[From Erling Jorgensen (2018.08.15 1830 EDT)]
Erling Jorgensen (2018.08.14 1545 EDT)
Adam Matic (Wed, 15 Aug 2018 09:56:24 +0200)
EJ: Is there anything that can be borrowed from this understanding, beyond
Bill's procedural method, for applying to the speed-curvature power law
phenomenon? In the speed-curvature literature, an ellipse is taken as a
simple prototype of curved lines. What if a first approximation for a
reference specification to produce an ellipse comes from the two points of
greatest inflection at the ends of the ellipse? Wouldn't a step-change
reference for position, that alternated between those two inflection points,
lead to a velocity profile that sped up between the two points and slowed down
on the curves?
AM: � I have been trying some point-to-point control schemes in 22D, did not
get far with them, don't have much to say definitively.
EJ: So when someone scribbles between two points, it’s not known whether the
deceleration and reversal of direction is linear, power law, or some other
function? I’m wondering whether change of direction along a line is a
simplified paradigm for then extrapolating to velocity along a curve. It
sounds like it’s been hard to specify the controlled variables for 2D
drawings.
EJ: I am suggesting a different PCT explanation than the statistical-
artifact/illusion one that Rick is proposing. Drawing on Bill Power’s
demonstrations with the Little Man V2 model, I am wondering about Position
control -- not instantaneous position, but a step-change in reference
position. Bill found that a simplified control model of angular position,
angular velocity, and angular acceleration, together with environmental arm
dynamics, could reproduce the detailed movements and velocity profiles
enumerated by Atkeson & Hollerbach (1985), without tangential-velocity and
other calculations specifically being controlled. Bill used a step-change in
the reference for position, and the behavioral dynamics emerged from there.
EJ: I suppose it wouldn’t work to have the Little Man follow an ellipse
target, and see what happens. If power-law dynamics are involved in mouse
movements to get the target going in an ellipse, that seems like putting the
dynamics into the reference itself. Maybe Bruce Abbott is the one to ask
here. Can a point moving in an ellipse be programmed as a target in Little
Man V2, for instance at a constant speed rather than a power-law varying one?
EJ: Admittedly, the sketch for a step-change-reference-position model
suggested above doesn't yet have an actual curve-generating portion. So a
second approximation might need to include some sine-wave reference generator,
to push the drawing away from the central axis as it passes each endpoint of
inflection. Does the literature suggest that sine-wave reproduction, at high
enough speeds, also shows a power law relationship?
AM: Parametric formula for the ellipse (with the orthogonal sine waves) does
produce a power law, at any overall speed.
EJ: I don’t understand the “at any overall speed�? portion. Your illustration
of the three ellipses with equal time-distant points showed an ellipse on the
left where the points were also equally distant spatially (beta of 1), so it
seems an ellipse can be traveled at a constant speed without it being forced
into a power law relationship. But you also emphasized, in your discussion
with Rick, that the 2/3 power law regularity emerged when the speed of the
drawing was fast enough to not deliberately counteract what may be happening
on the curves.
EJ: I’m trying to consider what happens when various repetitive phenomena --
e.g., a line between two points, a sine wave, an ellipse -- are produced not
by a formula but by a living control system. The formula for an ellipse comes
out of the two radii. A drawn version of an ellipse may relate to the
endpoints of greatest inflection on the curve. I don’t have equipment to
measure those things myself, so I have to rely on what others may have done.
I am trying (perhaps simplistically) to apply step-changes in Position
control, with two articulation points, to see whether a power-law relationship
‘falls out’ of the model as an un-controlled outcome, i.e., a side effect.
EJ: From what I’ve heard so far in this discussion, this possibility is not
(yet?) ruled out.
All the best,
Erling