Ton Van Den Bogert

08-05-2003, 04:37 AM

Rene Ferdinands formulated some interesting questions on power analysis

using inverse dynamics.

As moderator, I would like to suggest that this topic is of general

interest and quite suitable for a public discussion. So, please post your

responses directly to Biomch-L@nic.surfnet.nl.

Below follows my own response.

> (a) From the inverse dynamics solution of two or more coupled rigid bodies, what

> would the value of just the net joint torque multiplied the corresponding

> segment angular velocity compute (i.e. just P = Tw)? Anything meaningful?

This quantity would represent the mechanical power delivered by a hypothetical

motor that drives the joint. If the joint is driven by muscles that span only

one joint, the quantity P = Tw therefore directly represents the total

mechanical power delivered by those muscles. If some of the driving muscles

are biarticular, muscle mechanical power is much harder to estimate. It will

require estimates of individual muscle forces and lengh changes.

The fact that any muscle, even a single-joint muscle, causes motion in

all other joints, is not relevant here. P = Tw is part of an inverse

analysis, and still accurately represents the power required at that one

joint. When doing an inverse analysis, the accelerations induced elsewhere will

also be measured and thus automatically be accounted for in the power

calculations for those other joints.

> (b) If the net joint torque (from inverse dynamics) multiplied by the difference

> in angular velocities of the adjacent segments was calculated would this

> satifactorily give the values of THE TOTAL SUM OF active muscle power flows in

> or out of the segment (i.e. P=T(w2-w1)). Is this always necessary or could

> sometimes P=Tw used?

If I understand your definition of these variables, the joint angular

velocity w is simply the relative rotation between the two segments (i.e. w =

w2-w1). So both equations would give the same value for joint power. I

do not have Winter's book here to confirm this definition.

Power flow (in and out of the segment) is a tricky concept. There is also

power flow due to the resultant joint force (P = F.v). Results will depend

on which reference frame is used to measure v. This was discussed on

Biomch-L some time ago in relation to analysis of treadmill locomotion. I

prefer to stay away from the concept of power flow. It can be useful but

only if interpreted carefully.

> (c) Can joint muscle power or absorption be calculated accurately using Winter's

> approach using the joint torques found from an inverse dynamics solution? Is

> this what Winter meant or did he mean as in (ii) above?

Apart from the issue of bi-articular muscles, yes, I think that net muscle

power generation and absorption is equal to the joint power measured from

inverse dynamics. Note that I use "net muscle power". If there is co-contraction

of antagonistic muscles, there is simultaneous positive and negative muscle

power. Only the total net value is "seen" by the inverse dynamic analysis.

> Is the methodology for

> calculating power flows correct in Winter?

Again, without looking at Winter's book, I do not think that Winter proposed

calculating power flow. I think his analysis was limited to calculation of

joint power.

> (d) Can the power flow equations easily applied to the 3D case since power is a

> scalar quantity?

If you mean joint power, yes. One way to do this is to model the joint as a

mechanism with three successive hinge joints. This is the classic Grood-Suntay

joint coordinate system (JCS) which gives cardanic angles from the kinematic

analysis. The time derivatives of these angles are the joint angular velocities.

If you use the same JCS to represent the joint moments, you will have a scalar

moment in in each of the three hinge joints and the power calculation

works exactly as in 2-D. You will get three joint powers, which can be added up

if you want the total for the entire cardanic joint complex. The three

individual joint powers can be interpreted as the power required for,

respectively, flexion, abduction, and rotation.

However, inverse dynamics software usually does not give joint moments

decomposed along the JCS axes. Instead, you may get Mx,My,Mz expressed

in an orthogonal segment-fixed XYZ reference frame, or in a global reference

frame. To use this for calculation of joint power, you need an angular

velocity vector expressed in the same reference frame. 3-D kinematic

analysis gives, at each time, a rotation matrix R that describes the relative

orientation between the two segments. The angular velocity vector w =

(wx,wy,wz) can be estimated from R and its time-derivative Rdot:

( 0 -wz wy )

( wz 0 -wx ) = inv(R)*Rdot

( -wy wx 0 )

Now joint power will be the dot product of M and w: P = Mx.wx + My.wy + Mz.wz.

You can interpret these three terms of the dot product as flexion, abduction,

and rotation power, as long as you are aware that these components do *not*

correspond to standard JCS definitions.

I worry a bit that some people may be mixing these two approaches and

multiply JCS angular velocities by the "corresponding" segment-fixed XYZ

components of the joint moment vector. This is only correct when the

JCS axes are aligned with the XYZ axes, i.e. when two of the three

joint rotations are zero.

The ISB has proposed standards for the reporting of joint motion (Wu

et al., J Biomech 35:543-548, 2002; http://www.isbweb.org/standards ) but has

not directly made recommendations on the reporting of joint moments. This

issue needs to be resolved. My suggestion would be to report joint moments

in the same JCS as joint motion, so that power calculations can be done as in my

first suggestion above. I would welcome some comments on this, especially from

the ISB standardization committee. Please change the "Subject" line when

responding to this standardization issue rather than to Rene Ferdinands'

questions.

> (e) How should power flows into or out of a segment be described taking into

> account statement (iv) by Zajac et al. Does this mean that P=T(w2-w1) give the

> net joint power which represents the SUMMED power by the net joint moment

> to/from ALL the segments?

I agree with Zajac's statement and I think this is one more reason why the

concept of "power flow into a segment" is not useful. Joint power itself is

still perfectly valid, I think, as long as it is interpreted with an awareness

of biarticular muscles and antagonistic co-contraction.

--

Ton van den Bogert

--

A.J. (Ton) van den Bogert, PhD

Department of Biomedical Engineering

Cleveland Clinic Foundation

9500 Euclid Avenue (ND-20)

Cleveland, OH 44195, USA

Phone/Fax: (216) 444-5566/9198

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To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

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using inverse dynamics.

As moderator, I would like to suggest that this topic is of general

interest and quite suitable for a public discussion. So, please post your

responses directly to Biomch-L@nic.surfnet.nl.

Below follows my own response.

> (a) From the inverse dynamics solution of two or more coupled rigid bodies, what

> would the value of just the net joint torque multiplied the corresponding

> segment angular velocity compute (i.e. just P = Tw)? Anything meaningful?

This quantity would represent the mechanical power delivered by a hypothetical

motor that drives the joint. If the joint is driven by muscles that span only

one joint, the quantity P = Tw therefore directly represents the total

mechanical power delivered by those muscles. If some of the driving muscles

are biarticular, muscle mechanical power is much harder to estimate. It will

require estimates of individual muscle forces and lengh changes.

The fact that any muscle, even a single-joint muscle, causes motion in

all other joints, is not relevant here. P = Tw is part of an inverse

analysis, and still accurately represents the power required at that one

joint. When doing an inverse analysis, the accelerations induced elsewhere will

also be measured and thus automatically be accounted for in the power

calculations for those other joints.

> (b) If the net joint torque (from inverse dynamics) multiplied by the difference

> in angular velocities of the adjacent segments was calculated would this

> satifactorily give the values of THE TOTAL SUM OF active muscle power flows in

> or out of the segment (i.e. P=T(w2-w1)). Is this always necessary or could

> sometimes P=Tw used?

If I understand your definition of these variables, the joint angular

velocity w is simply the relative rotation between the two segments (i.e. w =

w2-w1). So both equations would give the same value for joint power. I

do not have Winter's book here to confirm this definition.

Power flow (in and out of the segment) is a tricky concept. There is also

power flow due to the resultant joint force (P = F.v). Results will depend

on which reference frame is used to measure v. This was discussed on

Biomch-L some time ago in relation to analysis of treadmill locomotion. I

prefer to stay away from the concept of power flow. It can be useful but

only if interpreted carefully.

> (c) Can joint muscle power or absorption be calculated accurately using Winter's

> approach using the joint torques found from an inverse dynamics solution? Is

> this what Winter meant or did he mean as in (ii) above?

Apart from the issue of bi-articular muscles, yes, I think that net muscle

power generation and absorption is equal to the joint power measured from

inverse dynamics. Note that I use "net muscle power". If there is co-contraction

of antagonistic muscles, there is simultaneous positive and negative muscle

power. Only the total net value is "seen" by the inverse dynamic analysis.

> Is the methodology for

> calculating power flows correct in Winter?

Again, without looking at Winter's book, I do not think that Winter proposed

calculating power flow. I think his analysis was limited to calculation of

joint power.

> (d) Can the power flow equations easily applied to the 3D case since power is a

> scalar quantity?

If you mean joint power, yes. One way to do this is to model the joint as a

mechanism with three successive hinge joints. This is the classic Grood-Suntay

joint coordinate system (JCS) which gives cardanic angles from the kinematic

analysis. The time derivatives of these angles are the joint angular velocities.

If you use the same JCS to represent the joint moments, you will have a scalar

moment in in each of the three hinge joints and the power calculation

works exactly as in 2-D. You will get three joint powers, which can be added up

if you want the total for the entire cardanic joint complex. The three

individual joint powers can be interpreted as the power required for,

respectively, flexion, abduction, and rotation.

However, inverse dynamics software usually does not give joint moments

decomposed along the JCS axes. Instead, you may get Mx,My,Mz expressed

in an orthogonal segment-fixed XYZ reference frame, or in a global reference

frame. To use this for calculation of joint power, you need an angular

velocity vector expressed in the same reference frame. 3-D kinematic

analysis gives, at each time, a rotation matrix R that describes the relative

orientation between the two segments. The angular velocity vector w =

(wx,wy,wz) can be estimated from R and its time-derivative Rdot:

( 0 -wz wy )

( wz 0 -wx ) = inv(R)*Rdot

( -wy wx 0 )

Now joint power will be the dot product of M and w: P = Mx.wx + My.wy + Mz.wz.

You can interpret these three terms of the dot product as flexion, abduction,

and rotation power, as long as you are aware that these components do *not*

correspond to standard JCS definitions.

I worry a bit that some people may be mixing these two approaches and

multiply JCS angular velocities by the "corresponding" segment-fixed XYZ

components of the joint moment vector. This is only correct when the

JCS axes are aligned with the XYZ axes, i.e. when two of the three

joint rotations are zero.

The ISB has proposed standards for the reporting of joint motion (Wu

et al., J Biomech 35:543-548, 2002; http://www.isbweb.org/standards ) but has

not directly made recommendations on the reporting of joint moments. This

issue needs to be resolved. My suggestion would be to report joint moments

in the same JCS as joint motion, so that power calculations can be done as in my

first suggestion above. I would welcome some comments on this, especially from

the ISB standardization committee. Please change the "Subject" line when

responding to this standardization issue rather than to Rene Ferdinands'

questions.

> (e) How should power flows into or out of a segment be described taking into

> account statement (iv) by Zajac et al. Does this mean that P=T(w2-w1) give the

> net joint power which represents the SUMMED power by the net joint moment

> to/from ALL the segments?

I agree with Zajac's statement and I think this is one more reason why the

concept of "power flow into a segment" is not useful. Joint power itself is

still perfectly valid, I think, as long as it is interpreted with an awareness

of biarticular muscles and antagonistic co-contraction.

--

Ton van den Bogert

--

A.J. (Ton) van den Bogert, PhD

Department of Biomedical Engineering

Cleveland Clinic Foundation

9500 Euclid Avenue (ND-20)

Cleveland, OH 44195, USA

Phone/Fax: (216) 444-5566/9198

---------------------------------------------------------------

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl

For information and archives: http://isb.ri.ccf.org/biomch-l

---------------------------------------------------------------