Study of abstract spaces is accomplishing real-world feats, such as operating robots remotely.
A snake-like robot worms its way through holes and inaccessible areas at a disaster site, looking for survivors and sending back images to relief workers at a remote location. As the snake-like robot reaches a wall, it suddenly changes shape, becoming the kind of robot that can climb walls and other barriers.
For such transformers to become a reality, one of the key challenges will be shape planning, or figuring out how a robot can efficiently move from one shape to another. To pull off such a technological feat, the esoteric science of “topology” has answers, says Robert Ghrist, a professor of mathematics in LAS.
Topology, which is the study of abstract spaces and shapes, has been around for 100 years as a theoretical tool. But thanks to the work of researchers like Ghrist, topology is making the crossover into the world of real-life engineering problems.
Ghrist is leading a project dubbed “Sensor Topology and Minimal Planning,” or SToMP, which is looking at how topology can be used for a multitude of purposes, such as creating shape-shifting robots, developing sensor networks, or coordinating teams of robots working on a factory floor.
To use topology, Ghrist says, researchers take a real, three-dimensional space and convert it into an abstract space—the mathematical equivalent to “trading spaces.” Then they use tools from topology and geometry to gather information about that abstract space. The final step is to translate this information back to the real-world space, using the information gathered to solve various problems.
To illustrate, let’s say you have a security network of sensors that is monitoring movement in a specific space—a field, for example. It is critical that there be no gaps or holes in the sensor network where a person in the field can go undetected, Ghrist says. To pinpoint gaps, mathematicians can use topology tools such as homology theory, which is effective in determining whether shapes have holes.
Ghrist says to think of each sensor as being surrounded by a disk representing the area of the field it covers. The union of all of these disks forms what is called the “sensor cover.” Mathematicians can use homology theory to study the shape of the sensor cover and determine whether there are any holes in the network—spots on the field where an intruder might go undetected.
“This works surprisingly well,” he points out.
Not only does it work well, it simplifies the process at the same time. Ghrist calls it “minimality” because topology cuts out much of the superfluous information about the space being studied.
“Sometimes more information is harmful,” he says. “You waste resources processing unnecessary information.”
Ghrist is working with fellow U of I mathematicians Stephanie Alexander and Richard Bishop, whom he describes as “absolutely the two best people around who know these geometric techniques.”
The beauty of topology is that it can be used with many types of complex systems and problems. It can even make it possible to keep track of robots’ locations without using expensive, bulky, global-positioning systems.
“All of these problems look really different,” Ghrist says. “But when you translate them into an abstract space, they are remarkably similar and amenable to the same tools.”
By Doug Peterson