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kwon3d
01-27-2004, 12:37 PM
Dear Richard and all,

I apologize for not carefully reading the previous postings and causing
confusion in the discussion.

>Young-Hoo also raises the issue of "power flow" - the idea that joint
power represents the flow of energy through a joint. I think this
concept can be traced back to the work of both Fenn and Elftman in the
first half of the last century but was later popularised by Winter.
>However what I call the "New Biomechanics" (as reviewed in the two Gait
and Posture articles I cited in yesterday's e-mail) makes it clear that
the power considered as being generated in a muscle (muscle power) or at
a joint (joint power) can ....

I don't believe I used the term 'power flow' in my previous posting
because I precisely know the definition of power given my background in
astronomy (^_^). It is the rate of energy flow or work. In the segment's
perspective, it is the rate of energy inflow (positive) or outflow
(negative). In the muscle's perspective it can be the rate of energy
generation (positive) / absorption (negative). PF in my previous posting
is the rate of energy transfer through muscle and PW is the rate of work
(energy generation/absroption). I am rather seeing a misconception in
Richard's statements quoted above. Power can't be generated. What is
generated is the energy and the generation rate is the power (PW).

>PS I think it was John Paul who first pointed out to me that "power
flow" is tautological. Power being the time derivative of energy is by
definition a flow. Our understanding of the terms may be clarified by
dropping the word "flow" when associated with "power". "Energy flow" may
be a preferable term but should not mask the fact that muscles are
capable of generating or absorbing energy as well as redistributing it.

Again, as I mentioned in my previous posting, muscles have two kinds of
power: PF and PW. PF is due to the energy transfer through the muscle
and PW is due to the work done by the muscles. These two terms encompass
what Richard stated above.

>However what I call the "New Biomechanics" (as reviewed in the two Gait
and Posture articles I cited in yesterday's e-mail) makes it clear that
the power considered as being generated in a muscle (muscle power) or at
a joint (joint power) can change the energy of any segment (depending on
the characteristics of the whole biomechanical system) not just the
segments proximal and distal to the joint. It thus strikes me that the
concept of either quantity representing power flow THROUGH a joint is
highly mis-leading (I'm very tempted to use the term "wrong" here rather
than "highly mis-leading").

There are two interaction media between two directly involved segments:
the joint and the muscles. The muscle power (my definition of muscle
power) can be computed by two different methods: M dot w (Richard's
definition of muscle power) or F dot v (Richard's definition of joint
power). I assume here that Richard's w is the joint angular velocity
(the difference between the angular velocities of the two segments) and
v is the contraction velocity of the muscle along the line of pull of
the muscle (the component of the difference in the linear velocities of
both muscle attachments along the line of pull).

Let me now try this. Imagine there is only one muscle connecting the
shank and the thigh at the knee for simplicity. In the shank's
perspective, energy inflow/outflow occurs at both the muscle attachment
and the knee joint due to the interaction between the shank and the
thigh. Say FJs is the force acting on the shank at the knee while FMs is
the force acting at the muscle attachment by the muscle. Velocity of the
joint is VJk while the velocity of the muscle attachment is VMs. The
rate of energy inflow/outflow of the shank at the knee due to the
interaction between the shank and the thigh is then

PJs = FJs * VJk. [1]

where s = shank, k = knee, VJk = the velocity of the knee, and * = dot
product operator. The energy in/outflow of the thigh due to the
interaction at the knee is then

PJt = FJt * VJk, [2]

where t = thigh. The force acting on the two segments at the knee due to
the interaction through the joint has the following relationship:

FJt = -FJs. [3]

Thus

PJt = -PJs. [4]

Inflow of energy into the shank means outflow of the energy from the
thigh at the same rate through the joint (energy flow or transfer from
one segment to the other through the joint).

On the other hand, the energy in/outflow into/from the segments due to
the interaction through the muscle can be described as

PMs = FMs * VMs, [5]

and

PMt = FMt * VMt, [6]

where FM = force acting at the muscle attachment by the muscle, and VM =
muscle attachment velocity. The muscle attachment velocity can be
divided into two terms:

VM = VJk + VM', [7]

where VM' = the relative velocity of the muscle attachment to the knee
joint. Eqs. 5 and 6 can be rewritten as

PMs = FMs * (VJk + VM's) = FMs * VJk + FMs * VM's [8]

PMt = FMt * (VJk + VM't) = FMt * VJk + FMt * VM't [9]

Let PM's and PM't be

PM's = FMs * VM's [10]

PM't = FMt * VM't. [11]

Eqs. 10 and 11 basically show the power terms at the muscle attachments
due to the relative motions of the muscle attachments to the knee joint.


In theory, the relative velocity of the muscle attachment to the joint
can be further divided into two terms: the velocity of the attachment
due to the common rotation of both segments as one unit (VR) and the
velocity induced by the muscle contraction (VC):

VM' = VR + VC [12]

(I am not going to discuss how to compute these terms here.) Therefore

PM's = FMs * (VRs + VCs) = FMs * VRs + FMs * VCs [13]

PM't = FMt * (VRt + VCt) = FMt * VRt + FMt * VCt. [14]

There exists a direct relationship between the muscle forces acting at
the attachments:

FMt = -FMs [15]

A simple geometric endeaver will prove the following relationship
between the 1st terms of Eqs. [13] and [14]:

FMt * VRt = -FMs * VRs, [16]

since VR is the linear velocity of the attachment due to the rotation of
both segments as one unit. The total power due to the interaction
through the joint and the muscle between the shank and thigh are then

Ps = PJs + PMs
= FJs * VJk + FMs * VJk + FMs * VRs + FMs * VCs
= (FJs + FMs) * VJk + FMs * VRs + FMs * VCs [17]

Pt = PJt + PMt
= FJt * VJk + FMt * VJk + FMt * VRt + FMt * VCt
= (FJt + FMt) * VJk + FMt * VRt + FMt * VCt [18]

The first terms are due to the linear interaction between the two
segments. FJ + FM in the first terms is the so-called net joint force in
the inverse dynamics. The second terms are due to the common rotations
of the segments as one unit. The third terms are due to the muscle
action. From Eqs. 4, 15, and 16:

Ps + Pt = (0) + (0) + (FMs * VCs + FMt * VCt)
= FMs * (VCs - VCt)
= FMs * v, [19]

where v = the muscle contraction velocity. The first two terms vanish
because there are only energy flows from one to another. The third term
is what Richard defines as the joint power (although I still want to
call it muscle power).

The point is that what Richard looked at was the third terms in Eqs.
17-19 only. When I said energy transfer (flow), it refers to the first
two terms. Eqs 17-19 can be also written as

Ps = Fk * VJk + Mk * wRk + Mk * wCs [20]

Pt = (-Fk * VJk) + (-Mk * wRk) + (-Mk * wCt) [21]

Ps + Pt = (0) + (0) + Mk * w [22]

where Fk = net joint force at the knee (= FJs + FMs), Mk = net joint
torque at the knee, and w = relative angular velocity of the shank to
the thigh (= wCs - wCt). Eqs. 20-22 are what we derive from the inverse
dynamics.

Whether it is the OLD biomechanics or the NEW biomechanics, we still
talk about the same thing in different forms. There are energy transfers
(flows) from one segment to the other through the joint and muscle (the
first two terms in Eq 17 and 18). The muscle also generates/absorbs
energy by doing a positive/negative work (the third terms in Eqs.
17-18).

>... that the power considered as being generated in a muscle (muscle
power) or at a joint (joint power) can change the energy of any segment
(depending on the characteristics of the whole biomechanical system) not
just the segments proximal and distal to the joint.

Assuming all muscles are uniarticular (as stated in Richard's earlier
posting), the shank only affects the thigh directly (Eqs. 17-19). What
affects the trunk directly is the thigh, not the shank. Of course the
shank will affect the trunk indirectly through the thigh, however,
contrary to Richard's statement, there is no conceptual problem in the
idea of energy flow between the shank and thigh through the knee joint
and muscle (the first two terms of Eqs. 17-19).

I hope I did not make any mistakes in running down the equations and
developing the linkage between the OLD biomechanics and the NEW
biomechanics. Thank you for reading this lengthy posting. Good night!

Young-Hoo
------------------------------------------------------
- Young-Hoo Kwon, Ph.D.
- Biomechanics Lab, Texas Woman's University
- kwon3d@kwon3d.com
- http://kwon3d.com
------------------------------------------------------

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