CGA FAQ: Inverse Dynamics & Control of Locomotion

Dear All,

Happy New Year!

Chris Kirtley switched the topics of “centrifugal forces” to a more
important question on how the nervous system perceives and controls
movements. I reproduce here his question:
“As far as I know, we have no sensors for segment acceleration - only
(conceivably) joint angular acceleration, via spindles, joint afferents
and skin receptors. Would this variable be sufficient, I wonder, for the
CNS to compute the inverse dynamics?”

I would like to point out that the term “inverse dynamics hypothesis”
(in the most explicit form formulated by Hollerbach but the idea is as
old as Newton’s mechanics) implies that the nervous system pre-plans the
desired movement kinematics and then, based on some intrinsic
representation of equations of motion, computes and specifies the
electromyographic activity, muscle forces and torques, which are
necessary to actualise the movement plan.

Chris’s question implies that the nervous system does compute the
“inverse dynamics”. I suppose that the majority of those who work in the
field of biomechanics and maybe somewhat smaller %  of physiologists
share this view. I would be pleased to know if this is an exaggeration
since I belong to those who, following the implicit arguments of Von
Holst (1969/1973) and very explicit arguments of Bernstein (1967), are
convinced that the nervous system cannot and does not need to compute
inverse dynamics to produce perfect movements.

 The inverse computational strategy works well for robotics. I would
like to point out some simple physical and physiological principles that
bring us directly to the conclusion that the inverse dynamics control
strategy cannot be realised in biological systems.
     1. Many human actions consist of movements from one stable posture
to another. 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.
    2. According to a general rule of physics (e.g., Glansdorff &
Prigogine 1971),  the spatial coordinates at which the equilibrium is
established in any physical system are determined not by output
variables (like EMG, forces, torques) but independently of them, by the
system’s parameters. For example, in a pendulum, the equilibrium
(vertical) position is determined not by variable forces but the
parameters of the pendulum, such as the length of the rope at which the
pendulum’s mass is suspended, the coordinates of the suspension point,
and the direction of gravity. Since these parameters are constant, the
equilibrium position of the pendulum remains the same even when the
system is put in motion. Thus, the ability of the nervous system to
change the equilibrium position implies that the nervous system has the
capacity not only to maintain but also to change appropriate parameters
or determinants of the equilibrium position and thus produce active
movements.
     3. Suppose control levels computed and specified EMG signals and
forces according to the planned kinematics, as suggested in the
inverse-dynamic approach. If the system left the parameters that
determine the equilibrium position unchanged, the programmed forces
would drive the system from the existing (initial) equilibrium position.
The inverse dynamic approach does not account for the fact that, like in
a pendulum, the system will produce additional, resisting forces trying
to return the system to the initial position and thus destroy the
programmed action. Even if the inverse-dynamic specifications of the
computed forces were combined with a shift in the equilibrium position,
the emerging, additional forces arising due to the difference between
the initial position and the new equilibrium position would also
interfere with the computed forces. As a result, the programmed motion
would again be destroyed. The idea of EMG and force programming thus
conflicts with the natural physical tendency of the system to generate,
without programming, muscle activity and forces associated with
deflections from equilibrium. Briefly, the inverse dynamic approach
conflicts with the natural dynamics of biological systems.
      4. Experimentally, one parameter (lambda) controlling the
equilibrium position of the system has been found by Asatryan and
Feldman (1965).

    I am looking forward to seeing reactions to my comments. In my view,
the inverse-dynamic approach is just a revival of our old illness - the
mechanistic tradition of thinking inspired by remarkable successes in
robotics. The robotics view may, however, be misleading in educating,
especially, young scientists on how movements are controlled in living
systems.

Best wishes in the New Millenium!

--
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/

 Dear all,

Just to paddle into the debate.....

How would you compute the inverse dynamics of the tongue, considering it has
no joints?

Cheers,
Gary.
*******************************************************
Gary Ellem.

School of Science & Technology
Central Coast Campus
University of Newcastle
Ourimbah
NSW 2258
Australia

Ph:    +61 2  43494489
FAX:+61 2  4348 4145


Hello,

I think perhaps this conversations started on CGA and is wandering.  I have
refrained from entering debates on religion, politics and dynamical systems,
but I resolved to be less shy in the new millennium.

1.  Where are the equilibrium states in walking and running?

2 and 3. I think control theory has long recognized that the dynamics of the
system are part of the control loop.

3, 4 and follow-up. Stonehenge, ancient diagrams of circles and epicycles,
and Keppler's equations attest to the fact you can describe undisturbed
motion accurately without Newtonian mechanics.  If everything worked as well
as the moon and stars, science could be an intellectual exercise.  However,
fortunately, we live in a world where we want to do things that have never
been done before; and, unfortunately, we live in a world where many people
cannot move according to the heavenly plan.  For both problems, you need
f=ma physics.  I doubt that the Druids or Ptolemy or Keppler or the lambda
hypothesis could have put people on the moon and got them back.  I hope we
do not update the education of young scientist and engineers to the point
that we totally loose the ability to solve real problems.


As to Chris's question.  I doubt that you need to measure accelerations.
What you get out of the various joint organs appears to be some sort of
modulated measure of velocity and force.  Sounds like momentum to me.
Impulse and momentum control determines where you're going, not where you
are.  Advantageous for a control system with long delays.  Watch a child
learning to walk or a quasi-adult (athlete) learning to bat, pitch or swing
a golf club.  They practice to get the feel of the movement.  That "feel" is
a sense of the dynamics not the kinematics.


Yours,
Pat
--------------------------------------
Patrick O. Riley, PhD
Harvard/SRH CRS
Ph.:   (617) 573 2731
FAX: (617) 573 2769
Email: priley@partners.org <mailto:priley@partners.org>

Dear all,

I think Pat Riley has had a touch too much Christmas spirit! This
discussion actually started on BIOMCH-L - to recap, it sort of went:
centrifugal force -> end-point hypothesis -> inverse dynamics. I'm not
complaining - this is turning into a classical BIOMCH-L debate a la
Woltring. Sorry for the cross-posting - and I suggest we keep the debate
on BIOMCH-L after this.

As it happens, I'm a bit of fan of megaliths, and a member of the
archaeoastronomy group at the University of Maryland
<http://www.wam.umd.edu/~tlaloc/archastro> and I have to say that we
don't really know how much the builders of these extraordinary monuments
knew about physics. I suspect it was more than we think, and I do know
that we haven't been to the moon for a generation - I'm not entirely
sure that we will ever manage to go back there. After the quite
embarassing series of Mars missions, with one crash due to incorrect
conversion of inches to millimeters, I agree with Pat that we are in no
position to be complacent about the state of our engineering skills -
this from an Englishman, who can't even build footbridges or run
railways anymore :(

My point about the segmental accelerations was simply that Ton and Paolo
had discussed the problems of measurement in non-inertial frames, and I
wondered how the nervous system copes with this problem since whilst we
have (perhaps) joint position/velocity sensors, we have no natural
segmental accelerometers. Ton responded that, although we don't
conceivably possess segmental acceleration receptors in the limbs, we
have one in the head. I confess I wasn't really convinced by this reply
- it seems to place a lot of reliance on one sensor, and people with
vestibular disorders, whilst clearly inconvenienced, aren't as disbleed
as you'd expect if the vestibule were so vital.

So, we are left with the original question, and now I suppose Pat and
Anatol are both saying that the CNS doesn't use inverse dynamics. I'm
not in one camp or the other, but I think the question gets to the heart
of motor control. For too long people have been bluffing over this
question - one reads, for example, in most textbooks, that the
cerebellum is responsible for "coordination" of movements. What does
that mean?

What have we learned from a century of motor control since Sherrington?
In my view (and I expect to be flamed for this!) not very much. We seem
to be still discussing controversies that William James contemplated. I
will go further and suggest that it is only when we engineer humanoid
robots that we will really understand the nature of the problem. I am
thinking of some work I saw last week from the MIT group (Gill Pratt and
Hugh Herr) on walking robots using series-elastic actuators and
actin-myosin machines. I know that Pat is working with this group, so
perhaps he can tell us more?

I'll now standby for the flames...

Happy New Year to all BIOMCH-L and CGAers!

Chris
--
Dr. Chris Kirtley MD PhD
Associate Professor
HomeCare Technologies for the 21st Century (Whitaker Foundation)
NIDRR Rehabilitation Engineering Research Center on TeleRehabilitation
Dept. of Biomedical Engineering, Pangborn 105B
Catholic University of America

Here are my responses to the comments (quated here) by Pat Riley, point
by point.

1. Where are the equilibrium states in walking and running?

Response. 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 shifts in the equilibrium body configuration and are not programmed
by the nervous system. The faster the shifts, the faster the step. If
you repeat the control shifts, you get walking. By speeding the shifts,
you get running. For more details, please consult section Response in
our article (Feldman & Levin 1995).

2 &3.  I think control theory has long recognized that the dynamics of
the system are part of the control loop.

 Comment. Generally, you are right but we discuss not a general but a
specific theory called the inverse dynamics. My point was that this
theory is inconsistent with the intrinsic dynamics of the neuromuscular
system and, therefore, it is an example of a theory which failed to make
“the dynamics as a part of the control loop”.

3, 4 Stonehenge, ancient diagrams of circles and epicycles, and
Keppler's equations attest to the fact you can describe undisturbed
motion accurately without Newtonian mechanics.  If everything worked as
well fortunately, we live in a world where we want to do things that
have never been done before; and, unfortunately, we live in a world
where many people cannot move according to the heavenly plan.  For both
problems, you need f=ma physics. I doubt that the Druids or Ptolemy or
Keppler or the lambda hypothesis could have put people on the moon and
got them back.  I hope we do not update the education of young scientist
and engineers to the point that we totally loose the ability to solve
real problems.

Comments. I did not understand the point of this philosophy. The famous
names you listed  were the precursors of  Newton’s and modern science.
In particular, Newton first tested his laws by looking whether or not
they were consistent with Keppler’s laws. So to be fare, we should blame
not only Newton but also Kepler for what “we could put people on the
moon and get them back”. Concerning the lambda hypothesis, do not you
want to use this theory to simulate locomotion?  Maybe, this is a way to
advance our knowledge on how the brain controls it? This might be not
only theoretically significant but might give you the practical ability
you want -  “to solve real problems”, such as the understanding of basic
movement pathologies (I heard the opinion that this is more essential
than putting people on the moon). Somebody can use inverse dynamics to
create walking robots. Fine. In fact, we have already thechnical
realizations- airplans-  immitating  locomotion of birds. Beyond
airodynamics, the creation of airplains did not advance our
understanding of how the brain of birds controls flight.  This is an
essential lesson showing that technical immitations of biological
phenomena are devices that are helpless without the brain of the
driver.

All best
--
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/

Anatol Feldman wrote:

    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.


Mariano Garcia responds:

If this were true, then one would expect people to walk much differently
on treadmills than they do on a hard surface, which is probably not true.

Mariano Garcia, Ph.D.
2250 North Triphammer Rd. Apt H-2B
Ithaca NY 14850
ph: 607-257-3509, email: garcia@tam.cornell.edu
still associated with the
Polypedal Lab, Dept. Of Integrative Biology
University of California Berkeley

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/

I would like to comment on the remarks of Professor Feldman
regarding the equilibrium states in walking and running.

>
> 1. Where are the equilibrium states in walking and running?
>
> Response. 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.

The equilibrium in running/walking is a dynamic one. One cannot describe it
as a transition from one postural equilibrium to another. The postural
equilibrium
is a static one (I assume that  it means the biped is standing). By
definition, a static
equilibrium  is "static". So, once a system is in a static equilibrium , it
should not be moving
anywhere.

According to the theory of nonlinear dynamical systems, the dynamic
equilibria in running and walking are described as limit cycles. The limit
cycle
is a periodic motion that can repel (unstable) or attract (stable) other
motions that start
from  neighboring  initial conditions.

In addition, there is one set of parameter values that are associated with
each equilibrium
state. The only things that vary in time are the state variables. Anything
else that changes can
be expressed as a function of these variables. One cannot have an
equilibrium state that
consists of transition from one equilibrium state to another by varying the
parameters.

Best regards,

Yildirim Hurmuzlu


===================================
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/



PART 2 - Inverse dynamics.

 Chris Kirtley asked if the CNS could do inverse dynamics to control
 the motor system. My first response was the same as Ton's but having
 thought about it some more, let me offer a different perspective. The
 CNS not only can do that but does do that. But it does not do it
 explicitly, the way we do it with our equations and Matlab. It does
 it implicitly in the patterns of forces it generates in the muscles.

 Think back, those of you old enough, to the time of analog computers
 when a few amplifiers, diodes, resistors and capacitors and wires
 could generate the solution to any nonlinear equation. Could not one
 do that with a few million neurons? And I suggest that that is how
 one should look at the force pattern generators in my schemes, the
 forward models in Shadmehr's, the force fields of
 Mussa-Ivaldi and Giszter and the dynamic forces of Ghez, Sainburg,
 Bastian and others (This is my opinion and I am not speaking for any
 of the above).
 The joint torques are the solutions to the inverse dynamic equations,
 even if we never explicitly write them out.

 The immediate objection to this is that even (or especially) analog
 computer solutions are only as accurate as their ability to integrate
 accurately which always is difficult and especially in a noisy
 environment. They drift and require frequent calibration. That is not
 a problem because we can recalibrate every time we make contact with
 an object or look at our limbs. Just as importantly, our
 neuromuscular system is not an ideal force generator but one with
 intrinsic elastic properties created especially by muscle
 co-contraction and also by reflex action. If you look at the block
 diagrams (or consider the implications of a converging force field),
 there are two inputs; the force input (the solution to the ID
 equations) and a position input that combines with length feedback.
 If the force input is well planned, that second input does nothing.
 But this elastic property makes the endpoint insensitive to errors in
 that dynamic force input.

 So I think Chris's question ties in very nicely with the issues of
 what are real forces. If you think that there is a fundamental
 difference between classes of forces, that some are real and some are
 reactions to real forces, then you come to a different conclusion
 about control and about pathology than you do if you think all force
 components have similar stature. This is another long and potentially
 interesting discussion but not this year.

 CONSUMER WARNING: The above views express the opinions of the writer
 and are not universally shared. Some of this was discussed  here a
 couple of years ago and we ended up agreeing to disagree. That has
 not changed.

 However, the evidence from this discussion is that educated people
 can make careers in this line of work on either side of the fence so
 I will just wish all you monists and dualists out there, Happy
 Chanukah, Merry Christmas and May the Force be with you.
 --
   ___________________________________________________________________
 |   Gerald Gottlieb                          (617) 358-0719
 |   NeuroMuscular Research Center                  353-9757
 |   Boston University                         fax  353-5737
 |   19 Deerfield  St.
 |   Boston MA 02215

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