Mathematical Sciences: Applications of Sweep Differential Equations to Automated Manufacturing

小行星9500808 该项目的目标是：1）进一步发展扫描 微分方程（EFD）的方法，以提高其适用性， 扫描体积的分析和表示;以及2）将 改进的算法和软件工具， 解决自动化中的运动验证和规划问题 制造业在该方法中， 在空间中移动的对象（对象的扫描体积）是 用微分方程（Differential Equation）表示。的一个主要目标 该项目是为了获得一个普遍的， 物体的几何形状-具有其轨迹叶状的属性 扫描体积的大部分边界。这导致减少一个 在生成扫描体积的集合中的维度，它承诺 在计算复杂度上产生一个数量级的降低， 基于这种新方法的算法。基于这种新方法的算法 应开发，方法的独特性应 用于获得近似的数值和图形表示， 扫描体积。特别是，边界表面的叶理应 纳入新的插值程序，和一个自然的功能 修剪（或排除）不属于扫描的点的标准 应采用体积边界。算法应用于 开发计算机程序，用于自动验证和生成 数控（NC）加工程序和机器人运动 规划问题。 该项目代表了一个跨学科的合作， 数学家和机械工程师，旨在发展新的理论， 其将被实施用于实际的制造相关问题， 从而推进先进制造业的技术水平。 一 主要目标是开发新技术，使人们能够 基本上通过边界表面上的曲线来表征扫掠体积 物体在其初始位置的位置-生成扫描体 通过微分方程的解。这项创新将用于 开发精确的计算机辅助方法， 由移动物体扫过的图形表示 穿越太空基于这个理论的算法 框架具有 将现有算法的计算量减少至少一个 数量级。由于扫掠体积在这种情况下起着重要作用， 自动化制造的不同领域，如数控加工，机器人运动 规划和计算机辅助设计，研究在这方面的应用 项目应该对现代制造理论产生重大影响 和实践

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.