Pushing a shovel through the snow, planting an umbrella on the beach, wading into a ball pit, and driving over gravel has one thing in common: they are all exercises in sneaking, with an intruding body applying some force to move through soft and granular material.
Predicting what it takes to propel through sand, gravel, or other soft media could help engineers pilot a rover over Martian soil, anchor a ship in rough seas, and walk a robot through sand and mud. But modeling the forces involved in such processes is a huge computational challenge that often takes days to weeks to solve.
Now, engineers at MIT and Georgia Tech have found a faster and simpler way to model infiltration through any soft, flowable material. Their new method quickly quantifies the forces it would take to push, oscillate, and etch an object through granular material in real time. The method can be applicable to objects and grains of any size and shape, and it does not require complex computational tools as other methods do.
“We now have a formula that can be very useful in settings where you have to check out many options as quickly as possible,” says Ken Camren, a professor of mechanical engineering at MIT.
“This is particularly useful for applications such as real-time route planning for vehicles traveling across vast deserts and other off-road terrains, which cannot wait for the current slower simulated routes to determine their path,” adds Shashank Agarwal SM ’19, Ph.D. 22.
Kamrin and Agarwal detail their new method in a study published this week in the journal Proceedings of the National Academy of Sciences. The study also includes Daniel I. Goldman, professor of physics at Georgia Tech.
In order to figure out how much pressure is on an object to move it through the sand, one can turn to bead by grain, using discrete element modeling, or DEM — an approach that systematically calculates the motion of each individual grain in response to a given force. DEM is subtle but slow, and it can take weeks to fully solve a practical problem involving just a handful of sand. As a faster alternative, scientists can develop continuum models, which simulate granular behavior in generalized chunks, or clusters of grains. This simpler approach can still create a detailed picture of how the grain is flowing, in a way that can shave a weeks-long problem down to days or even hours.
“We wanted to see if we could do a better job than that and cut this process down to seconds,” says Agarwal.
The team looked at Goldman’s previous work. In 2014, he was studying how animals and robots move through dry, granular materials such as sand and soil. In his search for ways to quantitatively describe their movements, he found that he could do so with a quick relation originally intended to describe fluid swimmers.
The formula, Resistive Force Theory (RFT), works by considering the surface of an object as a collection of small plates. (Imagine representing the ball as a soccer ball.) When an object moves through a fluid, each plate is subjected to a force, and RFT claims that the force on each plate only depends on its direction and local motion. The equation takes all of this into account, along with the individual properties of the fluid, to ultimately describe how the object as a whole moves through the fluid.
Surprisingly, Goldman found that this simple approach was also accurate when applied to granulomatous infiltrates. Specifically, he predicted the forces exerted by lizards and snakes to glide across the sand, as well as how the tiny two-legged robots would walk over the soil. The question, Camryn says, was why?
“It was a strange mystery why this theory, which was originally derived for movement through a viscous fluid, would work at all in granular media, which have a completely different flow behavior,” he says.
Camren took a closer look at the mathematics and found a relationship between RFT and a continuum model he derived to describe granular flow. In other words, the physics checked, and RFT could indeed be an accurate method for predicting granular flow, in a simpler and faster way than conventional models. But there was one big limitation: the approach was mainly practical for two-dimensional problems.
To model the intrusion using RFT, one needs to know what would happen if one moved a plate in every way possible – a task that can be managed in two dimensions, but not three. Then the team needed some shortcuts to simplify the 3D complexity.
In their new study, the researchers adapted RFT to 3D by adding an additional element to the equation. This component is the torsion angle of the plate, and measures how the plate’s orientation changes as the entire object is rotated. When they combined this extra angle, as well as the plate’s inclination and direction of motion, the team had enough information to determine the force acting on the plate as it moved through the 3D material. Importantly, by exploiting contact with continuous modeling, the resulting 3D-RFT can be generalized, and easily recalibrated for application to many dry granular media on Earth, and even to other planetary bodies.
The researchers demonstrated the new method using a variety of 3D objects, from simple cylinders and cubes to more complex rabbit- and monkey-shaped geometric shapes. They first cut the objects up, representing each one as a collection of hundreds to thousands of small plates. They then applied the modified RFT formula to each panel individually and calculated the forces that would be required over time to etch each panel, and eventually the entire object, down through a layer of sand.
“For more of the weird, bunny-like stuff, you can imagine having to constantly change your loads to keep digging straight down,” Camryn says. “And our method can even predict those tiny vibrations, the force distribution throughout the rabbit, in less than a minute.”
The new approach provides a fast and accurate way to model granular infiltration, which can be applied to a range of practical problems, from driving a spacecraft through Martian soil, to characterizing animal movement through sand, and even predicting what it takes to uproot a tree.
“Can I predict how hard it will be to uproot the natural vegetation? You might want to know, Is this tree going to be bombed by this storm?” Camryn says. “This is a way to get a quick answer.”
This research was supported in part by the Army Research Office, the US Army’s DEVCOM Ground Vehicle Systems Center and NASA.