[From Erling Jorgensen (2018.08.14 1545 EDT)]
“Adam Matic” (adam.matic@gmail.com via csgnet Mailing List) csgnet@lists.illinois.edu 8/14/2018 5:34 AM
From the beginning, the phenomenon of the speed-curvature power law in movement trajectories is meant as a test of any model that draws curves. If the model is good, it would show the same invariances in output that humans show, without (probably) having those invariances in the reference. That is what Bill did with velocity profiles - he demonstrated that step-changes in position control, plus arm dynamics, are enough to replicate the bell-shaped velocity profiles found in real data.
Hi Adam,
EJ: I’ve been thinking about this matter of the speed-curvature power ‘law’ (i.e., correlation) emerging from something else that control systems are doing when curves are produced. About 10 days back, Rick posted – in [Rick Marken 2018-08-03_22:07:26] – a discussion Bill Powers had back in 1993 with Greg Williams – [From Bill Powers (931029.0750 MDT)]. As an aside, that’s about a year after I first got to meet both Bill & Greg, at the 1992 CSG Conference in Durango. Bill’s post contained an observation about the Little Man-version 2 model:
BP: It just happens that when a control system is
given a step-change of reference-signal, the trajectory of the
controlled variable naturally scales up or down so that the
velocity rises and falls along the same generic curve. This is
purely a consequence of the mathematical relationships of control
and the passive dynamical properties of the arm; nothing is
acting to make sure that the trajectory follows any particular
path. The trajectory is a side-effect, not a planned movement.
EJ: I notice that Rick recently posted – in [Rick Marken 2018-08-13_18:02:27] – a fuller discussion Bill had in 1995 about controlled variables vs. side-effects – [From Bill Powers (950527.0950 MDT)]. There he reiterated:
the system’s operation are basically side-effects; they are not planned
and they are constant only in a constant environment. Furthermore, they
are unknown to the behaving system and play no part in the production of
behavior. We can deduce from the model of the behaving system what the
observable side-effects would be in a given environment, and so can
compare those side-effects with our external observations of the
behavior. But our explanation of the behavior is not based on those
side-effects.
BP: The trajectories of movement that result from
EJ: This is how I understand your comment above, that the appearance of a speed-curvature power relationship is a rigorous enough descriptive regularity that it can serve as a test of curve-producing models – i.e., ‘Does a given model generate that outcome, without (necessarily) controlling for that result?’
EJ: What struck me about Bill’s descriptions of the Little Man V2 model is this step-change in the reference signal for target position. Because of the proportional and integral nature of the output function, the end of the Little Man arm begins to accelerate in the needed direction, but the velocity is slow to scale up. Similarly, as the error decreases with approach to the target, the (integrated) velocity is slow to scale back down again. So the position reference produces a step function, the arm position follows a sigmoid function, and the velocity profile is bell-shaped, (if I’ve got the pieces arranged correctly.)
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?
EJ: I have no idea whether a power law relationship would emerge. I suppose one test might be to see whether drawing a straight line back and forth between two points (in effect, ‘collapsing’ the ellipse) showed any power law dynamic in relation to each change of direction.
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?
EJ: These suggestions are in partial agreement that the speed-curvature power law relationship may well be a side-effect of other factors in play with control system operation. I don’t agree with the Marken & Shaffer (2017) formulation of the issue. Basically, I cannot trust their treatment of the mathematics, because I have yet to see an adequate response to the tautology critique, nor even an understanding of it as a potential problem.
EJ: I’d be interested in whether any of your research would tend to rule out or rule in the above suggestions.
All the best,
Erling
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