A faster and more graceful robot, with the help of calculus

The problem:

Robots often struggle with inefficient or ineffective motions due to slow derivative computation. “Over the years, doing many problems like this in robotics and beyond, derivative computation was always a pain point,” said Rakita, assistant professor of computer science. “It was always a central bottleneck holding back the kinds of applications that we wanted to build. Eventually, I just got really determined and thought, ‘It’s time to finally figure this out.’”

The solution:

Typically, robots calculate each derivative from scratch, treating every computation as an entirely separate process. Researchers in Rakita’s lab, including PhD students Chen Liang and Peter Wang, recognized that derivatives do not have to be computed in isolation; it’s common to calculate a sequence of derivatives, each one building on the last. One of the key insights of this work is that derivative computation can be accelerated by reusing information from previous, related calculations—a strategy known as coherence. The team’s novel approach, called the Web of Affine Spaces (WASP) Optimization, enables robots to calculate derivatives about seven to 10 times faster than traditional methods.

“When you put them side by side, it really feels like the others are stuck in the mud,” he said. “They just take a lot longer to converge on good motions in comparison.”

The speed-up isn’t just a matter of convenience—it fundamentally changes the robot’s capabilities. “The robot can now consider more options and plan further into the future,” Rakita explained. “For example, in a cluttered environment, if there are obstacles in its path, it can proactively choose motions that gracefully reach around them, something that would be too computationally expensive to consider in real time with slower methods.”

Why this matters: 

Robots are increasingly becoming a part of everyday life, in workplaces, and eventually in the unstructured environments of homes. To succeed in these settings, they’ll need to operate with unprecedented speed, agility, and precision. The WASP derivative approach developed by Rakita’s lab represents a significant step toward making that future possible.

Going forward: 

Rakita said his lab is now working on ways to scale up the method. 

“Right now, this strategy works well for functions with up to about one thousand combined inputs and outputs,” Rakita said. “By comparison, modern neural networks often have millions or even billions of parameters. So, we need to come up with some new math tricks to scale WASP up to these larger functions.”

And the approach has broad potential applications, ranging from aeronautics and computational biology to weather prediction. Even within robotics, there’s much more to explore. For example, Rakita notes that WASP derivatives could help robots better reason through the physics-based outcomes of their own motion, such as anticipating when they might knock something over and adjusting their movement in advance to intentionally brush the object aside in a controlled way.

“We’re only beginning to scratch the surface of where these derivatives can be applied—even within robotics,” he said. “They open up new possibilities for robots to reason about how their actions affect the physical world.”