View Full Version : Inverse Dynamics

Anatol G. Feldman
01-22-2001, 04:17 PM
Previously, I outlined an explanation of locomotion (a single step,
walking and running) in terms of the equilibrium point (EP) hypothesis:
“A single step is a transition from one postural (equilibrium) state to
another. One can also say that a step results from changes in specific
parameters that transform the equilibrium configuration of the body in
such a way that eventually the body establishes approximately the same
(initial) posture but in another part of external space. All forces
(torques) required for such a transition emerge in response to the shift
in the equilibrium body configuration and are not programmed by the
nervous system. The faster the shift, the faster is the step. If you
repeat the control shift, you get walking. By speeding the shift, you
get running. For more details, please consult section Response in our
article (Feldman & Levin 1995)”.
Yildirim Hurmuzlu's reaction to this explanation is an interesting
example of a rejection of the EP hypothesis. I think his arguments are
largely flawed since they are based, in particular, on the misconception
that the term “equilibrium position” implies static. Dynamic essence of
the concept equilibrium position has been emphasized in one of my
previous messages posted on the Biomech-L:
“A stable posture is associated not only with the equilibrium position
at which all forces are balanced but also with the ability of the system
to generate forces resisting deflections from this position”.
If one thinks that “equilibrium position” implies “static", please
make the following exercise. Show that for a pendulum (take for
simplicity a pendulum without friction),

[equilibrium position] = [actual position] - k [acceleration],

where coefficient k is proportional to the squared period of
oscillations. This equation implies, first, that the equilibrium
position may be considered a virtual position that exists at any moment
of the pendulum's motion. Second, it implies that the equilibrium
position is an invariant of motion, meaning that kinematic variables are
summed in some way to produce the same value (= equilibrium position) at
any instant of motion. In fact, knowing that the equilibrium position is
an invariant, one can derive the equation of motion of the pendulum and,
as a consequence, other invariants of motion (e.g., energy). Static is
just a specific case of this law, when
[equilibrium position] = [actual position].
In general, with some reservations, the concepts of equilibrium
position and EP (they are not identical) resemble the concept of “point
attractor” in dynamic systems theory and as such they are not less
“dynamical” than, say, the concept of limit cycle.
The EP hypothesis strengthens the dynamical essence of the EP
concept by suggesting that the nervous system may change system's
parameters to shift the EP of the body and thus produce active
movements, an idea fully applied to locomotion as was outlined in the
beginning of this message.
I also disagree with Yildirim Hurmuzlu’s suggestion that locomotion
is a limit cycle. A specific case of locomotion - a single step - is not
a periodic process, which conflicts with the notion of limit cycle.
During continuous walking or running, a state variable - the position of
the body in space - changes monotoniously, which also conflicts with the
notion of limit cycle. If you consider only relative motion of the body
segments you may reduce the phenomenon of locomotion to a limit cycle
but in this case you ignore the explanation of the most important aspect
of locomotion - the displacement of the body in the environment.
On a positive side, the criticisms of Yildirim Hurmuzlu and my
response may be helpful in clarification of some aspects of the EP
hypothesis. I feel that clarifications are necessary since, at least for
now, all rejections of the EP hypothesis have been based on
misconceptions. Some of them are discussed in our paper (Feldman et al.
Are there any other rejections of the EP hypothesis in the cyberspace?
We can discuss them!
Best wishes to All!

Dr. Anatol Feldman
Neurological Science Research Center
Department of Physiology
University of Montreal and
Rehabilitation Institute of Montreal
6300 Darlington, Montreal, Quebec, Canada H3S 2J4
Tel (514) 340 2078 ext. 2192
Fax (514) 340 2154
Web Site: http://www.crosswinds.net/~afeldman/

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