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Thanks to all who replied regarding my original query. The consensus is
that power should be calculated from F*v. I also made an error by
calculating work from F*d whereas I should have integrated F with
respect to displacement. Others suggested that errors could be
associated with the method of derivation and integration. Below is my
original post followed by the reponses I received.
Regards,
Loren Chiu
Graduate Assistant
Exercise Biochemistry Laboratory
Human Performance Laboratories
The University of Memphis
Biomech-L subscribers,
I have a question about the mathematical calculation of mechanical work
and power during resistance exercise.
We have a set-up that involves using a force platform to measure force
and a linear position transducer to measure displacement. All data are
sampled at 500Hz.
>From the vertical force data and the displacement data, we'd like to
calculate mechanical work and power of the movement of the centre of
mass. The question is which approach to use:
1. Differentiate displacement wrt time to obtain velocity. Multiply
force and velocity channels to obtain power. Integrate power wrt time
to obtain work. Winter (1990) suggests this integration method to
calculate work.
or
2. Multiply force and displacement channels to obtain work.
Differentiate work wrt time to obtain power.
I've calculated work and power using both approaches and overlayed the
respective curves (work-time(1) vs. work-time (2), etc.) and they do
not replicate each other, indicating that the two approaches have
different results.
************************************************** ***********************
"D. Gordon E. Robertson, Ph.D."
I am not sure how your displacement transducer operates but I use a
force
platform to calculate displacement, velocity, work and power. The
system
only works for movements that start statically, for example, vertical
jumps, standing broad jumps, squats, etc. The equations operate in
three
dimensions so you can even evaluate lateral movements. The theory is
based
on single and double integration of the force signals. An important
requirement is that the body weight of the person is calculated from a
brief interval before the movement starts and when the person is
motionless. The program also computes the projection height and takeoff
velocity. We have tested it and it works for relatively small time
intervals. If the person stands for too long the double integral (to
obtain displacement) becomes unreasonably high. How high depends on the
drift in your force platform and other factors (lab. vibrations, heat
..).
************************************************** ***********************
"Milad GA Ishac"
Some of the difference is due the numerical techniques.
Numerical differentiation is an approximation of the tangent at
a point on
a supposedly continious data. The tanget value depends on the relative
value
of the data point with respect to the adjacent points. If all data
points
are equally biased, the bias will have no effect on the result.
On the other hand, integration value at a point is simply the
summation of
the segmented areas under the curve from an initial point to the
integration
point. If you take the data produced by the differentiation process and
integrate it you should get, theoritically, the original data. However,
it
is seldom the case. One source of error is due to the value of the
initial
condition. Another is due to accumulation of the errors inherited by the
approximation used to obtain all the tangents at all points from the
initial
point to the current point.
************************************************** ***********************
William Megill
I'd be curious to know what others have to say. I've never bothered to
look at option (1), I've just always gone with (2). Though I'm not
familiar
with Winter's work (animal biomechanist/ecologist!), I'd be worried
about
extra numerical steps - lots of potential to introduce/amplify noise. In
theory, both approaches should be identical. My guess at your problem is
either a syntax error (sorry to suggest it, but I've done that I don't
know
how many times) or it's in the way you're doing the calculus (what
method
are you using to differentiate, and more particularly to integrate -
Simpson's
rule, trapezoidal, ... - they all introduce different levels of
uncertainty. Do
you have a filter - even a running average - in your system somewhere?
That might be part of the problem.).
************************************************** ***********************
"At Hof "
In principle both methods should give the same results, BUT very
often the displacement data are smoothened before differentiation,
while the force data are not. This, and numerical inaccuracy in the
differentiatin, may be the source of your problems.
I could not make out what you measure the displacement of. Is it
a point on the trunk, more or less representative for the whole body
CoM?
In any case, it may be a suggestion, to determine the velocity of
the CoM from the acceleration, = ground reaction force/mass - g
Ref: Cavagna, Force platforms as ergometers. J. appl. Physiol 39:
174-179 (1985).
************************************************** ***********************
"Heinz-Bodo Schmiedmayer"
To get work you have to integrate F ds over the displacement to get
work. Just multiplying the force and the displacement chanel is wrong!
dW F ds
P = ---- = ------ = F v
dt dt
s1
/
W = | F ds with s0 and s1 being start and end displacements,
respectively
/
s0
(If F and ds and v are parallel)
************************************************** ***********************
"Z. Hasan"
The first method is correct (assuming you are measuring the component of
force in the direction of the displacement).
The second method gives you (d/dt) (f * x), which equals f * dx/dt + x *
df/dt; only the first term represents power.
************************************************** ***********************
"Aguinaldo, Arnel"
The choice of which mechanical work calculation to use has been the
subject
of debate for quite some time now. There are about 4 or 5 approaches
advocated (and disputed) by different people. Winter, as you know,
suggests
using the integration of power to get mechanical work at the joints.
Other
similar approaches such as the one used to calculate the transfer of
energies between joint segments rely on the assumption that the inertial
properties of each segments are known, which can vary from person to
person.
Another problem with these approaches is the propagation of errors from
the
kinematic data (ie, angular displacement and velocity) needed in these
calculations, which can result in unreliable or inconsistent data. There
is
a consensus, however, that different approaches will yield different
findings.
We use the method proposed by Cavagna and his colleagues based on ground
reaction forces. Since we are mainly interested in the mechanical work
on
the body center-of-mass during various activities, we feel this method
is
the most reliable because 1) none of the above assumptions are needed in
the
calculation and 2) the results are primarily dependent on the accuracy
of
the force platform measurements, which, safely to say, are highly
accurate.
The issue with this method is that it doesn't account for the work in
the
individual segments. However, there is an extension of this which
calculates
segmental mechanical work relative to the body COM based on kinematic
data.
So, it is not so much which approach is most "accurate," because no gold
standard exists to compare the results of these various methods
(although
some have used metabolic energy as a basis of comparison), but mainly
"how
accurately a given parameter can be measured given the equipment that is
available" (Burdett et al. JOR, 1(1), 1983). In the case of body COM
mechanical work, force platforms are the key!
************************************************** ***********************
Kevin Ness
In your second method below I assume you ment that you integrated force
with respect to displacement? Work is the definite integral of force
with
respect to displacement - it only equals the product when the force is
independent of displacement - which will NOT be so in your case.
************************************************** ***********************
Nitin Moholkar
The first way is correct. Multiply force and velocity to get power,
then integrate to get work.
The second way may be incorrect, depending on how you did the
calculation.
While work does equal force times distance, this is a simplification
used when force is constant, or when you are using an average force.
When force is changing, you need to integrate force with respect to
position to get work as a function of time, W(t)=Integral(F(t)dx).
(Sorry, I don't know how to make the integration symbol on Eudora.)
This should give you work as a function of time, since Force is a
function of time. When you take the derivative of this with respect
to time, you will get the same power curve as when you multiply force
and velocity.
I think the first way is easier, though both should give correct
answers.
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that power should be calculated from F*v. I also made an error by
calculating work from F*d whereas I should have integrated F with
respect to displacement. Others suggested that errors could be
associated with the method of derivation and integration. Below is my
original post followed by the reponses I received.
Regards,
Loren Chiu
Graduate Assistant
Exercise Biochemistry Laboratory
Human Performance Laboratories
The University of Memphis
Biomech-L subscribers,
I have a question about the mathematical calculation of mechanical work
and power during resistance exercise.
We have a set-up that involves using a force platform to measure force
and a linear position transducer to measure displacement. All data are
sampled at 500Hz.
>From the vertical force data and the displacement data, we'd like to
calculate mechanical work and power of the movement of the centre of
mass. The question is which approach to use:
1. Differentiate displacement wrt time to obtain velocity. Multiply
force and velocity channels to obtain power. Integrate power wrt time
to obtain work. Winter (1990) suggests this integration method to
calculate work.
or
2. Multiply force and displacement channels to obtain work.
Differentiate work wrt time to obtain power.
I've calculated work and power using both approaches and overlayed the
respective curves (work-time(1) vs. work-time (2), etc.) and they do
not replicate each other, indicating that the two approaches have
different results.
************************************************** ***********************
"D. Gordon E. Robertson, Ph.D."
I am not sure how your displacement transducer operates but I use a
force
platform to calculate displacement, velocity, work and power. The
system
only works for movements that start statically, for example, vertical
jumps, standing broad jumps, squats, etc. The equations operate in
three
dimensions so you can even evaluate lateral movements. The theory is
based
on single and double integration of the force signals. An important
requirement is that the body weight of the person is calculated from a
brief interval before the movement starts and when the person is
motionless. The program also computes the projection height and takeoff
velocity. We have tested it and it works for relatively small time
intervals. If the person stands for too long the double integral (to
obtain displacement) becomes unreasonably high. How high depends on the
drift in your force platform and other factors (lab. vibrations, heat
..).
************************************************** ***********************
"Milad GA Ishac"
Some of the difference is due the numerical techniques.
Numerical differentiation is an approximation of the tangent at
a point on
a supposedly continious data. The tanget value depends on the relative
value
of the data point with respect to the adjacent points. If all data
points
are equally biased, the bias will have no effect on the result.
On the other hand, integration value at a point is simply the
summation of
the segmented areas under the curve from an initial point to the
integration
point. If you take the data produced by the differentiation process and
integrate it you should get, theoritically, the original data. However,
it
is seldom the case. One source of error is due to the value of the
initial
condition. Another is due to accumulation of the errors inherited by the
approximation used to obtain all the tangents at all points from the
initial
point to the current point.
************************************************** ***********************
William Megill
I'd be curious to know what others have to say. I've never bothered to
look at option (1), I've just always gone with (2). Though I'm not
familiar
with Winter's work (animal biomechanist/ecologist!), I'd be worried
about
extra numerical steps - lots of potential to introduce/amplify noise. In
theory, both approaches should be identical. My guess at your problem is
either a syntax error (sorry to suggest it, but I've done that I don't
know
how many times) or it's in the way you're doing the calculus (what
method
are you using to differentiate, and more particularly to integrate -
Simpson's
rule, trapezoidal, ... - they all introduce different levels of
uncertainty. Do
you have a filter - even a running average - in your system somewhere?
That might be part of the problem.).
************************************************** ***********************
"At Hof "
In principle both methods should give the same results, BUT very
often the displacement data are smoothened before differentiation,
while the force data are not. This, and numerical inaccuracy in the
differentiatin, may be the source of your problems.
I could not make out what you measure the displacement of. Is it
a point on the trunk, more or less representative for the whole body
CoM?
In any case, it may be a suggestion, to determine the velocity of
the CoM from the acceleration, = ground reaction force/mass - g
Ref: Cavagna, Force platforms as ergometers. J. appl. Physiol 39:
174-179 (1985).
************************************************** ***********************
"Heinz-Bodo Schmiedmayer"
To get work you have to integrate F ds over the displacement to get
work. Just multiplying the force and the displacement chanel is wrong!
dW F ds
P = ---- = ------ = F v
dt dt
s1
/
W = | F ds with s0 and s1 being start and end displacements,
respectively
/
s0
(If F and ds and v are parallel)
************************************************** ***********************
"Z. Hasan"
The first method is correct (assuming you are measuring the component of
force in the direction of the displacement).
The second method gives you (d/dt) (f * x), which equals f * dx/dt + x *
df/dt; only the first term represents power.
************************************************** ***********************
"Aguinaldo, Arnel"
The choice of which mechanical work calculation to use has been the
subject
of debate for quite some time now. There are about 4 or 5 approaches
advocated (and disputed) by different people. Winter, as you know,
suggests
using the integration of power to get mechanical work at the joints.
Other
similar approaches such as the one used to calculate the transfer of
energies between joint segments rely on the assumption that the inertial
properties of each segments are known, which can vary from person to
person.
Another problem with these approaches is the propagation of errors from
the
kinematic data (ie, angular displacement and velocity) needed in these
calculations, which can result in unreliable or inconsistent data. There
is
a consensus, however, that different approaches will yield different
findings.
We use the method proposed by Cavagna and his colleagues based on ground
reaction forces. Since we are mainly interested in the mechanical work
on
the body center-of-mass during various activities, we feel this method
is
the most reliable because 1) none of the above assumptions are needed in
the
calculation and 2) the results are primarily dependent on the accuracy
of
the force platform measurements, which, safely to say, are highly
accurate.
The issue with this method is that it doesn't account for the work in
the
individual segments. However, there is an extension of this which
calculates
segmental mechanical work relative to the body COM based on kinematic
data.
So, it is not so much which approach is most "accurate," because no gold
standard exists to compare the results of these various methods
(although
some have used metabolic energy as a basis of comparison), but mainly
"how
accurately a given parameter can be measured given the equipment that is
available" (Burdett et al. JOR, 1(1), 1983). In the case of body COM
mechanical work, force platforms are the key!
************************************************** ***********************
Kevin Ness
In your second method below I assume you ment that you integrated force
with respect to displacement? Work is the definite integral of force
with
respect to displacement - it only equals the product when the force is
independent of displacement - which will NOT be so in your case.
************************************************** ***********************
Nitin Moholkar
The first way is correct. Multiply force and velocity to get power,
then integrate to get work.
The second way may be incorrect, depending on how you did the
calculation.
While work does equal force times distance, this is a simplification
used when force is constant, or when you are using an average force.
When force is changing, you need to integrate force with respect to
position to get work as a function of time, W(t)=Integral(F(t)dx).
(Sorry, I don't know how to make the integration symbol on Eudora.)
This should give you work as a function of time, since Force is a
function of time. When you take the derivative of this with respect
to time, you will get the same power curve as when you multiply force
and velocity.
I think the first way is easier, though both should give correct
answers.
---------------------------------------------------------------
To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl
For information and archives: http://isb.ri.ccf.org/biomch-l
---------------------------------------------------------------