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

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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.