[Roger Moore 2016.09.15.22.12 BST]
- THE MATHEMATICS
Equation 8 states that the instantaneous velocity V is a function of the first-derivative of any movement decomposed into X and Y coordinate space. This means that V can be calculated from the velocity in X (Xdot) and the velocity in Y (Ydot). It also means that V is independent of any other variables - i.e. V is independent of any acceleration parameters and hence, by definition, INDEPENDENT OF CURVATURE.
Equation 9 states that the radius (inverse curvature) R of a curve is a function of the first and second derivatives of any movement decomposed into X and Y coordinate space. This means that R can be calculated from the velocity in X (Xdot), the velocity in Y (Ydot), the acceleration in X (Xdoubledot) and the acceleration in Y (Ydoubledot).
Rick’s equation relating V to R and D is correct in that the derivation contains no errors. However, it is NOT CORRECT to interpret V as a function of R only. In Rick’s equation, V is a function of R and D. This means that D CANNOT BE IGNORED. For all values of X and Y, D has a value which cancels R exactly, and leads to a true result - V=V (which has already been pointed out).
Hence, the mathematics tells us that V and R vary independently, exactly as one would expect. There is NO hidden “law” relating V and R.
- THE PROBLEM
If the observed effect - slowing during curved trajectories - is the result of a closed-loop negative-feedback control system, then it is possible to hypothesise an explanation as follows:
a) Let Vr be the intended (reference) velocity.
b) Let Vp be the perceived velocity.
c) Let e=Vr-Vp
d). Let Vm be the output velocity, where Vm is some function of e.
e) If Vp=Vr, then e=0 and Vm will remain unchanged.
f) If Vp>Vr, then Vm will be reduced until e=0.
So, given that an increase in ‘curvature’ is observed to elicit a reduction in ‘speed’, it is possible to deduce that Vp>Vm in the presence of curvature. I.e. the ‘perception’ of velocity is affected by the ‘disturbance’ of curvature. In other words, when going around a curve, an organism perceives itself to be going faster than it really is, and it slows accordingly. (This certainly concurs with my own experiences when driving with the cruise control switched on - a speed that feels comfortable on the straight can feel very uncomfortable when you encounter even the slightest bend.)
Of course, this only explains the slowing, it doesn’t explain the so-called ‘Power Law’. In order to derive the necessary relations, it would be necessary to analyse the particular perceptual system(s) involved in estimating speed. E.g., for a visual perceptual system, I would look to an estimate of velocity based on something like ‘optical flow’. In other cases one might have to consider the characteristics of proprioceptive feedback. In each case, the hypothesis is that the perceived speed is OVERESTIMATED and hence gives rise to a compensatory reduction in velocity.
This is a scientific hypothesis that can be tested by modelling/simulation of perceptual systems with the appropriate characteristics, and then comparing those characteristics with those found in living systems.
Prof ROGER K MOORE* BA(Hons) MSc PhD FIOA FISCA MIET
Chair of Spoken Language Processing
Vocal Interactivity Lab (VILab), Sheffield Robotics
Speech & Hearing Research Group (SPandH)
Department of Computer Science, UNIVERSITY OF SHEFFIELD
Regent Court, 211 Portobello, Sheffield, S1 4DP, UK
- Winner of the 2016 Antonio Zampolli Prize for "*Outstanding Contributions *
*to the Advancement of Language Resources & Language Technology *
Evaluation within Human Language Technologies"
Editor-in-Chief: COMPUTER SPEECH AND LANGUAGE