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All of the examples of keeping a double/triple inverted pendulum balanced using a PID controller I've seen seem to be on a cart. Like this one https://www.youtube.com/watch?v=cyN-CRNrb3E

How come the PID controller always controls a cart rather than a servo that holds the first pendulum? The second/third pendulum could be connected loosely on the first pendulum and the PID controller controls the first pendulum. Is it because servos tend to be too slow or are there other reasons?

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There's no reason why you couldn't do it with a servo on the first pendulum, but the problem is very different. Instead of motion constrained on a line (cart), motion is constrained along an arc. The dynamics change dramatically due to both vertical and horizontal motion along the arc, but that's not to say that it couldn't be done. As far as I know, it's just more complicated and not as well studied a problem as the cart problem.

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