Exactly the same way as the middle layers. A motor rotates each outer layers.
On the other hand, the complicated thing to understand is that the cube has 6 outer layers (left, right, front, back, top, bottom). But in this kinematics, only 3 outer layers move (left, front, top) and the 3 others remain fixed during the resolution, hence the fact that we see the fixed corner common to all these 3 fixed layers. If the algorithm calculates that we must move a fixed layer, the 3 others will move in the other direction. That's why, although it's a 4x4x4 cube, there are 3 motors per dimension, so 9 motors in total.
It was for the exercise of filing a patent without the help of a lawyer. I've always wanted to see how difficult it is. Although this is a novel manipulation device, I'm not sure if it will ever have an industrial application, but it's been accepted so it's cool, I'm glad. I don't care about intellectual property and I wouldn't sue anyone.
It looks like a nice exercise, but for a real patent you sadly need a real layer. There are a lot of standard patenting tricks to ensure nobody can make a stupid generalization of your patent and get a patent that includes your idea. Does you patent apply to cubes with N layers? Does your patent apply with cubes painted with less (or more?) colors?
I worked very hard to get it accepted as is. But it's obvious that if a professional had written this patent, it would have been more general and robust. But it's expensive, especially to make it world-wide, and that's not at all what I want. This patent was a non-technical aside in this project.
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