Tops, yo-yos, and other spinning toys are amongst the oldest playthings created by man, with the earliest examples dating back to 3,500 BC. Paradoxically, they’re not very easy to make with their design requiring a lot of trial and error. One mistake and, instead of a pirouetting plaything, you get a clattering paperweight. That’s why spinning toys tend to be symmetrical – until now. In a blow for symmetry, Disney Research Zurich and ETH Zurich have developed a computer algorithm that can take any shape, no matter how cock-eyed, and make it spin like a top.
The crux of the problem is the "moment of inertia." Without getting into some tedious and eye-watering mathematics, the moment of inertia is one of the main properties of a spinning object that determines if it falls over or not. If the moment of inertia is optimized the object rotates at right angles to a selected axis, and it stays up. If not, the object is off balance and the result is a bit like a washing machine that tries to walk across the room during the spin cycle.
How it worksThe Disney algorithm takes a solid, asymmetric object that should fall down in a second and optimizes its rotational dynamics. In practical terms, it works a bit like repositioning the wet duvet so the washing machine gets back into balance.
The algorithm maps the interior of the object and redesigns it using "adaptive multi-resolution voxelization." The algorithm then digitally removes material from the interior until the object counterbalances against the asymmetry, so it will spin around any selected axis. The new design is then run through a 3D printer to produce a new object with the right shaped voids inside and a lower center of gravity that should spin quite nicely.
If the object is so eccentric that putting voids isn't enough to get it to spin, or the central body of the shape is too flat for any voids, the algorithm falls back on a process called cage-based deformation. In this, the algorithm alters the shape of the object slightly, so it spins along the desired axis.
Finally, if neither of the above processes does the job or the shape can’t be altered, the algorithm uses "dual-density optimization." This means that the algorithm solves the problem along the lines of loading a dice. Instead of hollowing it out or tweaking its shape, the algorithm adds a second, heavier material to the inside of the object as a counterbalance, so it spins properly. It then designs a void for the new material, and a mold for the heavier material that can be cast and inserted into the spinning object to balance it.
"Our approach is effective on a wide range of models, from characters such as an elephant balancing on its toe, or an armadillo break-dancing on its shell, to abstract shapes," says Moritz Bächer, a post-doctoral researcher at Disney Research Zurich. "It’s well-suited to objects that can be produced with a 3D printer, which we used to make tops and yo-yos with unusual shapes but remarkably stable spins."
According to Disney Research, the moment of inertia is fundamental to mechanical systems, so the algorithm is useful for more than just toys. By optimizing the inertial properties of individual parts of a machine, the entire device can operate with greater efficiency.
The Disney research will be presented at ACM SIGGRAPH 2014 this week in Vancouver, Canada.
The video below outlines the algorithm’s process.
Source: Disney Research
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