Bill Sellers wrote:

> In 2D, moment arm is the perpendicular distance of the line

> of action of a force from the fulcrum. In 3D, as calculated

> using the delLen/delTheta formula this isn't the case at all.

> It will give the same answer if the line of action is at

> right angles to the axle but otherwise the value it gives is

> less (potentially very much less) than the perpendicular

> distance. In the extreme when the line of action is parallel

> to the rotation axis there is no change of length with change

> of angle and the result is zero. Mechanically this is

> obviously a very useful value to calculate since it is

> certainly the effective moment arm in terms of the torque

> generated by contraction of a muscle but I'm not sure that

> moment arm is the correct term.

Thanks for bringing this up, it is a great topic for Biomch-L.

I have no answer but some comments that may be helpful for the

discussion.

In 3D, moment is a vector and defined using a cross product:

M = d x F

where d is a vector from the origin of the reference frame to the point

of application of the force vector F. To get moment with respect to a

joint center, as we usually do, we just have to define that point to be

the origin. For this choice of origin, we use the term "joint moment".

So far, there is no concept of "moment arm" at all. That requires a

more specific situation.

If the joint is a hinge, we can project the joint moment vector onto the

hinge axis, so we get a scalar joint moment value. Then, you can define

"moment arm" to be this scalar moment, divided by the magnitude of the

force.

And this is exactly the same moment arm that you would get with the

partial derivative method.

The remainder of the moment vector (total moment, minus the projection

on the axis) is a "constraint moment" which are related to tissue

loading but not movement. This constraint moment vector can be

decomposed further into two components that are perpendicular to the

joint axis. Each of these would then have its own "moment arm" also,

being the moment divided by muscle force.

Confusion may arise from the fact that term "moment arm" on its own is

meaningless, it must be defined with respect to a specific rotational

degree of freedom. In 2D, this is always the joint angle, so no further

specification is needed. In 3D, you can define three joint angles, in

many different ways (Euler/Cardan sequences, choice of axes), so there

is no longer a unique "moment arm". Not even a unique set of three

moment arms. You just have coefficients that, when multiplied by the

muscle force, produce the generalized forces ("moments") for each of the

kinematic variables that you have chosen. These coefficients can be

found from An's partial derivative method. This is in fact how the SIMM

and Opensim software systems calculate moment arms for muscles. These

moment arms will depend not only on the anatomy and skeleton posture,

but also on the choice of rotational variables! This is one important

reason why it is important to standardize the kinematic description of

joint motion (see ISB standardization documents).

Also, in the general case, kinematic variables can be translational as

well as rotational. For translational variables, you also have a

generalized force Q which is the muscle force F multiplied by a

coefficient. This coefficient is now dimensionless and certainly the

term "moment arm" is not appropriate there. Rather, it is the direction

cosine between muscle force vector and translation vector.

If you have a musculoskeletal model with N muscles and M kinematic

degrees of freedom, you have a matrix of M x N coefficients (most of

which will be zero). This matrix is used to transform muscle forces

into generalized forces in the dynamic equation of motion, and usually

termed the "moment arm matrix". For didactic reasons (e.g. Zajac's

papers, or Erdemir et al. Clin Biomech 2007), the equations of motion

are often presented as if there are only rotational degrees of freedom

("joint angles"). This certainly simplifies the terminology but it is

not completely general. For the general case, with rotational as well

as translational degrees of freedom, I am not sure that the correct term

for the coefficient matrix would be. It would be nice if we had one!

Ton van den Bogert

--

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

Department of Biomedical Engineering

Cleveland Clinic Foundation

http://www.lerner.ccf.org/bme/bogert/

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