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

Smith College, Northampton MA

Investigators

Abstract

Abstract for DMS - 0138374 This incubation project plans to start an interdisciplinary collaboration whose goal is the understanding of folding and unfolding processes for polygonal linkages and other related structures. We plan to investigate the topology and the algebraic-geometric structure of configuration spaces of two- and three-dimensional serial linkages, via generalizations of recent successful rigidity-theoretic approaches based on pseudo-triangulation mechanisms, and to advance the understanding of the global structure of the associated configuration spaces. The rigidity-theoretic techniques have allowed the definition of 2-d motions that are purely expansive or contractive: the former can guarantee that a chain will not self-collide. We plan to extend these techniques to three dimensions and subsequently develop efficient data structures and algorithms for planning, analyzing, approximating, tracking and querying such motions. We expect that folding problems, beyond their intrinsic mathematical interest, will lead to techniques that will impact areas such as molecular biology, where the protein folding problem is of central significance, as well as robotics and micromechanics, where modular kinematic mechanisms (with possibly large numbers of links) are often employed. The common denominator of these endeavors, and our main theme, is the general question of planning and reasoning about collision free motions for serial linkages of various kinds.