College of LAS « Illinois

Mathematics

Shaping (and Reshaping) Tomorrow's Robots

Trapped beneath the days-old rubble of the collapsed building, an injured child cries faintly for help. Unable to reach her, firefighters send in the rescue robot. The insect-like robot climbs through huge boulder fields of debris—then its legs melt away, its body elongates, and it slithers snake-like to the child, handing her lifesaving food and water. While there's still much work to be done, new mathematics from the U. of I. could one day help make such rescue robots a reality.

As depicted in movies like Terminator 2: Judgment Day, morphing robots are already taking baby steps in robotics labs worldwide. Many consist of a movable stack of identical cubes, each of which moves through the stack like a square in a sliding-tile puzzle. As cubes move to new locations, the robot changes shape. Like early automotive engineers, builders of today's morphing robots use a variety of designs, and each must figure out from scratch how to program the cubes to move without colliding with each other. Now Robert Ghrist, an LAS associate professor of mathematics, is using topology, the study of shapes, to develop a better way to program morphing robots.

To illustrate the math, Ghrist offers a simple example from a different type of robot: factory robots that carry objects from one workstation to the next. Each starts at one point and moves to another. The coordinates of the starting and ending points can be represented mathematically as two points on the surface of a donut-shaped object called a torus. By making sure the paths between the points don't intersect, you can "make sure they don't crash into each other or into walls," Ghrist says.

Similar mathematics underlies an algorithm Ghrist created to keep moving cubes from bumping into each other as a morphing robot rearranges. Unlike other algorithms that control such robots, Ghrist's algorithm also works for morphing robots composed of stacks of hexagons or dodecahedrons because the underlying mathematics "doesn't depend on the engineering details," Ghrist says. And perhaps one day it will direct morphing rescue-bots flexible enough to help rescue a trapped child.

Fall/Winter 2005–06