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

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