Researchers explore the phenomenon of holonomy

BB-8 of Star Wars fame is known for its adorable beeps, dome-shaped head and spherical body. But fighting alongside the Resistance is just one of its many talents. As this spherical robot rolls across surfaces, it’s exhibiting holonomy, a phenomenon in rigid body dynamics that Berkeley researchers think may have broad applications in real-life robotics.

In a study published in the Proceedings of the Royal Society A, co-authors Theresa Honein, a Ph.D. student in mechanical engineering, and Oliver O’Reilly, professor of mechanical engineering and vice provost for undergraduate education, calculated all the different orientations of a rolling sphere in a quest to better understand holonomy. Their findings may lead to new strategies for achieving more efficient navigation with spherical robots.

“Our goal was to compute the change in orientation and to see if it is possible to compute a rectangular path for the rolling sphere, so that any prescribed change in orientation can be achieved,” said Honein. “The results have application to path planning and reorientation for spherical robots such as BB-8.”

But what is holonomy? “Imagine a tennis ball rolling in a closed path, on a tabletop, so that the center of the ball returns to its original location. Surprisingly, the orientation of the ball will have changed, as you’ll notice by looking at the branding markings on the ball,” said O’Reilly. “This same phenomenon is exhibited by BB-8. As the robot moves around a room and returns to its original location, the orientation of its spherical part will have changed, albeit camouflaged by the ingenious magnetic mechanism used to orient its head.”

In the study, the researchers showed that a sphere tracing a closed path returns to its original position but not always to its original orientation. They also showed that any rotation of a sphere (in three dimensions) can be achieved by tracing rectangles in the plane — and presented dimensions of three rectangles that should be traced consecutively to obtain a desired change in orientation of the sphere.

“The surprising result was that we showed how any change in the orientation of a sphere [in three dimensions] can be achieved by tracing three interlocked rectangles on the plane,” said Honein. “This finding used one of three methods for describing rotations developed by the famous mathematician Leonhard Euler in the 18th century.”

The researchers now can explain the complicated concept of holonomy in straightforward terms, using a simple example. In addition, these same ideas can be applied to more complicated motions, opening the door to pedagogical applications.

“Our work can be easily used in courses to demonstrate and explain holonomy and to help students gain additional insight into more challenging examples, including parallel parking for self-driving cars,” said O’Reilly.

Spherical robots like BB-8 might seem like a novelty, but they have potential real-world applications in surveillance, environmental monitoring, and underwater and planetary exploration. Honein and O’Reilly’s work could help ensure that these robots are ready to roll, so to speak.

“By better understanding holonomy, researchers could design algorithms to optimize robots’ pathways, potentially saving them time and energy performing these tasks,” said Honein.

For more information
Berkeley Dynamics Lab Rotations: An online, free educational resource created and supported by the Dynamics Lab