Computational and Algorithmic Representations of Geonetric Objects - CARGO: Folding and Unfolding Processes for Polygonal Linkages, with Applications to Robotics and Biology

DMS -0138374摘要本孵化项目计划开始一个跨学科的合作，其目标是了解多边形连杆机构和其他相关结构的折叠和展开过程。我们计划通过推广最近成功的基于伪三角形机构的刚性理论方法，研究二维和三维串联连杆机构位形空间的拓扑和代数几何结构，并促进了对相关位形空间的整体结构的理解。 刚性理论技术已经允许定义纯膨胀或压缩的二维运动：前者可以保证链不会自碰撞。 我们计划将这些技术扩展到三维空间，并随后开发有效的数据结构和算法，用于规划、分析、近似、跟踪和查询这些运动。我们预计折叠问题，除了其内在的学术兴趣，将导致技术，将影响分子生物学等领域，其中蛋白质折叠问题具有核心意义，以及机器人和微观力学，其中经常采用模块化运动机构（可能具有大量的连杆）。 这些努力的共同点，我们的主题，是generalquestion的规划和推理的碰撞自由议案seriallinkages的各种。

Abstract for DMS - 0138374This incubation project plans to start an interdisciplinary collaborationwhose goal is the understanding of folding and unfolding processes forpolygonal linkages and other related structures.We plan to investigate the topology and the algebraic-geometric structureof configuration spaces of two- and three-dimensional serial linkages, viageneralizations of recent successful rigidity-theoretic approaches basedon pseudo-triangulation mechanisms, and to advance the understanding ofthe global structure of the associated configuration spaces. Therigidity-theoretic techniques have allowed the definition of 2-d motionsthat are purely expansive or contractive: the former can guarantee that achain will not self-collide. We plan to extend these techniques to threedimensions and subsequently develop efficient data structures andalgorithms for planning, analyzing, approximating, tracking and queryingsuch motions.We expect that folding problems, beyond their intrinsic mathematicalinterest, will lead to techniques that will impact areas such as molecularbiology, where the protein folding problem is of central significance, aswell as robotics and micromechanics, where modular kinematic mechanisms(with possibly large numbers of links) are often employed. The commondenominator of these endeavors, and our main theme, is the generalquestion of planning and reasoning about collision free motions for seriallinkages of various kinds.