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

对于DMS-0138374，这个孵化项目计划开始一个跨学科的合作，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。我们计划研究二维和三维串联连杆机构的位形空间的拓扑和代数几何结构，推广最近成功的基于伪三角机构的刚性理论方法，并促进对相关位形空间的全局结构的理解。刚体理论技术允许定义纯膨胀或收缩的二维运动：前者可以保证链条不会自碰撞。我们计划将这些技术扩展到三维，并随后开发有效的数据结构和算法来规划、分析、近似、跟踪和查询此类运动。我们预计，折叠问题超出了它们固有的数学兴趣，将导致一些技术的影响，如分子生物学，蛋白质折叠问题是核心意义，以及机器人学和微观力学，其中经常使用模块化运动学机制(可能有大量的链接)。这些努力的共同点，也是我们的主题，是关于各种系列链接的无碰撞运动的计划和推理的一般问题。

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.