kasey rios asberry
Leonardo Digital Reviews
mitpress.mit.edu/e-journals/Leonardo/ldr.html
On Tue, 14 Apr 1998, Darren Kelly wrote:
>
> Dear tricky Prof. Liz,
>
> > (Incidentally, these papers are from nonlinear dynamics journals like
> > _Chaos_ and _Nonlinearity_, so I would not recommend them for the
> > non-mathematicians in the audience.)
>
> I'm a mathematician in the audience. Besides, why not discuss such matters
> conveniently on dance-tech if they lead to interesting dance ? There are
> discussion groups for algorithmic music composition, so dance-tech
> must surely encompass nitty-gritty algorithmic dance-composition and
> discussions. I suggest we keep chatting via private email from here on
> as regards inverted pendulums (such as below), but discussions on
> algorithms for generating dance could happily reside here.
>
> ---
>
> > Incidentally, local linear control works just fine on the inverted
> > pendulum, and the textbook controllers so constructed are not chaotic.
>
> I've done this with a ruler with physics students, and you have to be
> rather unlucky for one to get who can move their finger fast enough to
> keep it up. I wouldn't say that "linear control works just fine", it
> depends entirely on regime, and for household items like a wooden ruler
> it's much easier to use chaos than high-frequency linear control to keep
> it up.
>
> Also, one must distinguish between the 1D case and the 2D case. If you
> try to balance a 2D rod on your finger it is much harder than trying to
> balance a rod constrained to move in a plane, such as an inverted pendulum
> held by a (possibly moving) axle moving through its base. In the 2d case
> the finger (controller) has to move in all directions. If you choice a
> suitably high (single) frequency for your finger (which will have to move
> in two dimensions) you end up moving in a small circle very quickly, a
> difficult motion (this has nothing to to do with the mathematical
> solution, but with fingers). The chaotic 2D motion is far easier and
> successful, and that's why the demonstration works.
>
> Still, you're right that I oversimplified it. I'll dig up some
> hard numbers.
>
> > (Haim Bau and collaborators suppress chaos with a NON-chaotic
> > controller perturbation, BTW: see Phys Rev Lett 66:1123.)
>
> Interesting twist. This reciprocal behaviour makes intuitive sense, but
> sometimes such intuition can be badly wrong.
>
> > The driven pendulum on my desk, for instance, balances easily at the
> > inverted point if the drive frequency is high enough - a phenomenon called
> > parametric resonance.
>
> What do you drive your inverted pendulum with ? How long and how heavy is
> it ? You can construct a cute diagramm showing the solution regions as
> function of a dimensionless frequency against a scaled mass-to-length
> variable. I can't remember where it is for a light wooden ruler, but the
> cutoff is safely above what anyone will manage. The cute thing is that it
> is much easier to keep it up with modest chaotic motion.
>
> I'm off home to prevent some chaos there.
>
> Darren
>
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