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Thank you for all the excellent replies.
My original question asked how one should treat positive
and negative joint power occurring simultaneously in the different planes of
the anatomical coordinate system. I asked whether the best 'approximation' of
muscle-tendon power would be to calculate joint power as the dot product of the
moment vector and the angular velocity vector:
Power = [Mx,My,Mz] . [wx ,wy,wz],
=Mx.wx + My.wy + Mz.wz
or whether the three power terms should be left separate.
Several people responded that, because power is a scalar, one should simply add
the three power terms. While mathematically correct, there exists some debate
as to whether this approach is functionally the most appropriate. On one hand,
adding the three power terms may underestimate muscle power since, at some
joints, different muscle groups are responsible for the positive and negative
joint power occurring in different planes (e.g. hip extensors and abductors).
On the other hand, keeping the three power terms separate may overestimate
muscle power since power may be transferred between different planes if the
muscle has an action about more than one axis. The comparison was made to
two-joint muscles where simultaneous positive and negative joint power can
occur at adjacent joints with no muscle power. Due to the lenght and number of
replies I have not included all but have chosen sections that best summarize
these views.
(I also had a second question regarding the mathematical calculation of joint
power using different coordinate systems; the anatomical coordinate system vs
the joint coordinate system. I have not included a summary of these replies
here since many are very long mathematical proofs and since they have already
been posted on the list. If anyone would like a summary of these please let me
know).
*******************************************
Replies:
A.J. (Ton) van den Bogert, PhD
Department of Biomedical Engineering
Cleveland Clinic Foundation
Jonas Rubenson wrote:
> Power = [Mx,My,Mz] . [wx ,wy,wz].
Which leads to: Power = Mx.wx + My.wy + Mz.wz
> But perhaps this will lead to an underestimate of the true muscle powe
I think this is very well possible. For example, if you have hip
extensors and abductors active, during a movement that is a combination
of flexion and abduction, the extensors will do negative work and the
abductors will do positive work. If extensors and abductors are
different muscles (which is an approximation, see below), some muscle
work would not be seen if you add the three terms, because positive
and negative terms are partially canceling.
To some extent, the three degrees of freedom of the hip are separate
joints. You would never add hip and ankle power, and for the same
reason you should not add hip extension power and hip abduction power,
if different muscles are involved.
Of course many muscles span two or three joints. There are muscles
that are at the same time a hip extensor and hip abductor, just as there
are muscles that are a knee flexor and hip extensor. In those cases,
you may see positive power at one joint and negative power at another
joint, when there may not be any muscle power at all! This is a reason
for adding two joint powers, but only to the extent that they have
a common source.
But, for consistency, if you keep knee and hip separate, you should
also keep the degrees of freedom within each joint separate. This is
the standard way of reporting joint power, see, for example
Ferber R, Davis IM, Williams DS (2003) Gender differences in lower
extremity mechanics during running. Clin Biomech 18: 350-357.
In the knee, the ab-adduction power will include elastic energy
storage and release in ligaments and cartilage. It would not be correct
to add this to the flexion-extension power, which has a completely
muscular origin.
The correct way to account for muscle work is to calculate power for
each muscle as a product of force and shortening velocity. But
in the real world, we don't know individual muscle forces.
*******************************************
Richard Baker
Gait Analysis Service Manager, Royal Children's Hospital
Flemington Road, Parkville, Victoria 3052
Ton, Jonas, subscribers
I'd like to express an opposing point of view to Ton's. I'm replying though
partly to try and sort out my own thoughts on what joint power actually is
in the light of the light of recent articles (principally those by Felix
Zajac, Rick Neptune and Steve Kautz, Gait and Posture 2002;16:215-232 and
2003;17:1-17) and I'd be most open to comment, criticism and correction of
anything I write.
[...] There is another approach to calculating the changes in the total
mechanical energy of the system. This is to work out the joint moments
using inverse dynamics and the joint angular velocities. The dot product of
these is the quantity we generally refer to as "joint power" (although I am
becoming increasingly convinced that this expression is highly misleading).
The sum of the joint powers for all of the joints also represents the total
rate of change of mechanical energy in the entire system and is thus equal
to this quantity as calculated by either of the two methods outlined in the
above paragraph.
Zajac et al. show very neatly that if all muscles crossing a joint are
uniarticular then the joint power at any joint must be equal to the muscle
power. However in the presence of bi-articular muscles this is not the case
and the joint power at any joint is not equal to the combined muscle power
of all muscles crossing that joint. Thus the stated aim of Jonas' analysis,
" I am ultimately using the joint power to get an estimate of the amount of
power that muscle-tendon units must generate or absorb at a joint" is not a
valid one.
Leaving this aside, the question asked was how to interpret the
"components" of power in the different planes. I use the parenthesis
because power is a scalar (it is the time derivative of energy, another
scalar) and, unlike a vector, does not have components. The identity that
the scalar product of muscle force (F) and contraction velocity (v) is
equal to that of joint moment (M) and angular velocity (w) does not imply
the equivalent relationship for the individual components i.e.
F.v = M.w
can be expanded to
Fx.vx + Fy.vy + Fz.vz = Mx.wx + My.wy + Mz.wz (1)
but this does not imply that Fx.vx = Mx.wx and, indeed, this is generally
not the case. (This is different to the relationship between moment and the
vector product of force and moment arm. As this is a relationship between
vectors, it can be broken up into an individual analysis of the components
in different directions.) I'd thus argue that it is not correct to separate
out the "components" of joint power as Ton recommends. I think this is
generally inappropriate but think that in this particular case where joint
powers and muscle powers are being compared it has the potential to be
extremely misleading.
I'd thus tip what Ton has said on its head. Rather than, "if you keep knee
and hip separate, you should also keep the degrees of freedom within each
joint separate", I suggest, particularly in looking at the relationships
between joint powers and muscle powers, that you cannot look at the hip in
isolation (because this will miss out the effects of bi-articular muscles
crossing both the hip and knee and knee and ankle) and that you cannot
consider the "components" of power separately (because it is a scalar and
does not have components).
The biggest question in my mind is why the association between "muscle
power" and "joint power" has persisted for so long in our collective
sub-conscious (including mine). Anyone like to defend it? (indeed has
anyone managed to read this far?!).
Richard
*******************************************
Frank L Buczek Jr, PhD
President-Elect, Gait & Clinical Movement Analysis Society
Director, Motion Analysis Laboratory
Shriners Hospitals for Children
To Jonas, Ton, Richard, Young-Hoo:
I have several comments to add to your very good discussion, and will group
them according to topic areas:
SCALAR NATURE OF WORK AND POWER
Richard, you may remember asking a similar question a couple of years ago,
regarding the scalar nature of kinetic energy. I responded then with a
thought experiment that leads to physically meaningful information (see
BIOMCH-L archives, 23 March 2000). Briefly, each term (not component) in
the dot product equation for work can be used to determine changes in
kinetic energy for motion along its respective reference axis. This step is
perfectly consistent with the scalar nature of work and the relationships
stated in the work/energy theorem (i.e., the work done on a rigid body
equals its change in kinetic energy). It seems to me that a problem would
arise only if we tried to perform vector addition using the terms of the dot
product. Since my thought experiment didn't do that, nor does it need to,
there is no problem assigning physical meaning to work (or power) terms.
In this regard, I agree with Ton. Individual terms in the dot product for
joint power do have physical meaning, and are useful in understanding motion
of a multi-link system. (I could even argue that, in the general case, all
six degrees-of-freedom would be useful, but in deference to Ton, I'll keep
the discussion focused on three rotational degrees-of-freedom, only.)
JOINT POWER VERSUS MUSCLE POWER
Regarding the present discussion, I have long since stopped equating "joint
power" with "muscle power" when these are derived through Inverse Dynamics.
(Important qualifier, see below.) I'll skip details of my learning curve,
and focus on the gait analyses we perform on children with cerebral palsy.
A good example involves a child who goes into recurvatum at the knee
approximately at mid-stance. It is typical to see a profound intrinsic knee
flexion moment at this time, in the presence of power absorption. Often,
there is also knee flexor EMG activity, but not always. When knee flexor
EMG is absent, the moment arises due to deformation of soft tissues in the
joint capsule and muscle/tendon unit; in extreme cases, it may arise from
bone-to-bone contact.
It seems to me that the terms "joint moment" and "joint power" are always
correct. They express two mechanical characteristics of the joint motion.
On the other hand, since it is likely that passive contributions to the
moment are always present in addition to active muscle contributions, the
terms "muscle moment" and "muscle power" are less likely to be accurate when
obtained via Inverse Dynamics.
[...]
INVERSE DYNAMICS essentially describes motion that we've already observed,
strictly as an engineering mechanics problem. Yes, we use anthropometry to
define some inertial characteristics, but once we've done that, the
equations we write would be the same for a person walking as they would be
for a machine. The joint moments describe the NET effect of all moments
that arise from active and passive structures, and in this regard, they do a
fine job of describing why we saw the observed motion.
[...]
Best regards,
FB
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My original question asked how one should treat positive
and negative joint power occurring simultaneously in the different planes of
the anatomical coordinate system. I asked whether the best 'approximation' of
muscle-tendon power would be to calculate joint power as the dot product of the
moment vector and the angular velocity vector:
Power = [Mx,My,Mz] . [wx ,wy,wz],
=Mx.wx + My.wy + Mz.wz
or whether the three power terms should be left separate.
Several people responded that, because power is a scalar, one should simply add
the three power terms. While mathematically correct, there exists some debate
as to whether this approach is functionally the most appropriate. On one hand,
adding the three power terms may underestimate muscle power since, at some
joints, different muscle groups are responsible for the positive and negative
joint power occurring in different planes (e.g. hip extensors and abductors).
On the other hand, keeping the three power terms separate may overestimate
muscle power since power may be transferred between different planes if the
muscle has an action about more than one axis. The comparison was made to
two-joint muscles where simultaneous positive and negative joint power can
occur at adjacent joints with no muscle power. Due to the lenght and number of
replies I have not included all but have chosen sections that best summarize
these views.
(I also had a second question regarding the mathematical calculation of joint
power using different coordinate systems; the anatomical coordinate system vs
the joint coordinate system. I have not included a summary of these replies
here since many are very long mathematical proofs and since they have already
been posted on the list. If anyone would like a summary of these please let me
know).
*******************************************
Replies:
A.J. (Ton) van den Bogert, PhD
Department of Biomedical Engineering
Cleveland Clinic Foundation
Jonas Rubenson wrote:
> Power = [Mx,My,Mz] . [wx ,wy,wz].
Which leads to: Power = Mx.wx + My.wy + Mz.wz
> But perhaps this will lead to an underestimate of the true muscle powe
I think this is very well possible. For example, if you have hip
extensors and abductors active, during a movement that is a combination
of flexion and abduction, the extensors will do negative work and the
abductors will do positive work. If extensors and abductors are
different muscles (which is an approximation, see below), some muscle
work would not be seen if you add the three terms, because positive
and negative terms are partially canceling.
To some extent, the three degrees of freedom of the hip are separate
joints. You would never add hip and ankle power, and for the same
reason you should not add hip extension power and hip abduction power,
if different muscles are involved.
Of course many muscles span two or three joints. There are muscles
that are at the same time a hip extensor and hip abductor, just as there
are muscles that are a knee flexor and hip extensor. In those cases,
you may see positive power at one joint and negative power at another
joint, when there may not be any muscle power at all! This is a reason
for adding two joint powers, but only to the extent that they have
a common source.
But, for consistency, if you keep knee and hip separate, you should
also keep the degrees of freedom within each joint separate. This is
the standard way of reporting joint power, see, for example
Ferber R, Davis IM, Williams DS (2003) Gender differences in lower
extremity mechanics during running. Clin Biomech 18: 350-357.
In the knee, the ab-adduction power will include elastic energy
storage and release in ligaments and cartilage. It would not be correct
to add this to the flexion-extension power, which has a completely
muscular origin.
The correct way to account for muscle work is to calculate power for
each muscle as a product of force and shortening velocity. But
in the real world, we don't know individual muscle forces.
*******************************************
Richard Baker
Gait Analysis Service Manager, Royal Children's Hospital
Flemington Road, Parkville, Victoria 3052
Ton, Jonas, subscribers
I'd like to express an opposing point of view to Ton's. I'm replying though
partly to try and sort out my own thoughts on what joint power actually is
in the light of the light of recent articles (principally those by Felix
Zajac, Rick Neptune and Steve Kautz, Gait and Posture 2002;16:215-232 and
2003;17:1-17) and I'd be most open to comment, criticism and correction of
anything I write.
[...] There is another approach to calculating the changes in the total
mechanical energy of the system. This is to work out the joint moments
using inverse dynamics and the joint angular velocities. The dot product of
these is the quantity we generally refer to as "joint power" (although I am
becoming increasingly convinced that this expression is highly misleading).
The sum of the joint powers for all of the joints also represents the total
rate of change of mechanical energy in the entire system and is thus equal
to this quantity as calculated by either of the two methods outlined in the
above paragraph.
Zajac et al. show very neatly that if all muscles crossing a joint are
uniarticular then the joint power at any joint must be equal to the muscle
power. However in the presence of bi-articular muscles this is not the case
and the joint power at any joint is not equal to the combined muscle power
of all muscles crossing that joint. Thus the stated aim of Jonas' analysis,
" I am ultimately using the joint power to get an estimate of the amount of
power that muscle-tendon units must generate or absorb at a joint" is not a
valid one.
Leaving this aside, the question asked was how to interpret the
"components" of power in the different planes. I use the parenthesis
because power is a scalar (it is the time derivative of energy, another
scalar) and, unlike a vector, does not have components. The identity that
the scalar product of muscle force (F) and contraction velocity (v) is
equal to that of joint moment (M) and angular velocity (w) does not imply
the equivalent relationship for the individual components i.e.
F.v = M.w
can be expanded to
Fx.vx + Fy.vy + Fz.vz = Mx.wx + My.wy + Mz.wz (1)
but this does not imply that Fx.vx = Mx.wx and, indeed, this is generally
not the case. (This is different to the relationship between moment and the
vector product of force and moment arm. As this is a relationship between
vectors, it can be broken up into an individual analysis of the components
in different directions.) I'd thus argue that it is not correct to separate
out the "components" of joint power as Ton recommends. I think this is
generally inappropriate but think that in this particular case where joint
powers and muscle powers are being compared it has the potential to be
extremely misleading.
I'd thus tip what Ton has said on its head. Rather than, "if you keep knee
and hip separate, you should also keep the degrees of freedom within each
joint separate", I suggest, particularly in looking at the relationships
between joint powers and muscle powers, that you cannot look at the hip in
isolation (because this will miss out the effects of bi-articular muscles
crossing both the hip and knee and knee and ankle) and that you cannot
consider the "components" of power separately (because it is a scalar and
does not have components).
The biggest question in my mind is why the association between "muscle
power" and "joint power" has persisted for so long in our collective
sub-conscious (including mine). Anyone like to defend it? (indeed has
anyone managed to read this far?!).
Richard
*******************************************
Frank L Buczek Jr, PhD
President-Elect, Gait & Clinical Movement Analysis Society
Director, Motion Analysis Laboratory
Shriners Hospitals for Children
To Jonas, Ton, Richard, Young-Hoo:
I have several comments to add to your very good discussion, and will group
them according to topic areas:
SCALAR NATURE OF WORK AND POWER
Richard, you may remember asking a similar question a couple of years ago,
regarding the scalar nature of kinetic energy. I responded then with a
thought experiment that leads to physically meaningful information (see
BIOMCH-L archives, 23 March 2000). Briefly, each term (not component) in
the dot product equation for work can be used to determine changes in
kinetic energy for motion along its respective reference axis. This step is
perfectly consistent with the scalar nature of work and the relationships
stated in the work/energy theorem (i.e., the work done on a rigid body
equals its change in kinetic energy). It seems to me that a problem would
arise only if we tried to perform vector addition using the terms of the dot
product. Since my thought experiment didn't do that, nor does it need to,
there is no problem assigning physical meaning to work (or power) terms.
In this regard, I agree with Ton. Individual terms in the dot product for
joint power do have physical meaning, and are useful in understanding motion
of a multi-link system. (I could even argue that, in the general case, all
six degrees-of-freedom would be useful, but in deference to Ton, I'll keep
the discussion focused on three rotational degrees-of-freedom, only.)
JOINT POWER VERSUS MUSCLE POWER
Regarding the present discussion, I have long since stopped equating "joint
power" with "muscle power" when these are derived through Inverse Dynamics.
(Important qualifier, see below.) I'll skip details of my learning curve,
and focus on the gait analyses we perform on children with cerebral palsy.
A good example involves a child who goes into recurvatum at the knee
approximately at mid-stance. It is typical to see a profound intrinsic knee
flexion moment at this time, in the presence of power absorption. Often,
there is also knee flexor EMG activity, but not always. When knee flexor
EMG is absent, the moment arises due to deformation of soft tissues in the
joint capsule and muscle/tendon unit; in extreme cases, it may arise from
bone-to-bone contact.
It seems to me that the terms "joint moment" and "joint power" are always
correct. They express two mechanical characteristics of the joint motion.
On the other hand, since it is likely that passive contributions to the
moment are always present in addition to active muscle contributions, the
terms "muscle moment" and "muscle power" are less likely to be accurate when
obtained via Inverse Dynamics.
[...]
INVERSE DYNAMICS essentially describes motion that we've already observed,
strictly as an engineering mechanics problem. Yes, we use anthropometry to
define some inertial characteristics, but once we've done that, the
equations we write would be the same for a person walking as they would be
for a machine. The joint moments describe the NET effect of all moments
that arise from active and passive structures, and in this regard, they do a
fine job of describing why we saw the observed motion.
[...]
Best regards,
FB
-----------------------------------------------------------------
To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl
For information and archives: http://isb.ri.ccf.org/biomch-l
Please consider posting your message to the Biomch-L Web-based
Discussion Forum: http://movement-analysis.com/biomch_l
-----------------------------------------------------------------