Hmm, this reminded me of an empirical discovery I made: When walking with an open coffee cup at hand oscillating your hand vertically in random fashion greatly reduces the coffee resonance and the sloshing effect.

This problem has been investigated before (http://www.sciencemag.org/news/2012/05/physics-spilled-coffe...) but AFAIK the above solution has not been proposed. I wonder if the theorem given can help motivate this finding?

Walk without rythm and we won't spill the coffee.

This is a funny example of spread spectrum modulation. You're decreasing the energy at the frequency of the sloshing mode by modulating it with random noise.

This isn't original to Acheson - the single pendulum case is widely known as the Kapitza pendulum.

https://en.wikipedia.org/wiki/Kapitza%27s_pendulum

If you speak Russian the following video with V.I. Arnold is nice

http://www.youtube.com/watch?v=ZBSLK3Jbd2I

This one is in English: https://www.youtube.com/watch?v=gnn21smGVrQ

The paper says in its first sentence “Stephenson (1908a, b) showed that it is possible ... He confirmed his theoretical predictions by a practical demonstration ...”

His derivation isn't nearly as complete as Kapitza's, though.

When that was discovered it was a shock to the controls community. Inverted pendulum stabilization was long thought to require active control.

Here's a video demonstrating passive stabilization.[1] It's so simple it's embarrassing.

[1] https://www.youtube.com/watch?v=rwGAzy0noU0

The last line of the paper mentions seeing a triple inverted pendulum recover from even very severe perturbations and go back to vertical. I would love to see a video of this stuff!

This is a different control mechanism but it's still a cool video :)

https://www.youtube.com/watch?v=cyN-CRNrb3E

Came across this video after watching that, somehow related but equally amazing. https://www.youtube.com/watch?v=Lt-KLtkDlh8

this one was linked by ars elsewhere in the comments here. references the paper as well. very cool! https://www.youtube.com/watch?v=gnn21smGVrQ

Sane link:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.205...

I don't understand how people put up with a wobbling "Click here to see more" on fermatslibrary.com.

What is next? Back to marquee?

Mark Levi has written about these things on SIAM and has a nice video up on,

http://www.personal.psu.edu/mxl48/Inverted_Pendulum.html

I think they've also shown that this is robustly stable for a particular family.

Makes me think about drones in particular. If you can control an inverted n-ary pendulum then controlling inverted thrust should be a straightforward extension.