[From Bruce Abbott (2016.07.30.0935 EDT)]

Iâ€™ve gone back to Rickâ€™s post of 7/24 because it contains a clear statement of his derivation of the relationship between velocity and the radius of curvature, which he believes to be a fixed relationship that holds for â€œall two-dimensional movement patterns so long as D is taken into account.â€?Â I will show that Rickâ€™s conclusion is based on a misunderstanding of the equation for the radius of curvature and explain (in plain English) why it is wrong.Â My analysis begins at the end of Rickâ€™s post copied below:

Rick Marken (2016.07.24.1220) –

Martin Taylor (16.07.20.21.19)

MT: If “V” is a true velocity, then the dots signify derivatives with respect to time. If “R” is the radius of curvature, it’s a length and the dots signify derivatives with respect to arc length along the curve. They aren’t the same thing, but you use your “kindergarten math” as though they are…

MT: I’m leaving the rest of my message in this one so you won’t have to look back a few hours in the archive if you ever decide you want to read the explanation of why I am quite certain your V = |dX

ddY|^{2}Y-d^{2}X^{1/3 }*R^{1/3}formula not just wrong, but just so nonsensical as to be neither right nor wrong.

RM: To do the regression of log(R) on log (V) the values of R and V are computed for each point in time during which the movement is made. I found the formulas for computing R and V in Gribble & Ostry (1996). J. Neurophysiology, 76(5), 2853-2860. The value of V (velocity) at each point in the movement pattern is computed as as follows:

and the value of R (curvature) is computed at each point as follows:

RM: These are formulas I used for computing V and R in my spreadsheet analysis of the power law. If R and V are related by a power law then there will be a linear relationship between log (R) and log (V) and the slope of the relationship will be a measure of the coefficient of the power function. Below is a graph of the relationship between log (R) and log (V) for an elliptical pattern of movement like that shown above:

RM: This graph is from Wann, Nimmo-Smith & Wing (1988).JEP: HPP, 14(4), 622. The two lines are for elliptical movements that were generated in two different ways: 1) using two sine waves (a Lissajous pattern) and 2) using a movement generation process called a “jerk” model. In both cases the relationship between log (R) and log (V) is precisely linear (R^2 = 1.0) and the coefficient of the power function is .33. I found the same relationship between log (R) and log (V) for ellipsoidal movement created by a control model, even for control models with different output and feedback functions.

RM: The fact that a power law (with a coefficient of .33) was found for elliptical movements that had been produced by many different processes suggested to me that the relationship between log (R) and log (V) may depend only on the nature of the movement pattern itself and not on how that movement pattern was produced. Since R and V are both measured from the movement pattern (the same values of the derivatives of X and Y movement are used in the computation of both R and V) I looked to see if there might be a mathematical relationship between V and R.

RM: Looking at the formulas for V and R I noticed that V^{2 }= X.dot^{2}+Y.dot^{2 }, which is a term in the numerator in the formula for R. See for yourself in the equations for V and R above. So the equation for R can be re-wriitten as

R = (V^{2})^{3//2 }|/ |X.dot*Y.2dot-X.2dot*Y.dot|

And from there it’s a couple steps to:

V = D^{1/3 }*R^{1/3}

where D = |X.dot*Y.2dot-X.2dot*Y.dot|

RM: So the math shows that there is a power relationship between R and V with a coefficient of .33 (1/3) which should hold for all two dimensional movement patterns as long as the variable D is taken into account. I tested this out for several movement patterns using multiple regression on the logs of variables R and D. The multiple regression equation was of the form:

^{log (V) = a * log (D)}^{ + b * log (R)}

where both a and b are predicted to be .33. The result for all movement patterns was that the regression picked up all the variation in log (V) (R^2 always = 1.0) and the a and b coefficients were .33.

RM: The research on the power law relationship between V and R has resulted in the finding of coefficients for a power law relationship between R and V other than .33 for patterns of movement that differ from an ellipse and for patterns produced by movements in contexts other than in air and by species other than humans. I thought this might be because these studies had included only log (R) as a predictor of log (V) in the regression analyses. And my initial spreadsheet analyses suggested that this was the case. But after rechecking and correcting some of my spreadsheet calculations I have found that regressing just log (R) on log (V) will result in an R^2 of 1.0 and a power coefficient of .33 for *any* pattern of movement as long as there is no point in the pattern where the first derivatives of X and Y are exactly equal. Why that should be, I don’t know. Perhaps someone who is more math savvy than I can figure it out.

. . . . .

RM: These results show that it is impossible to learn anything about how movements are produced by looking at the relationship between measures of the movements themselves. The observed relationship between variable aspect of the movement, such as V and R, reflects nothing more than a mathematical relationship which, in this case, can be written as

V = D^{1/3 }*R^{1/3}

RM: And this equation can be found by simply observing that the equation that defines V is part of the equation that defines R.

RM: So the big question is why did all the researchers in this area fail to see either that this mathematical relationship between V and R exists or, if they did know that it exists, why did they fail to see its implication, which is that any observed relationship between V and R is determined by math, not by anything about how the movement is generated? It’s certainly not because these researchers are not good at math; nor is it because they are not extremely intelligent. I think the only possible explanation – and the one that is very relevant to PCT – is that these researchers were (and still are) blinded by the wrong view of what behavior is: a view that sees behavior as a step – usually the last one – in a sequential causal process. They can’t help seeing a pattern of movement – in terms of the degree of curvature of that movement measured as the R (or C) – as caused by a pattern of forces that move a point though those curves at varying velocity (measured by V). I think this same view of behavior is the reason why my discovery of the mathematical relationship between V and R led to such dismay on CSGNet.

RM: The power law is important because it shows that you have to be able to look at behavior though control theory glasses – see that behavior is control – before you can correctly apply the theory of control – PCT – to behavior. The truth is that because I understood this fact about behavior I knew that the power law could tell us nothing about how organisms produce movement patterns before I discovered the mathematical relationship between R and V. When output is produced in a control loop you can’t “see” the output function that is producing the controlled result. This is one of the reasons why the conventional approach to research is guaranteed to produce results that are misleading (as per Powers 1978 Psych Review article). But it’s often very difficult to show this clearly to researchers. That’s why my discovery of the fact that there is a mathematical relationship between R and V was so exciting; it shows as clearly as I can imagine that looking at behavior – like drawing a squiggle pattern – as a step in a causal sequence is the wrong way to look at the behavior of a control system. And in this case, research based on this view of behavior is misleading in the worst possible way; it’s leading researchers to take a mathematical fact about curved movement as a fact of behavior to be explained.

RM: The power law research shows why PCT has had a hard time getting accepted (by mainstream psychologists) or understood (by many of its fans). PCT is a model of a different phenomenon than the one studied by mainstream psychologists. Mainstream psychologists, like the ones doing the power law research, are studying (and trying to explain) the phenomenon of output generation; control theorists (of the PCT persuasion) are studying (and trying to explain with PCT models) the phenomenon of input control. I think the power law can show why it’s hard for mainstream psychologists to get excited about a theory (PCT) that is not an explanation of the behavior they want to explain; indeed, PCT is not only an explanation of a phenomenon that is not the one that conventional psychologists want to explain; PCT also shows that the phenomenon that conventional psychologists want to explain is an illusion.

BA:Â The reason previous researchers did not see the relationship Rick discovered by means of a simple rearrangement of mathematical terms is that Rickâ€™s rearrangement does not show what he thinks it shows. To establish this, I begin by reproducing the two equations Rick uses in his derivation.Â The first is this:

V is the velocity of a moving point along a curve, such as a car going around a bend in the road.Â V would be the speed shown on the carâ€™s speedometer.Â The above equation is for motion in two dimensions, X and Y.Â The dots over X and Y indicate that these are the first derivatives of X and Y with respect to time, or in other words, velocities in the X and Y directions.Â The equation is simply the Pythagorean Theorem in which X.dot and Y.dot are the sides of a right triangle (the horizontal and vertical components of the velocity) and V is the hypotenuse.Â V is the instantaneous velocity along the curve at a given time T.

**Dimensional analysis**.Â X.dot and Y.dot are speeds along the X and Y dimensions, measured, say, in meters/second.Â Squaring each gives meters squared/seconds squared for each.Â As the units are the same for both, we can add them.Â Taking the square root gives V in meters/second.

The second equation used in Rickâ€™s derivation is this one:

This is the equation for the radius of curvature.Â It is the radius of a circle whose curvature matches that of the line being followed, at a given point on the line.

The numerator of this equation is simply the equation for velocity V, raised to the third power.Â (The portion inside the parentheses is raised to the Â½ power, which is the same as taking its square root.Â The result is then cubed.Â So we can rewrite the numerator as V^{3}.

In the denominator we have the velocities of the point along the X and Y dimensions (X.dot and Y.dot) and the accelerations of the point along those same dimensions.Â With respect to acceleration, picture a point moving horizontally at some speed X.dot.Â Vertical speed (Y.dot) is zero.Â If the point is following a circle, however, then the point must begin to accelerate in the vertical (Y) direction and decelerate in the horizontal (X) direction.Â When the point reaches the right edge of the circle it will be traveling vertically at some speed but its horizontal velocity will be zero.Â The denominator takes these velocities and accelerations into account.

Because V in the numerator is derived by adding the squares of X.dot and Y.dot and then taking the square root, V cannot be negative.Â As we want R, the radius of curvature, to be a positive number, we remove any negative value in the denominator by taking its absolute value.Â Rick refers to this quantity as â€œDâ€? in his analysis.

**Dimensional analysis**.Â Recall that V is the instantaneous velocity of a point along the curve and that we are assuming units of meters/second for this velocity.Â V cubed will this be in units of meters^{3}/seconds^{3}.

In the denominator, we multiply a velocity by an acceleration.Â If velocity is in meters/second, then acceleration is in meters/second/second, or meters/second^{2}.Â Multiplying meters/second by meters/second^{2} given meters^{2}/second^{3}.Â Thus both X.dot times Y.dot.dot and X.dot.dot times Y.dot are in these same units and can be added together.Â Taking the absolute value gives a positive value in meters^{2}/second^{3}.

Dividing the numerator by the denominator gives meters^{3}/seconds^{3} divided by meters^{2}/second^{3}, which leaves the dimension in meters (seconds cubed cancel and meters cubed-squared = meters).Â A unit length is just what is wanted for a radius.

So, what do we have?Â The first equation gives the instantaneous tangential velocity (speed) of a point along a curve at time T, in meters/second.Â The second equation gives the instantaneous radius of curvature, in meters, of the path the point is following at that same time T (corresponding to a given position along the path that has been reached at time T).Â ***This equation compensates for the velocity of the point (in meters/second) so that the value of R is independent of the velocity.***Â It will give the same radius of curvature for any velocity > 0, for any given curvature.

**Rickâ€™s mistake**.Â Rick noticed that the numerator of the equation for R is equivalent to V cubed.Â He then rearranges the equation to solve for V and discovers that the value of V is essentially determined by the value of R:Â

V = D^{1/3 }*R^{1/3} where â€œDâ€? is the denominator of the equation for R. Â

But what this equation tells us is the value of V at the instant that radius R was being computed.Â A moment later V could be higher or lower, and so would D (because D compensates for changes in V to give a radius of curvature that is independent of speed).Â Because Rick does not understand what D represents, he concludes that velocity is a function of R and that this relation will hold no matter what shape of figure is being followed by the moving point.

But in fact R is independent of V.Â Consequently, V can be any value whatsoever no matter how sharp or gentle the curve, in so far at the math is concerned.Â Yet empirically, biological organisms do tend to adjust their speeds depending on the radius of the curve, slowing for sharper curves and speeding up for more gentle ones.Â The empirical relationship has been found to follow the power law with beta = 1/3 in most cases (but with significant exceptions, such as when tracing a figure in a viscous medium.Â The question is, how can we account for this relationship?

Bruce