On Designing 2D Discrete Workspaces to Sort or Classify Polynminoes

This paper studies the general problem of physically sorting polyominoes according to shape using a 2D, rigid, grid-based workspace. The workspace is designed for sensorless operation, using a fixed set of open-loop force-field inputs that move a polyomino from an inlet port to an outlet port that corresponds to the polyomino's shape, and reset the workspace to classify the next polyomino. This paper proves that static workspaces can classify all orthoconvex polyominoes of width $w$ and height $h$ , and provides a motion sequence and required size of workspace as a function of wand $h$ . By allowing moving polyomino cams that assist in the sorting, we can design dynamic works paces that can sort all polyomi-noes that are “completely filled” using a constant number of force-field inputs. Hardware experiments using magnetic and gravity-based actuation demonstrate these static and dynamic sensorless classifiers at the millimeter scale.