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

几何重构问题是机器人、机械设计和结构工程中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（如蛋白质折叠、病毒组装、晶体材料的柔性研究和具有所需功能的高分子的合理设计），采用最初为抽象几何约束结构开发的方法。本项目将开发基于数学严格技术的新算法，通过利用三维铰接结构（例如连杆、面板铰链链和多面体）下的离散结构。 示例包括在其3D工作空间内重新配置机器人手臂，避免奇异和碰撞，在3D和周期性设置中扩展运动和伪三角形，以及折纸可折叠性。该研究项目建立在PI以前的工作基础上，并将其扩展到新的方向。它试图概括刚性理论（如点伪三角形），这是以前成功地应用于2维联动重构问题的概念。它解决了有关的极端配置问题的旋转关节机器人及其三维工作空间的内在数学结构。它旨在引进新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。有关折纸设计的分段线性表面的属性最近的结果也将扩展到折纸折叠性能的问题，面板和铰链结构的刚性问题。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，史密斯学院是一所全女子学院，在科学领域成功地教育少数民族本科生方面享有盛誉。

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.