Kennaway talk IAPCT 2019

Warren recently uploaded the talk from last year’s IAPCT conference, the first part of the video is Rick Marken’s talk, and the second part is Richard Kennaway, power law and gait planarity. The slides are also available somewhere on the CSGnet. I saw it and wanted to comment and maybe discuss a few things with Richard (in cc)

Some of the things I agree with, I like the conclusion - the first step toward understanding behaviors should be something like the test for the controlled variable, that is how we see what is controlled and get a mathematical definition of the controlled variable. Great.

The power law

I also agree with some of the slides about the power law - lots of movements in the physical world follow some speed-curvature power law - masses on springs, different 2d sinusoids, etc. Lots of smooth periodic functions satisfy the law too. Pure noise also. All good.

The first slide contains some errors:

The voluntary movements produced by people (and other living things) follow a power law
Speed = Curvaturebeta

Lots of movements don’t follow a power law - like movements along straight lines (although some people change the formula, but that is then a different law). For very slow movements where you can maintain constant speed trough any curve, there is also no power law. “Random” movements also don’t consistently produce a power law, but only under certain conditions.

From non-biological movements, you’ve got planets where the speed does not depend on the curvature of the orbit; or a pendulum if you look at it from the side - constant curvature, changing speed, etc, etc.

For the formula, there is generally also a parameter that depends on the average velocity, so the speed is nearly k * curvature beta.

Depending on how speed and curvature are measured, the coefficient, beta, of this power function is always approximately 1/3 or 2/3

This is not correct. Beta is not always approximately 1/3 or 2/3. For different shapes, at high speed, there is a whole range of betas (e.g. Huh and Sejnowski 2015, also reproduced in the reappraisal paper),.

Betas in tracing ellipses also change with speed, but after a point, they converge to 2/3. They change with the drawing medium - tracing ellipses in water is different from tracing them in air, different beta. Small children draw differently than adults, etc, etc.

So it is not a very general law, in the sense that the same exponent is appearing everywhere, but it is a very strong law because in the same conditions, drawing the same shape, you get very high correlations, and consistently the same betas . For example, drawing or tracing or tracking ellipses in a smooth and fast way, always produces a speed-curvature law with the same exponent and a high correlation. In other words - people cannot follow constant speed targets along elliptic trajectories with a constant speed - instead they always slow down in the areas of higher curvature, and hurry up a bit in the areas of lower curvature. This is empirical data from one such experiment, I hope the plot and the legends explain everything:

So, my first question is - do you think that the high correlations and consistency of finding the same exponent under the same conditions qualify the speed-curvature power law as a study-worthy behavioral law?

It is not a stimulus-response law, it is simply a feature, or invariance of movement. It is not claimed that the speed is a response to curvature, they are measures of the same instant of behavior. There are some people who theorize that the movement is planned in the brain to achieve the power law but not directly - rather by minimizing different cost functions, like jerk or torque, etc., here are other people who claim otherwise, that the power law is a side effect of some other process, of muscles smoothing the movement, etc. I don’t see nothing new coming from PCT in that debate - planned vs side effect - the solution will probably be in the side effect camp. The new thing will be one or several controlled variables.

In the conclusion on the power law:

Mathematical regularity of power law led researchers to assume that, like Kepler’s laws, it would reveal something about how movements are produced.

Did Kepler’s law reveal something about how planet movements were produced? I think they just described the movements, and then Newton explained them in some sense, with the law of gravitation.

The speed-curvature law is a descriptive law, nothing explanatory, unless, of course, someone explicitly claims that the brain plans the movements to comply with a power law. Generally, people don’t claim this.

PCT-based test for the controlled variable rules out power law as revealing anything about how movement is produced.

I don’t follow this logic. How does the TCV rule out the power law as revealing anything about how movement is produced? I think that a computational model that explains why the power law appears will have to take into account correct controlled variables and a correct ‘body’ element, output function, and then will have to reproduce the power law with the same exponent as the human in the same situation. Or equally, a lack of the power law where it does not appear in human movement.

And finally, a standard tracking model (cursor-target distance) does not follow the ellipse in the same way as humans do (target is red, model green):

Interestingly, it does produce a power law trajectory, but the size doesn’t match. So the “simple” model does not really explain much.

For the planarity of movement topic, I also have a comment. In the paper you cite, I can find no claim that the gait planarity is planned. From what I can see they also state it can be emergent from some oscillators as a side effect.

Yes. The new thing will be a function of more than one CV. Presumably one CV is forward velocity v, and the effect of the power law results from f(v,x), where in the simplest case x is one other controlled perceptual variable. One variable that can be perceived in curve-drawing is centripetal velocity c. So is there some function f such that f(v,c) is constant? What happens when c is disturbed e.g. by placing the subject on a moving platform?

In defense of Richard’s comment: Kepler’s law revealed something to Newton, who told us about what Kepler’s law reveals when you posit gravitation as a universal factor. (He might even have thought that Kepler’s data revealed gravitation–he did say “hypotheses non fingo”.) We are looking at the descriptive generalizations expressed by the power law to figure out what they might reveal when we posit that behavior is means of controlling specified perceptual inputs.

Presumably one CV is forward velocity v , and the effect of the power law results from f (v ,x ), where in the simplest case x is one other controlled perceptual variable.

I agree. I feel like I can control something like forward velocity, or maybe velocity averaged over a short time period. Drawing curved lines could be similar to driving a car on a wavy road. You can change your forward velocity, but there are limits to your acceleration - in one case limited by the muscles, in the other by the car properties. In drawing fast ellipses, I can control the average speed, or the rhythm/frequency/period, but not instantaneous speed.

One variable that can be perceived in curve-drawing is centripetal velocity c

Lot of questions for this one. How do you define c? And how do you know it can be perceived, are there some studies about it? And what do you mean by a moving platform? How about driving a car or a bicycle? (there are some beautiful papers about controlled variables in driving, but I don’t know about other types of moving platforms)

Centripetal acceleration is a change in speed along a vector that corresponds to the radius of curvature r (from the center of curvature through the instant location, at right angles to the tangent velocity at the instant location), and is inversely related to r, that is, the tighter the curve the greater the centripetal force required to depart from a straight line, and greater the ‘centrifugal force’. Centripetal force applied to the moving mass causes it to deviate from a straight path. A=F/M.

Centripetal acceleration can be sensed inertially as “centrifugal force”, by translation in the visual field of a future-target location ahead along the path, in perceptions of the diverse exertions maintaining posture if e.g. in a moving car, probably in other ways. I should say centripetal velocity when the radius of curvature is constant, though centrifugal force even with constant r has inertial effects that feel like acceleration.

Years ago, Isaak Kurtzer did his PhD work at Brandeis experimenting with motor control with subjects on a large rotating platform.

Something could be improvised on the kind of merry-go-round that is often among the equipment on childrens’ playgrounds.

Unfortunately, after getting his degree Isaak appears to have chosen a safer career path that does not involve PCT research.

Tom Bourbon was his teacher, so Bill’s principled but unfortunately tone-deaf dealing with the controversy about coercion and Ed Ford’s education program, and Tom’s subsequent alienation after Vancouver, may have had some influence. Search for Kurtzer in the archive.

BN Centripetal acceleration is a change in speed along a vector that corresponds to the radius of curvature r (from the center of curvature through the instant location, at right angles to the tangent velocity at the instant location), and is inversely related to r, that is, the tighter the curve the greater the centripetal force required to depart from a straight line, and greater the ‘centrifugal force’. Centripetal force applied to the moving mass causes it to deviate from a straight path. A=F/M.

BN Centripetal acceleration can be sensed inertially as “centrifugal force”, by translation in the visual field of a future-target location ahead along the path, in perceptions of the diverse exertions maintaining posture if e.g. in a moving car, probably in other ways. I should say centripetal velocity when the radius of curvature is constant, though centrifugal force even with constant r has inertial effects that feel like acceleration.

Thanks, I’ll look more into it, it might be a relevant variable. Another relevant variable, I think, is something like “distance from the desired path”.

BN Unfortunately, after getting his degree Isaak appears to have chosen a safer career path that does not involve PCT research.

I’ve read some of his posts in the archives before, seems like a reasonable guy. I would definitely not try guessing why he does not quote Bill. If I’ve learned anything from applying PCT, it is that finding out why people do stuff, what they want, or what they are trying to achieve is a mystery that cannot be easily resolved even by careful and systematic laboratory experiments, with rigorous TCV, much less by some superficial assessment.

So, my first question is - do you think that the high correlations and consistency of finding the same exponent under the same conditions qualify the speed-curvature power law as a study-worthy behavioral law?

I would not consider it worth my while to study. My reasons are these:

1. The “classical” 1/3 or 2/3 law (these are the same law, just expressed in terms of a different pair of variables) is not always observed, but instead a different power. (You mentioned examples of this yourself.) This gives an extra parameter to fit. It cannot hold when the trajectory is close to straight (as you also mention), so people respond to that in two ways. They may leave out the straight parts, divide the trajectory into segments and fit a curve separately to each segment. Or they may replace the curvature C by C+alpha, where alpha is another parameter inserted just to avoid zero curvature. One paper combined three different concepts of velocity (the ordinary sort, affine, and semi-affine) and mixed them in proportions that provided even more parameters.

With all this variation in what “the law” is, and the proliferation of parameters, it seems possible to fit some version to almost anything. As the title of one of my slides says, the power law is not much of a law.

2. My second reason is that the observed power law for repetitive ellipses inevitably arises from any method of producing these by “smooth” reciprocating movements in two directions X and Y. By “smooth” I mean roughly “one maximum and minimum for x, dx/dt, and d2x/dt2 per cycle, and the same for y”. The images below show the marked contrast between two ways of traversing an ellipse: simple harmonic motion in X and Y, out of phase with each other, and constant speed around the ellipse. Simple harmonic motion obeys the 2/3 power law exactly. Constant velocity motion obviously does not. If the acceleration of X and of Y are square waves, making an oval path, with graphs of X and Y against time consisting of parabolic approximations to sine waves, then the power law is followed quite well — as well as the experimental data of these studies.

Since it is so easy to generate power-law-like movement by multiple mechanisms, and so easy to fit power-law-like laws, and the data are so mushy compared with Kepler’s, it seems like a futile activity to me.

Did Kepler’s law reveal something about how planet movements were produced? I think they just described the movements, and then Newton explained them in some sense, with the law of gravitation.

Indeed. The difference between Kepler’s laws and the power law is that for the planets, there are no disturbances (at least, none well measurable in the times of Kepler and Newton). The system consisting of the sun, the planets, and the Moon, proceeds uniformly and exactly according to Kepler’s laws. Newton’s development of universal gravitation put a theory under them that predicted them exactly, and that theory then further permitted the analysis of those discrepancies that were measurable up to the late nineteenth century, leading to such things as computing the influence of Jupiter on the other planets’ orbits, the discovery of Uranus and Neptune, and explanations of fine details of the Moon’s orbit. Kepler’s laws enabled the discovery of Newton’s, and Newton’s enabled an even more accurate description of the heavens, and showed that their mechanics were the same as those on Earth (an issue of substantial religious significance in his time).

In contrast, what we see in the experimental work on movements of living creatures has no such precision, and exhibits a great deal of variation with the environment and the nature of the task. The papers I have seen amount to no more than a catalogue of curve-fitting exercises. I quote this in the slides for the talk:

“The power law is systematically violated with increasing pattern size, in both exponent and the goodness of fit. … (I)n unconstrained rhythmic movements, the power law seems to be a by-product of a movement system that favors smooth trajectories, and … it is unlikely to serve as a primary movement-generating principle.”

(Stefan Schaal, Dagmar Sternad “Origins and violations of the 2/3 power law in rhythmic three- dimensional arm movements”, Exp Brain Res (2001) 136:60–72)

Where is the Newton who will find such a principle? And what use will he make of those papers?

For the planarity of movement topic, I also have a comment. In the paper you cite, I can find no claim that the gait planarity is planned. From what I can see they also state it can be emergent from some oscillators as a side effect.

Indeed it can, and so easily that it seems hardly worth the bother of studying.

The paper cited is Catavitello, G, Ivanenko, Y & Laquaniti, F. (2019) “A kinematic synergy…”, which I’m not going to minutely search through, but it cites “Motor Patterns in Walking” (Lacquaniti et al, 1999) which says things like “In other words, a neural law that coupled together the angular motions at the individual limb segments would uniquely determine the spatial trajectory (as well as its time course) of the center of body mass.” This is immediately before the paper goes on to treat gait planarity. I cannot read this otherwise than suggesting that gait planarity is produced by such a neural law. A later paper, “Kinematic control of walking” (Lacquaniti et al, 2002) talks in its abstract in terms that I would almost agree with, i.e. planarity emerging from smooth oscillations of the same period. So maybe, and I can’t say without a more extensive familiarity with recent papers, maybe the field, or some researchers in it, have moved away from the idea that observed regularities have neural homunculi.

But it then seems redundant to even mention the “planar law”. Any collection of smooth oscillations with the same period will trace out a roughly planar trajectory, whether it’s a walking animal or a steam train. It would take an especially poorly designed robot not to.

Kelvin said that there are two sorts of science: physics, and stamp collecting. The planar law seems to me like a rather common stamp, hardly worth a place in the stamp album of animal kinematics.

– Richard

With all this variation in what “the law” is, and the proliferation of parameters, it seems possible to fit some version to almost anything. As the title of one of my slides says, the power law is not much of a law.

With four parameters I can fit an elephant. With five, I can make him wiggle his trunk. - Johnny von Neuman. I think I see what you mean, there isn’t a single “the power law of movement”, there are many instances with different parameters or expressions that can be fitted to just about any movement. There are also some pretty bad studies published on the topic.

I would also add that its generality is sometimes overstated, as in “all unconstrained movements follow a power law”, but this is not quite the case because if you move slowly, you can easily break the law. You just set your relative velocity higher in the curved parts, for example, or vary it randomly.

I agree with all that, but there is still the empirical fact of consistent behavior of humans when drawing certain shapes in certain conditions. If it is a fast ellipse, it is always going to be V ≈ kR1/3, Even if you instruct people to move at constant speed or follow a constant speed target, they will still slow down in curves, and speed up in flat parts. Equally for other shapes, as shown by Huh and Sejnowski I posted previously.

That fact that there are many instances of different power laws does not make them weak laws - there are high correlations, in the nineties. Even if the law is dismissed as having many instances, the question still remains - why do people follow different power laws in different conditions?

Since it is so easy to generate power-law-like movement by multiple mechanisms, and so easy to fit power-law-like laws, and the data are so mushy compared with Kepler’s, it seems like a futile activity to me.

It is also easy to generate non-power-law like movement by multiple mechanisms; including slow human hand movement, and the data can certainly be collected with a lot of rigor and care, instead of being mushy, especially with modern recording devices.

The “study of the power law”, as I see it, is not really about the power law. It is about studying how people move their hand when drawing, tracking targets, tracing lines; and it is about creating models that generate the same movement features as humans do, when faced with the same situation. One great take on the problem comes from optimal control folks; not that it doesn’t have its (many) problems, but they are using a model that is supposed to be minimizing a cost function; and then as a result the behavior comes out similar to human behavior.

The power law, whichever instance fits the conditions, is a check on the model, a feature that the model must reproduce if it is to be considered a good model of the human. Of course, along with other features, like drawing the shape as precisely as humans, etc. Whatever the human systematically does, the model must repeat.

The papers I have seen amount to no more than a catalogue of curve-fitting exercises.

I have seen a lot of those too, not all with great fit. Still, if the curve has a very good fit to the behavior of the human, then the behavior of the model has to fit the curve too. Otherwise, the model doesn’t explain the behavior of the human; it doesn’t matter how many curves there are for different conditions.
A good model can explain some of them, and a great model maybe all of them.

But it then seems redundant to even mention the “planar law”. Any collection of smooth oscillations with the same period will trace out a roughly planar trajectory, whether it’s a walking animal or a steam train. It would take an especially poorly designed robot not to.

Interesting. How would you design a robot to not follow gait planarity? Different frequency oscillators for each joint?

Interesting. How would you design a robot to not follow gait planarity? Different frequency oscillators for each joint?

Have it operate each of its joint motors one at a time. In configuration space, the trajectory will consist of line segments parallel to each of the axes.

– Richard