" The pendulum analogy also does not apply to double-limb support, where a pendulum (inverted or otherwise) clearly cannot swing."
This is actually a bad analysis on the part of the author. Double limb support can be treated as just an inverted pendulum from the metacenter located between the supports -- which are not normally nearly broad enough to create much propping moment against an applied overturning force -- and when they are-- they are only in one plane and so the treatment is a perfect inverted pendulum in the perpendicular to the plane running between the feet on a long-base kamae.
In other words -- the human body has effectively zero static stability -- all stability is dynamic -- like a bicycle.
Plus we are talking about a system where we have some other object as a link in the pendulum chains -- be it q weapon or opponent. This makes me chuckle when I see the video of Ueshiba with the boar spear form of chinkon kishin starting at 4:50 where he points at his center -- and then watch
what he does with the spear to illustrate the point he is making about his center. And the same again at 5:00. He also has this recurring little reflexive "pop and drop" to the balls of his feet and back to the flats. His performance of various techniques in other respects has always had a bit of "hop" to it, and I am hardly the first to have said it.
At 5:28 he shows a spherical spiral resolving to a vertical thrust up -- and then a reverse spiral ending in a vertical thrust down. Accident? OK, if you say so -- but I see what he is getting at.
It ends as it began in a vertical orientation of the body and spear. Zhan zhuang, anyone ? The entire exercise can be viewed as an exploded view of miniscule excursions of the balance center one can feel when doing zhan zhuang-- and which you will
notice if you stand there long enough -- and which -- might
just be the point of the exercise demonstrated....
What we don't usually recognize is that we have a normal tonic vibration which is -- as you say below -- normally highly damped. Essential tremor patients lose the neurological damping element of normal tone -- and so they shake involuntarily -- but at a characteristic frequency, ~5-10 Hz or so -- the resonant fundamental frequency and first harmonic of the human body -- and the latter is precisely the frequency of a proper furitama
. This is hardly a coincidence in its relation to aikido -- as this vibration is characteristic of all the shaking forms of chinkon kishin or aiki-taiso, call it as as you prefer. "Spirit of bees" Ueshiba called it.
"Although the pendulum analogy is important for understanding how walking can be economical, it does not explain why walking costs energy at all. In that respect, the inverted pendulum model is incomplete."
I believe that I said "some kinks remain." More importantly, the study was looking at energy-optimizing
aspects of this system for locomotion -- we are talking about optimizing exploits of instability and structural damage
-- and energy conservation is largely a secondary aspect.
For example, "In real oscillators, friction, or damping, slows the motion of the system."
Too true, but the existence of damping can be modulated-- and normally IS modulated. By modulation, a moment (a kinetic potential) can be created and stored -- in the structure
-- which I believe is a major point of the discussion ...
Control of the damping mechanisms is key to the release and deployment of such potentials. This has a strong component of reflex action that the chinkon kishin/aiki-taiso training is design to tune into the neuro-muscular system at a deep (spinal/cerebellar) level -- not voluntary motor control -- and thus extremely responsive -- spooky fast -- far faster than any voluntary action is capable of.
And no, Ueshiba didn't say so. Current literal translations show that Ueshiba stated and restated ancient martial training theories in regards to IP/aiki.
This is probably so. I don't think any one is disputing that this line of empirical development goes very deep and has many branches -- but we are only just now starting the analytical examination of it.
When people can move things without moving anything physically -- I'll drop talk about physics. But until then -- "Il muove.