EJ:Â So when someone scribbles between two points, itâ€™s not known whether the
deceleration and reversal of direction is linear, power law, or some other
function?Â Iâ€™m wondering whether change of direction along a line is a
simplified paradigm for then extrapolating to velocity along a curve.Â It
sounds like itâ€™s been hard to specify the controlled variables for 2D
Maybe that is a good idea. There is a lot of research on point-to-point movements, and Bill never published his analysis done with the LittleMan simulation.Â
EJ:Â I suppose it wouldnâ€™t work to have the Little Man follow an ellipse
target, and see what happens.Â 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.Â Maybe Bruce Abbott is the one to ask
here.Â 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?
Yes, you can also take a position control model (even without arm dynamics simulated) and see what happens. I’ve been doing that, and it seems position control is not enough here, I don’t have a complete solution.
EJ:Â I donâ€™t understand the â€œat any overall speedâ€? portion.Â 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.Â 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.
Movement at overall speed means that there might be variation in local speed, but the average speed of the total movement is relatively constant across cycles. People can traverse ellipses at low overall speed without following the power law, yes. For trajectories generated with orthogonal sinusoids, at any overall speed, speed will be correlated with curvature.
EJ:Â Iâ€™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.Â The formula for an ellipse comes
out of the two radii.Â A drawn version of an ellipse may relate to the
endpoints of greatest inflection on the curve.Â I donâ€™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
â€˜falls outâ€™ of the model as an un-controlled outcome, i.e., a side effect.
EJ:Â From what Iâ€™ve heard so far in this discussion, this possibility is not
(yet?) ruled out.
If you have only point-to-point movement, then there is no curvature, right? So, you’d need something else controlled to see what falls out.