Binary Transmission

So the problem is that you have a limited number of motors but you want to motorize a lot of functions.
I ask, how can you power the most axes with the least possible number of motors?

The usual parts of a gear box are given:

This setup is able to transmit the force from one axis to two switchable axes. There are three states of this little fellow:

• power to first axis
• neutral
• power to second axis

If you want an automatic gadget, then an additional motor is required, to operate the shift lever.

As a naive example, suppose that you have a car and you want to drive it, and steer it as well. This requires two motors, one for drive, one for steering. However, if you use a transmission box, like above, then the same motor can drive and steer, but the shift is up to you. The operation of the shift lever allows no spare of your motors. See the LDD file 00 for demonstration.

The minimum number of motors to power two separate axes is two, this is an information theoretical limit (with this setup).

The key idea is to use binary tree to extend the number of powered axes. Two gear boxes can be set to 2×2 = 4 states. Three of these can carry out 2×2×2 = 8 states and so on. In the LDD file 01 you can see a setup with 3 motors and 4 powered axes.

Note that the shift levers on the second stage are bound together, this is the key part why you can power more axes than the number of your motors.

With this binary tree stuff you can power 2^n axes with n+1 motors: one motor for the power, and n motors for the shifts. For example 4 motors can power 8 axes.

So, 3 motors can power 4 axes, and 4 motors can power 8 axes, it means that, in theory, 3 motors can power 8 axes as well. Moreover, this calculation is recursive.

I tried to construct such a 3 to 8 transmission, see the LDD file 02.
I would firmly say, that three motors can separately power any number of axes, however the structure is uneffectively large and complex. A better thing to do is to buy more motors.