Consider the wheel. Round. Dependable. Boring. Hasn’t been redesigned in centuries. Positively Neolithic.
Now consider the spiral. Eccentric. Open-ended. Captivating. Native Americans carved spirals as petroglyphs. The Greek mathematician Archimedes wrote about them around 225 B.C. Jacob Bernoulli, a 17th century mathematician, had a spiral engraved on his tombstone.
As an alternative to the lackluster wheel, the quirky spiral may be the key to forward motion for people who have trouble walking on their own.
Round and round
Logarithmic and Archimedean spirals have very different properties. The logarithmic spiral (above left) maintains a constant angle, the camming angle, between its tangent and its radial line. The Archimedean spiral (right) maintains a constant distance between coils.
Credit: E. Otwell
Bernoulli’s epitaph may capture the spiral’s essence best: Eadem mutata resurgo. “Though changed, I shall arise the same.” He meant that the spiral seems to grow and replenish itself endlessly as it spins around. But his motto could just as well describe the way that engineers keep discovering and rediscovering spirals.
Spirals come in different varieties. The Archimedean spiral looks like a coiled rope or the grooves on a vinyl record. This kind of spiral travels outward (or inward) an equal distance on each turn. Bernoulli was partial to the logarithmic or equiangular spiral, which spreads out as it travels outward, like a nautilus shell. Unfortunately, the stonecutter carving Bernoulli’s tombstone made a mistake and engraved an Archimedean spiral instead of the logarithmic one Bernoulli wanted.
In any spiral, a line drawn from the center of the curve to any given point makes an oblique angle with the tangent to the spiral at that point. In a logarithmic spiral, this angle, the spiral’s “camming angle,” never varies. In the Archimedean and other types of spirals, the camming angle is not constant.
Spirals had a modern-day rebirth among rock climbers in the 1970s. Former aerospace engineer-turned-rock climber Ray Jardine invented a mechanical device to break a climber’s fall. The device used four spring-loaded cams, each shaped like a logarithmic spiral. When placed into a vertical crack in the rock, the cams rest unless the hiker misses a step or slips, causing his or her weight to pull the cams down the crack. The cams grow in width as they rotate. Eventually, they jam tightly against the walls of the crevice, creating a frictional force that stops the climber’s fall.
Jardine’s device, aptly called the Friend, “changed the game of climbing,” says John Middendorf, a climber and engineer from Tasmania who wrote the first published article explaining how it works. “At first there was a huge controversy, because some people thought it made the sport too easy,” Middendorf says. Indeed, Jardine was able to climb some routes previously considered impossible. For example, he was first to ascend the sheer vertical Phoenix in Yosemite National Park.
A climbing buddy
A spring-loaded camming device (top) can save a climber from falling when inserted into a vertical crack in the rock. The climber’s weight (P in the illustration above) causes the spiral cams to rotate and expand until they jam against the sides of the rock. The resulting frictional force (F) stops the climber’s fall.
Credit: Jovino/Flickr (CC BY-NC 2.0); E. Otwell. Source: J. Middendorf
No matter how far the cam rotates before it jams in the crack, the force it exerts on the rock will be the same and it works reliably well within a range of crack widths. The design was supposed to be a trade secret, but Middendorf wrote about it in 1985 because he was concerned that climbers were beginning to use homemade imitations that were not safe. A camming device with the wrong camming angle, or even worse, the wrong kind of spiral, might fail to hold. It’s an easy mistake to make: Just ask Bernoulli’s stonecutter.
From vertical to horizontal
The spiral cam, for 40 years a friend to climbers, may soon become a friend to stroke victims as well. The principle is the same: to redirect a vertical force into a nearly horizontal one. The vertical force of a person’s weight, stepping on a shoe with spiral wheels, causes the shoe to roll either forward or backward, depending on the desired effect, and thereby to even out an asymmetric gait.
People with gait impairments limp in a predictable way. They spend more time on their strong leg, taking longer strides to step onto the weak leg. Amy Bastian, a neuroscientist at Johns Hopkins University, discovered that a specially designed treadmill could help limping patients improve their gaits. The treadmill has two separate belts; the one on the strong side can be programmed to go twice as fast as the one on the weak side.
“You exaggerate the asymmetry, to make them want to lengthen that short step,” says South Florida’s Reed. When the patients got off the treadmill, they would continue to exaggerate their shorter step, and their limp would be reduced. “After 15 minutes of practice on a treadmill, you can get large changes that you don’t see with any other treatment I know of,” says neuroscientist Erin Vasudevan, who worked in Bastian’s lab from 2007 to 2010 and collaborated with Reed while there.
But split-belt treadmills are expensive and not commonly available. Also, patients do not always retain their gains over the long term. Reed started thinking about a portable device that patients could take with them — a shoe that would offer the therapeutic benefits of a treadmill while allowing them to go about their daily lives.
The challenge was to make a shoe that glides backwards when stepped on, as if the walker were on a treadmill. Reed’s first model had ordinary circular wheels and wasn’t predictable enough for users to get the hang of. “It felt a little bit as if you were slipping on ice,” says Vasudevan, who is now at Stony Brook University in New York. When Handžić joined Reed’s lab as a student in 2009, Reed assigned him to improve the design. He suggested that Handžić try spiral wheels.
Handžić started with Archimedean spirals and logarithmic spirals. The Archimedean spirals didn’t work well because they didn’t generate a constant backward force. The logarithmic ones didn’t feel right either. To simulate the steady pace of a treadmill, the body expects the combined force of the foot and the spiral wheels to be constant. In a natural stride, the planted leg does not exert a constant backward force, so the design of the shoe must compensate.
If the shoe fits
The Gait-Enhancing Mobile Shoe (prototype right) is equipped with spiral wheels that simulate walking on a treadmill.
Heel strike Rear wheel touches down and begins rotating backward, redirecting the user’s weight to a mostly horizontal force.
Mid stance Front wheel touches down so both wheels contact the ground and rotate backward.
Terminal stance A properly designed spiral wheel must compensate for the variations in a human step to produce a steady backward force.
Toe off As the user lifts the foot, the spring-loaded spiral wheels reset themselves.
Credit: I. Handžić and K. Reed/USF
After months of frustration, Handžić finally had a revelation. Instead of prescribing the wheel shape, he could prescribe the forces. He had been stymied by the fact that the force variables (the desired force profile for his user) and the geometric variables (the shape of the wheel) were tangled up together in a complicated equation.
He had been trying to use a computer to find an approximate solution. But he realized he could use a mathematical method called separation of variables to disentangle the forces and the geometry, and translate perfectly from one to the other. Handžić had recently reviewed this trick while studying for an exam, never suspecting that it would have any use for his research.
There is no record of what the inventor of the wheel did when he saw his creation roll for the first time. But Handžić recorded in his Ph.D. thesis what he did when he discovered the separation of variables idea: “I ecstatically jumped, pumped my fist, [and] jump shot my soft drink can into the trash can, while repeating the words, ‘That’s it!’ ” He could finally custom-design a wheel for any application.
Still, spirals have one limitation as a locomotive device: They end. That’s why cars don’t travel on spiral wheels. For walking, however, spiral wheels can work. As the foot lifts up and the wheel is in the air, the wheel has time to reset itself to its starting position. Eadem mutata resurgo, indeed.
Handžić has taken the resetting idea to the extreme with a skateboard he calls the kinetic board. The rider starts leaning back on the rear wheels and rolling forward. (Unlike in the gait-enhancing shoe, where the wheels roll backward to simulate a treadmill, the skateboard’s spiral wheels are configured to redirect the rider’s weight into a forward force.) Just before reaching the end of the spiral, at which point the rear wheels are unable to roll forward any more, the rider rocks forward onto the front wheels, lifting the rear wheels off the ground so they can reset. Rock back, roll forward. Rock forward, roll forward.
Handžić and Reed have no plans to market their skateboard. However, other self-propelled skateboards, called RipStiks and waveboards, have done well on the toy market, and elementary school students who visit Reed’s lab want to give the rock-and-roll skateboard a try. “Their cam-powered scooter looks really fun,” says Andy Ruina, an engineer at Cornell University who works on robot and human locomotion. He was not involved in the research. “It’s always nice to see new ways to propel things.”
The gait-enhancing shoe, on the other hand, has a patent pending and has been licensed by a medical device company called Moterum LLC of Tampa. Reed and Handžić have tested it on healthy volunteers, and the first clinical trial with 12 stroke patients is planned for this fall. The stroke patients will wear the wheeled shoe on their healthy side (simulating the faster treadmill) and a height- and weight-matched shoe with no wheels on the weaker side (simulating the slower treadmill). The inventors hope it will retrain the patients more effectively than regular split-belt treadmill therapy so that the patients eventually won’t need the shoe.
“They’ve done their engineering homework, and it seems promising, but the studies haven’t been done yet,” says Darcy Reisman, a physical therapist at the University of Delaware in Newark who has researched the effects of split-belt treadmill therapy. “Patients always surprise us.”
A better crutch
Handžić and Reed also hope to reinvent the crutch. Normal crutches have a flat tip, and at the apex of the crutch’s motion, the user has to provide all of the propulsive force, as if pushing over a hill on every step. A spiral tip can make the job easier. At the apex, it converts vertical weight into a forward force. Or if the user is going downhill and wants to slow down, the crutch can be turned around to convert vertical weight into a backward force.
In experiments, volunteers said the spiral crutch required less effort than a conventional crutch. Eventually, the researchers envision making a crutch with a tip that could change its shape so the crutch could stand up on its own, freeing the user’s hands for other purposes. “Having a crutch or cane be able to stand up independently is a feature that several chronic users of canes are excited about,” Reed says.
Meanwhile, at least one person’s life has been changed by the research: Handžić himself. Growing up in Bosnia and Herzegovina, he had little idea of what research meant. He entered graduate school, he says, for the sole purpose of earning a higher salary. But the spirals have captured his imagination and showed him that research is about passion, not money. Says Handžić, who hopes to continue research in academia or industry, “It was a cool experience to put the mathematical theory together with practice and have something come out that is so beautiful.”
Dana Mackenzie is a freelance mathematics and science writer based in Santa Cruz, Calif.
This article appears in the November 15, 2014 issue with the headline, “Spiral Steps: The spiral’s unique characteristics may help disabled people get where they’re going.”