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View Full Version : Dynamics of Gait (was Re: Inverse Dynamics)

Dr. Yildirim Hurmuzlu
01-22-2001, 10:39 PM
In my previous message my main point was that the equilibrium in bipedal
locomotion
is a limit cycle, which is a dynamic equilibrium (as I specifically stated).
I cannot see how Prof. Feldman reached the conclusion that my arguments
wre based on "the misconception that the term "equilibrium position" implies
static". My entire argument was based on the dynamic nature of the
equilibrium
that arise in gait.

The pendulum example given by Prof. Feldman is a bad one here. The periodic
motions that arise in an undamped pendulum have zero measure on the Poincare
Map. Thus, they are highly improbable to observe in nature because they
are structurally unstable equilibria (an infinitesimal perturbation of the
parameters will result in a new equilibrium state which is static).
Physically, minute amount of added damping, such as existence
of joint friction or drag force, will destroy this dynamic equilibrium.

The more common type of dynamic equilibria arise through Hopf
bifurcations, that take place as a result of nonlinear effects. A stable
static equilibrium goes unstable (as a result of changes in the parameters),
repelling the state away from the equilibrium point. Yet, nonlinear effects
will force the state to establish a closed orbit that encircles the unstable
static equilibrium, which is termed a limit cycle. This type of equilibrium
can be
stable and structurally stable. Here, stability means that the limit cycle
attracts
the local trajectories in the state space. Structural stability on the other
hand,
means that the stability of the limit cycle is preserved for bounded
perturbations of the parameters. I can see now that Prof. Feldman
confuses the state variables with parameters. If a parameter continuously
varies during a specific motion, by definition, it should be called a state
variable! I believe that more proper sequence of events should be as
follows:

1) The biped is standing at a static equilibrium.
2) The brain changes a parameter such that this equilibrium becomes
unstable.
3) The biped undergoes a Hopf bifurcation that results in the emergence of a
stable
limit cycle that we call locomotion.

So instead of changing parameters at every step (as Prof. Feldman claims) to
jump from
one posture to the other, the biped changes the parameters once and
achieves a
dynamic equilibrium that results in continuos walking.

Also, the argument that the equilibrium in locomotion cannot be a
limit cycle because the biped translates horizontally is not a
valid one. One can always express the position of the biped in
terms of the joint coordinates (kinematics). Thus, despite
the monotonically increasing horizontal position of the biped,
the equilibrium in walking/runnung can be characterized
as a limit cycle. As a matter of fact, bipeds are not the only examples of
such systems. There are other physical systems that demonstrate
similar behavior such as bicycles, rimless wheels, etc.

In closing, I must admit that the response of Prof. Feldman did not clarify
the
EP hypothesis for me. On the contrary, now I am even more confused by his
explanations that are in my opinion are rather vague and do not fit into a
norm that
is encapsulated by the existing nonlinear dynamical systems theory. I
believe that,
dynamics of bipedal locomotion can be analyzed by employing simpler and more
coherent concepts and analytical tools such as limit cycles, Poincare Maps,
and
Floquet multipliers.

Best regards,

Yildirim Hurmuzlu

ps. I recommend the following related references to the interested readers:

[1] Garcia M., Chatterjee A., Ruina A. and Coleman M. (1997).
The Simplest Walking Model: Stability, and Scaling.
ASME Journal of Biomechanical Engineering, 120, 281--288.

[2] Guckenheimer J. and Holmes P. (1985), Nonlinear Oscillations, Dynamical
systems, and Bifurcations of Vector Fields, Springer-Verlag.

[3] Hurmuzlu Y. and Basdogan C. (1994),
"On the Measurement of Dynamic Stability of Human Locomotion.",
ASME Journal of Biomechanical Engineering, 116(1), 30--36.

[4] Hurmuzlu Y., Basdogan C. and Stoianovici D. (1995), "Kinematics and
Dynamic Stability of the
Locomotion of Polio Patients", ASME Journal of Biomechanical
Engineering, 118, 405--411.

__________________________________________________ _______________________
Yildirim Hurmuzlu
Professor of Mechanical Engineering
Southern Methodist University
Dallas, TX 75275

Phone : (214) 768-3498
Fax : (214) 768-1473
e-mail : hurmuzlu@seas.smu.edu
web : http://cyborg.seas.smu.edu/~hurmuzlu/
__________________________________________________ _______________________

----- Original Message -----
From: "Anatol G. Feldman"
To:
Sent: Tuesday, January 23, 2001 12:17 AM
Subject: Inverse Dynamics

> 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.
> 1998).
> Are there any other rejections of the EP hypothesis in the cyberspace?
> We can discuss them!
> Best wishes to All!
>
>
> --
> Dr. Anatol Feldman
> Professor
> Neurological Science Research Center
> Department of Physiology
> University of Montreal and
> Rehabilitation Institute of Montreal
> 6300 Darlington, Montreal, Quebec, Canada H3S 2J4
> feldman@med.umontreal.ca
> Tel (514) 340 2078 ext. 2192
> Fax (514) 340 2154
> Web Site: http://www.crosswinds.net/~afeldman/
>
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