CGA FAQ: Processing CoP for inverse dynamics

Hello CGA subscribers

I am currently refining software to calculate joint moments and powers
using the inverse dynamic approach. Of particular interest is how people
have processed centre of pressure (CoP) excursion data for this purpose and
also for determing the magnitude of the excursion of the CoP.

I have had difficulties with errors in CoP excusrion data for the first
and/or last value(s) during stance phase. I have made several checks to
ensure that the magnitude of the force at this time is well above the noise
level of the plate and of sufficient magnitude to not not be likley to
compound any erroe. I have also assessed the affect of noise and offsets on
CoP data and tried to minimise these within the software. I have also
recently tested the accuracy of the forceplate with particular attention to
the CoP and have found it to be accurate to a couple of millimeters. I have
also checked heel contact and toe off times using footswitch data, the
vertical force and the velocity of the toe marker in the X and Z directions
and have found good agreement between all methods.

I have tried several methods for firstly, identifying these incorrect
values and secondly recalculating or estimating these values.

Identification of these incorrect values has been done using the velocity
or acceleration of the CoP as well as comparison of the position of the CoP
relative to kinematic markers of the foot. I have also used the moment arm
about the ankle to assess any obvious abnormality.

I have tried to reestimate these values using several techniques including
interpolation using various methods, or reestimating the erronious point by
predicting its value using the velocity of CoP progression. Neither of
these methods seem to give reasonable results due to the non-systematic
nature of the error.

These CoP errors do not seem to affect joint moment or power estimates. I
assume this is because I'm dealing with one or two inaccurate points/moment
arms at either the beginning or end of the moment curve. These inaccuracies
in the CoP do affect the determination of the excursion of the CoP during
stance and as such I would like to eliminate these errors.

I would be interested in any contribution or simular experiences. Thankyou
for your time.

Kindest regards.

Mick Dillon

Concerning erroneous CoP values at initial and final stance phase of
walking (question posed by Mick Dillon):
It is not suficient that the force values well exceeds the noise in
order to get precise estimates of the CoP. Since the calculation of CoP
involves taking the difference between vertical force components and
then dividing with the total vertical force, Fz, the error tends to be
large whenever Fz is small. This error can be further increased by
vibrations in the plate due to the impact at heel strike.
This erroneous location of CoP, together with the error in the direction
of the force vector can, according to my experience, indeed give quite
large errors in estimated joint moments and powers (Even if the forces
are small, their moment arms can become quite large with respect to the
knee and hip joint, when the vector has a wrong direction).
In my opinion, this error is due to the nature of the problem, and
cannot be overcome by any simple intepolation techniques. We simply
cannot measure the force components with sufficient precision during the
impact phase. Also at toe off we are approaching a situation where we
try to calculate noise divided by noise, which is a highly ill posed
I am interested to hear other peoples experinence and opinion about

Best regards - Håkan
* Håkan Lanshammar, Dept of Systems and Control, Uppsala University
* P.O. Box 27, S-751 03 Uppsala, SWEDEN
* E-mail:, Tel: +46-18-471 30 33, Fax: +46-18-50 36 11
* WWW: 

Hello all CGA readers:

Concerning erroneous CoP values at initial and final stance phase of walking (question posed by Mick Dillon):  I concur with
Hakan Lanshammar's comments about resolution of center of pressure (CoP) at small forces levels.  On way to think of the
problem is that if the digitized force value can only resolve a force to, say, one part in ten, then the center of pressure can only be defined to within one tenth of the distance between the vertical force sensors that support the plate.   Errors of several centimeters are certainly possible, particularly in a force platform that is designed for a large total force level, and consequently has a relatively large value of force corresponding to the smallest bit of the analog to digital converter.

Another possible problem is that a force platform does not measure the center of pressure at the surface of the platform, but at some level below the surface.  This distance below the surface depends on the construction of the force platform, and can be as much as several centimeters.  Unless the force data processing software calculates the intersection of the force vector with the surface of the force platform, there will be errors in the location of the CoP for any force that is not vertical.  These errors become larger as the force vector becomes more inclined away from the vertical, as at the beginning and end of stance phase.  In the early days of force platform use, this problem was widely overlooked, and gave rise to odd results, such as the CoP moving anteriorly out to the toe toward the end of stance phase and then moving posteriorly  again just prior to toe-off.  Some interesting arguments were offered to explain how this unlikely result could occur, but I suspect it was a simple measurement artifact.

Larry Lamoreux
Lafayette, CA  USA

Dear colleagues:

I have a web page dealing with the forceplate-related issues: It currently contains
issues such as action-oriented vs reaction-oriented coordinate system, CoP
position & free vertical torque, and padding on the plate. I have included
several equations which might be useful to the readership. It is still under
construction and I welcome your input.

I agree with Hakan Lanshammar & Larry Lamoreux on that the erroneous CoP
position at the beginning and at the end of the foot contact phase is due to
the error in moment & force components with a small vertical force
component. I have experienced the same problem occasionally with my
software. Equations 5-3a & b and 5-7a & b on the above-mentioned webpage
show this point rather well. These equations use the location of the true
origin of the plate for the computation of the CoP position.

Looking forward to getting valuable feedback.

Young-Hoo Kwon

- Young-Hoo Kwon, Ph.D.
- Ball State University
- The Human Performance/Biomechanics Lab.
- Office: +1 (765) 285-5126
- Fax: +1 (765) 285-8596
- URL:

Dear all,
I fully agree with the comments of both Haakon Lanshammar and Larry
Lamoureux on this subject, but to add to the complexity, I would
like to call to your attention a third effect.
        In fast walking and running, very especially when barefoot, there is
a distinct 'impact' force: a fast and high transient force, with a
direction often grossly deviating from the 'normal' direction of the
GRF, which is more or less directed through the hip joint.
When fed into an inverse kinetics calculation this can lead to very
high peaks in the moment, especially the hip  moment.
In my opinion these impact force are probably correctly measured
by the force plate,(when your sample frequency is sufficient!) but
the problem is in the inverse kinetics. Before calculating the
acceleration, position data need to be strongly smoothed, at 5 Hz or
even lower, otherwise the acceleration data are much too noisy.
Haakan has in the past pointed to this necessity.
        As I see it, the impact force is the result of a very sudden
deceleration of the leg at foot contact.  In a true inverse kinetics
analysis the impact force should thus be cancelled by the m*a due to
this deceleration. The problem is that due to the filtering the very
short and high acceleration is smoothed so much that it does not fit
the GRF recording. The result is an unrealistic high peak in
(especially) the hip moment.    Possible solutions might be
  1. 1) trying to measure the relevant acceleration with accelerometers
  2. 2) filtering the force signal with the same filter as the position data. In that case  the impact will be vanished, and it can give you an idea of what filtering at 5 or 6 Hz does to your data......
        Awaiting furter comments,

Best wishes,

At Hof
Department of Medical Physiology &
Laboratory of Human Movement Analysis AZG
University of Groningen
Bloemsingel 10
Tel:   (31) 50 3632645
Fax:   (31) 50 3632751

To all concerned

Thank you to all who replied to my question concerning error in the calculated centre of pressure during the beginning and end
of stance phase. I shall post a summary in due course.

There seems to be some consensus that erroneous CoP data at the beginning and end of stance phase occurs when the vertical
force component is small. This argument seems reasonable given that the vertical force component forms the denominator of the
CoP calculation.

I'm not sure I completely agree with this view or perhaps I'm just missing something.

A few weeks ago we checked the accuracy of the force plate prior to coding software for process the raw voltage signal and
calculate CoP. One of the checks was to assess the accuracy of the CoP. Using a screwdriver a load was applied in a number
of positions over the force plate. The position of the force was measured using a ruler and compared to the calculated values.
The average RMS error observed was about 2mm. The magnitude of the vertical forces applied during these experiments was
significantly less than that which I am measuring during the initial contact.

I would have to agree with Chris Kirtley in saying that the forces at initial contact are appreciable and I am keen for someone to
explain why my view point is somewhat different from the consensus.

A number of other responses to my posting have described errors in CoP due to the filtering of CoP data. For the software I've
developed I have chosen not to filter CoP, but rather to filter the raw moment and force data prior to calculating CoP. As CoP
is a calculated parameter, such as a joint moment etc., I was under the assumption that it should not need further signal

>From the responses I've received I've reached the conclusion that it is currently a problem in CoP calculation which has not
been resolved. In terms of estimating CoP excursion, I've chosen to perform this task manually by drawing comparisons to foot
length and kinematic data.

Thank you for your thought provoking correspondence.

Mick Dillon

Michael Dillon
B P&O Hons. PhD Student
Centre for Rehabilitation Science and Engineering
School of Mechanical, Manufacturing and Medical Engineering
Queensland University of Technology
GPO Box 2434
Brisbane. 4001.
Ph. +61 07 3864 5151 E-mail:
Fax. +61 07 3864 1469 

Michael Dillon's obtaining accurate determinations of center of
pressure on a force platform applied by a screw driver may be due to
the small area of the force applied. The tip of a screw driver may be
likened to a star, whereas the heel is more like a cloud descending on
a given (larger) region of the force platform.

In the beginning of the first chapter of his book, Cybernetics,
Norbert Wiener discussed how celestial mechanics is one of the oldest
sciences in history. The movements of the visible stars are
predictable because of their huge distances from one another, whereas
meteorology is a very young science, hardly able to satisfy us when we
are wondering whether or not to carry an umbrella. Admittedly, the gap
described by Wiener is probably greater than the one between a screw
driver and a heel, but the idea should be clear enough.

My hunch can be converted into a testable hypothesis. Replace the
screw driver with a padded object, about the size of a human heel,
built symmetrically enough that you can easily identify the "point" on
the force plate that you are aiming for. Press the artificial heel
slowly downward and see if the force platform's determination of
center of pressure becomes gradually less noisy as the pressure

Unfortunately, I do not have a force platform at hand. Could someone
else try this?

Cheers from Hiroshima
Paul Andrew

Hello Chris and CGA readers,

No, I did not mean to suggest that the heel-strike transient does not exist,
but that errors in the CoP will occur when the ground reaction force is not
vertical, if the measured CoP is not corrected up along the force vector to
the surface of the plate.  As far as I know, all forceplates measure the CoP
at a plane below the surface, down inside the forceplate structure.

The heel-strike transient is not easy to measure, however, because it occurs
so quickly.  I'm reminded of a memorable class in Berkeley called
"Experimental Mechanics" where one of the experiments measured impact
deceleration of a dropped mass with two different accelerometers at the same
time.  One was a piezoelectric accelerometer with very high frequency
response, the other was a strain-gage accelerometer with much lower
frequency response.  At first glance, the two deceleration records bore no
resemblance to each other whatsoever.  The strain-gage accelerometer showed
a smooth, gentle curve, while the peizoelectric accelerometer showed high
amplitude, high frequency vibrations.  The purpose of the experiment was to
determine what the true deceleration was.  It turned out that neither
instrument represented the true deceleration very well.  The low frequency
accelerometer smoothed out the rapid changes, while the high frequency one
mainly showed the internal vibrations, or "ringing," of the mass that both
accelerometers were mounted on.  Incidently, the course was taught by Don
Cunningham and Wayne Brown, who designed and built two strain-gauge force
platforms in Berkeley in 1946.  If I'm not mistaken, those two platforms
were the first electrical output forceplates.

Accurate recording of the heel-strike transient requires both a force
platform with high resonant frequency, perhaps 1000 Hz, and an
analog-to-digital converter with high sampling rate, that is, greater than
2000 Hz.  Motion measurements usually are not measured at anything like
these frequencies, giving rise to the discrepancies so aptly described by At
Hof in his CGA contribution earlier today.

Best regards,

Larry Lamoreux
Lafayette, CA, USA

Dear At Hof, and others,

That's an interesting observation about the GRF force not being
cancelled out by the inertial forces. I am wondering, however, if there
would be any difference whether the GRF or the calculated moments are
smoothed - surely any high frequencies from the GRF which sneak into the
inverse dynamics should be filtered out when the moments are filtered.

I guess that brings up the interesting question again - when should we
filter? I remember a discussion some time back on BIOMCH-L in which
someone pointed out that, being a linear operation, filtering could be
done at any point in the calculation chain. I suspect that's not
entirely true, though, now that you've pointed this problem out.

So, I ask again - when should we filter? Both GRF and kinematics BEFORE
doing any calculations, or at the final stage? I'm still not sure when
and how the Vicon Clinical Manager filters, for example - anybody know?
What about Kintrak/Orthotrak?

Anybody done any experiments on possible differences between each

Dr. Chris Kirtley
Dept. of Rehabilitation Sciences
The Hong Kong Polytechnic University

Dear Michael and colleagues:

I believe the problem is more serious than the star-cloud analogy (Paul
Andrew). Sometimes the CoP position is way out of the footprint. Nor is it
due to the direction of the force (Larry Lamoreux). I have encountered this
problem even though I accounted for the real origin of the plate in my
software. (The equations can be found at

Here I'd like to add one more possibility: the A/D board. Each plate
provides 6-8 channels but the A/D board generally does not acquire all 6 or
8 channels at the same time. Instead, it multiplexes. Say you have a
straingauge force plate (6 channels) and use 200-Hz sampling rate. Then the
A/D board multiplexes to the next channel once every 1/1200 s. So a complete
sampling cycle of 6 channels will be done in 1/200 s. But the channel data
have intrinsic time differences due to multiplexing. If the force changes
rapidly, as is the case in the heel strike, this time difference may cause
relatively large error especially when Fz is relatively small. Of coures,
assuming that you did not pre-interpolate the raw force and moment data.

Following this scenario, you can get accurate CoP estimates for a small Fz
if the force is not changing that rapidly. But you may have problem in a
dynamic situation where the force changes rapidly.

It really depends on the A/D board. Check if your A/D board acquires all the
channels at the same time and hold them until the data are completely read.
You are likely to end up a multiplexing A/D board such as DT2801A used by

I am not quite sure if the commercial software packages pre-interpolate the
force (and moment) data so that correct the time differences in the raw
data. I'd like to hear from the companies such as Kistler, AMTI & Bertec
about how they handle this problem.

Hope to receive some feedback from the readership on this view.

- Young-Hoo Kwon, Ph.D.
- Ball State University
- The Human Performance/Biomechanics Lab.
- Office: +1 (765) 285-5126
- Fax: +1 (765) 285-8596
- E-mail:
- URL:

Dear all,

I was a bit disappointed not to get any responses to my question about
filtering. To try to stimulate some, I've plotted some spiky data (a
series of pulses) both in Vicon Clinical Manager and in MLSViewer from
Motion Lab Systems (

You'll see that there is some rippling and smoothing of the raw data in
the VCM plot. Basically, what this little test shows is that there is
some filtering going on in the display of the GCD gait data - i.e. right
at the end of the process. I wonder if is widely known, and whether
anyone knows precisely what the filter characteristics are?

Look forward to your comments!

Dr. Chris Kirtley
Dept. of Rehabilitation Sciences
The Hong Kong Polytechnic University

Back to FAQ