。这将导致生成扫描体的集合中的一维减少，并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。应开发基于这种新方法的算法，并应利用该方法的独特特性来获得扫掠体积的近似数字和图形表示。特别是，边界表面的分层应纳入新的内插程序，并应采用自然函数准则来修剪(或排除)不属于扫掠体积边界的点。这些算法应用于开发自动验证和生成数控(NC)加工程序和机器人运动规划问题的计算机程序。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，旨在开发新的理论，并将其应用于与制造相关的实际问题，从而促进先进制造的最先进水平。一个主要的目标是开发新的技术手段，使人们能够基本上通过物体在其初始位置的边界表面上的一条曲线来表征扫掠体积-其中扫掠体积是由微分方程式的解产生的。这项创新将被用来开发精确的计算机辅助方法，以获得物体在空间中移动时所扫过的形状的数字和图形表示。基于这一理论框架的算法有可能将现有算法的计算量减少至少一个数量级。扫掠体在数控加工、机器人运动规划、计算机辅助设计等自动化制造领域具有重要作用，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.