AF: Small: Collaborative:RUI: Mathematical foundations of reconfiguration algorithms for geometrically constraint structures

AF：小：协作：RUI：几何约束结构重构算法的数学基础

• 批准号: 1319389
• 负责人: Ciprian Borcea
• 金额: $ 12.08万
• 依托单位: Rider University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319389&HistoricalAwards=false
• 关键词: AF; Small; Collaborative; RUI; Mathematical

Geometric reconfiguration problems underlie modern mathematical investigations in robotics, mechanical design and structural engineering. In recent years, challenging questions in computational biology and computational materials science (such as protein folding, viral assembly, flexibility studies of crystalline materials and the rational design of macromolecules with desired functionality) are approached with methods originally developed for abstract, geometrically constraint structures.This project will develop novel algorithms based on mathematically rigorous techniques, by exploiting discrete structures underlying three-dimensional articulated structures (such as linkages, panel-and-hinge chains and polyhedra). Examples include reconfigurations of robot arms within their 3D workspace, with singularity- and collision-avoidance, expansive motions and pseudo-triangulations in a 3D and in periodic settings, and origami foldability. The research project builds upon the PIs' previous work, and extends it in new directions. It seeks to generalize concepts from Rigidity Theory (such as pointed pseudo-triangulations), which were previously applied successfully to 2-dimensional linkage reconfiguration problems. It addresses problems concerning extremal configurations of revolute jointed robotic manipulators and the intrinsic mathematical structure of their 3D workspace. It aims at bringing in novel algebraic and combinatorial rigidity-theoretic methods, and at applying them to periodic and crystalline structures. Recent results relating origami design to properties of piecewise linear surfaces will also be extended to questions concerning origami folding properties to rigidity questions of panel-and-hinge structures.The grant provides funding for the training of graduate and undergraduate students. In particular, REU projects emerging from these topics will involve students from Smith College, an all-women college with a sustained reputation for successfully educating minority undergraduate students in the sciences.

几何重构问题是机器人技术、机械设计和结构工程领域现代数学研究的基础。近年来，计算生物学和计算材料科学中的挑战性问题（例如蛋白质折叠、病毒组装、晶体材料的灵活性研究以及具有所需功能的大分子的合理设计）通过最初为抽象、几何约束结构开发的方法来解决。该项目将通过利用三维基础上的离散结构，开发基于数学严格技术的新颖算法 铰接结构（例如连杆、面板铰链和多面体）。 示例包括在 3D 工作空间内重新配置机器人手臂、避免奇点和碰撞、在 3D 和周期性设置中进行扩展运动和伪三角测量，以及折纸可折叠性。该研究项目以 PI 之前的工作为基础，并将其扩展到新的方向。它试图概括刚性理论的概念（例如尖点伪三角剖分），这些概念先前已成功应用于二维连杆重构问题。它解决了有关旋转关节机器人操纵器的极值配置及其 3D 工作空间的内在数学结构的问题。它的目的是引入新颖的代数和组合刚性理论方法，并将其应用于周期性和晶体结构。将折纸设计与分段线性表面特性相关的最新成果也将扩展到有关折纸折叠特性的问题以及面板和铰链结构的刚性问题。该赠款为研究生和本科生的培训提供资金。特别是，从这些主题中出现的 REU 项目将涉及史密斯学院的学生，史密斯学院是一所女子学院，在成功教育少数族裔本科生的科学方面享有盛誉。