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Katrina Mcdonald
12-01-2004, 09:55 AM
Dear All,

Thank you to everyone that responded to my query about how to
calculate the centre of mass from kinetic data. And a
particular thanks to Drew Smith who gave me much help. From
the responses, I found the best was to calculate the COM
using kinetic data without knowing the initial position or
velocity is to calculate the acceleration from the force
data, and then use double integration, finding the area under
the graph to give velocity, and then finding the area under
that graph to give the displacement. Finding the area can be
done using the trapezoidal rule (2-point Newton-Cotes rule).
It should be noted this only gives the COM in relation to the
centre of the force plates, not in relation to starting
point. There seems to be a number of variations on how you
can calculate COM, with each method making different
assumptions. However, the method I used appeared to be the
simplest whilst still giving a reasonable answer.

Below is the initial question I posted, followed by the
individual responses I received. Thank you all again, it was
of great help.

Katrina McDonald

Initial Posting:

Hello All, I am trying to calculate the centre of mass of
sprinters running over the first two steps after exiting the
blocks. I am using kinetic data from 2 force plates, having
only obtained force, moment and centre of pressure data
collected at 990Hz. I have calculated the centre of mass
using the kinetic double integration method considering the
initial velocity and initial displacement to be zero. I used
this method as it was described in the biomch L archives:
Chris Kirtleyâ€™s - CoM from force plate: Summary of responses.
The results I obtained seem unrealistic and I am having
trouble interpreting them. I would very much appreciate any
help with a correct method to use, how to calculate the
integration constants (if they should be non-zero) and any
information on what to expect or how to interpret them.
Thanks for any help.

From: Danik Lafond, Ph.D < dlafond@ergometrix.ca >

see Lafond et al. (2004) Journal of Biomechanics.

From: Young-Hoo Kwon, Ph.D. < kwon3d@kwon3d.com>

It is impossible to compute accurate CoM position from
acceleration if the initial position and velocity are
unknown. When you integrate acceleration twice, you only get
the position change due to acceleration. You are missing the
contribution of initial velocity and initial position.
It may be OK, however, not knowing the initial position if
the main focus is on the position change (displacement)
during the two steps. In this case, simply start the
integration from the stationary on-block position (initial
velocity = 0). This will give you the initial velocity of the
CoM at the beginning of the two steps. You will be able to
get the displacement due to acceleration and that due to the
initial velocity. I hope it helped.

From: Omar Feix do Nascimento

Try this Reorganisation of human step initiation during acute
experimental muscle pain. Gait Posture. 1999 Dec;10(3):240-7.

From: Drew Smith, PhD

It's difficult to assess where your errors/problems may lie
from your posting, so forgive me if I am stating some things
you consider obvious. The double integration method is good
in that integration will also smooth your COM kinematic data.
So, the resulting curves are likely to be fairly smooth. The
downside of using this method is that the COM displacement
data are in force platform coordinates, ie, the 0,0,0 will be
the centre of the force platform and typically some 20mm
below the surface (depending on the type of force platform -
the specs will confirm this). This means you will need to
know where the centre of the force platform is relative to
your actual measurement space, eg, the blocks or the start
line, to make sense of the kinematics. This is especially
true for multiple platforms, since each will have its own
0,0,0. A second problem is that as soon as your sprinter
touches the second platform (I am assuming from your posting
that you have 2 platforms) while still having a foot on the
first platform, things get a bit hairy, since the forces are
now being distributed two platforms. There are methods for
determining the overall COP from two platforms, but I have
never seen this method combined with determining COM
displacement data. However, if you in fact have only one
platform, then your COM data will only be valid until just
before the other foot lands on the ground. If you have some
actual GRF data that you could send me, eg, in a spreadsheet,
I'd probably be able to give you some more specific advice.

From: Aguinaldo, Arnel"

I encourage you to read the methods described by Donelan JM,
Kram R, Kuo AD. Simultaneous positive and negative external
mechanical work in human walking. J Biomech. 2002, 35(1):117-
24 Traditional kinetic methods of COM estimation often
underestimate the double integrals due to summation of GRFs
of both the trailing and leading limbs before integration.
Donelan et al. describe a more logical calculation that may
give you a more appropriate answer. Also, check out the
methods outlined by Eames et al. Comparing methods of
estimating the total body centre of mass in three-dimensions
in normal and pathological gaits. Human Movement Sci 1999,
18: 637-646