With that said, though, I believe that the movement of the actual atoms making up the mechanical hands is still happening in an "infinitely smooth" sense, although the way we measure and observe that movement (via reflections of light or perhaps some other type of electromagnetic radiation) may be limited.
You don't even need to appeal to Planck length here, because it's clear from more basic quantum mechanics that particles simply do not have classical paths.
Please don't misinterpret the Planck length. Please don't interpret it in any way, it's right in the wiki article:
>There is currently no proven physical significance of the Planck length;...
It's just dimensional analytical playing with some fundamental physical constants. Physicist believe that this scale length may have some connection with the yet to be discovered laws of quantum gravity. It has nothing to with the "finite resolution of the universe". There is no evidence that the universe has finite resolution.
With that said, though, I believe that the movement of the actual atoms making up the mechanical hands is still happening in an "infinitely smooth" sense, although the way we measure and observe that movement (via reflections of light or perhaps some other type of electromagnetic radiation) may be limited.