Walking and slithering aren’t as different as you think

New mathematical model links up slithering with some kinds of swimming and walking, and it could make programming many-legged robots easier.

Abrahamic texts treat slithering as a special indignity visited on the wicked serpent, but evolution may draw a more continuous line through the motion of swimming microbes, wriggling worms, skittering spiders and walking horses. A new study found that all of these kinds of motion are well represented by a single mathematical model.

“This didn’t come out of nowhere—this is from our real robot data,” said Dan Zhao (MSE Electrical and Computer Engineering, PHD Mechanical Engineering), first author of the study in the Proceedings of the National Academy of Sciences.

“Even when the robot looks like it’s sliding, like its feet are slipping, its velocity is still proportional to how quickly it’s moving its body.”

Unlike the dynamic motion of gliding birds and sharks and galloping horses—where speed is driven, at least in part, by momentum—every bit of speed for ants, centipedes, snakes and swimming microbes is driven by changing the shape of the body. This is known as kinematic motion.

The expanded understanding of kinematic motion could change the way roboticists think about programming many-limbed robots, opening new possibilities for walking planetary rovers, for instance. Shai Revzen, a professor of electrical and computer engineering at U-M, senior author and director of the study, explained that two- and four-legged robots are popular because more legs are extremely complex to model using current tools. 

“This never sat well with me because my work was on cockroach locomotion,” said Revzen. “I can tell you many things about cockroaches. One of them is that they’re not brilliant mathematicians.”

And if cockroaches can walk without solving extremely complex equations, there had to be an easier way to program walking robots. The new finding offers a place to start.

Slipping feet complicate typical motion models for robots, and the assumption was that it might add an element of momentum to the motion of many-legged robots. But in the model reported by the U-M team, it is not so different from lizards that “swim” in sand or microbes swimming in water. Because microbes are small, the water seems a lot thicker and stickier—as if a human was trying to swim in honey. In all of these cases, the limbs move through the surrounding medium, or slide over a surface, rather than being connected at a stationary point.

The team discovered the connection by taking a known model that describes swimming microbes and then reconfiguring it to use with their multi-legged robots. The model reliably reflected their data, which came from multipods—modular robots that can operate with 6 to 12 legs—and a six-legged robot called BigAnt. 

The team also collaborated with Glenna Clifton, an assistant professor of biology at the University of Portland in Oregon, who provided data on ants walking on a flat surface. While the robot legs slip a lot, up to 100% of the time for the multipods, ant feet have much firmer connections with the ground, slipping only 4.7% of the time. Even so, the ants and robots followed the same equations, with their speeds proportional to how quickly they moved their legs. It turned out that this kind of slipping didn’t alter the kinematic nature of the motion.

The charting of an ant in motion to show its parts
Data gathered from ants helped to show that many-legged walking, slithering and some kinds of swimming follow the same mathematical model.
Credit: Clifton Group, University of Portland

As for what this suggests about how walking evolved, the team points to the worm believed to be the last common ancestor for all creatures that have two sides that are mirror images of each other. This worm, wriggling through water, already had the foundations of the motion that enabled the first animals to walk on land, they propose. Even humans begin learning to propel ourselves kinematically, crawling on hands and knees with the three points of contact on the ground at any time.

The skills of managing momentum—running with four legs or fewer, walking or running on two legs, flying or gliding—those ladder on top of that older knowledge about how to move, the researchers suggest.

This research was supported by the Army Research Office (grants W911NF-17-1-0243 and W911NF-17-1-0306), the National Science Foundation (grants 1825918 and 2048235), and the D. Dan and Betty Kahn Michigan-Israel Partnership for Research and Education Autonomous Systems Mega-Project.

Zhao is now a senior controls engineer at XPENG Robotics.