"But not too far out of line or else you'd use gears or chains, right?"
Yeah... don't think there is a lot of forgiveness in an Oldham coupler.
How efficient is this? As opposed to cut gears. Friction loss (depending on materials, bearings) appears to be larger, plus you do not reverse direction...
In addition to coupling shafts together, there was once a scheme for using an Oldham arrangement (also known as a "trammel of Archimedes") to do elliptical turnings on a lathe.
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