Dear Biomch-L readers,

The posting on joint centers of rotation by Ian Stokes is interesting;

a (literal) shift of perspective from the center of rotation can be useful.

I tend to disagree however, with his final conclusion that "...neither

biomechanical theory, nor practical considerations support it (the use

of joint centers as reference points)". If you do not make assumptions

about the line of action of the joint force (JF), the JF must be described

by 3 variables (2D) instead of 2: two components of force, and one for

the line of action. This introduces one extra unknown variable into the moment

equilibrium equation, and typically there are too many unknowns already.

So you must make an assumption about the line of action of the JF. But there

is only one thing a priori known about this: the JF goes through the

instantaneous center of rotation (ICR).

This can be proved using the principle of virtual work. A joint is defined

as a 'kinematic' connection, i.e. the force associated with this connection

generates or absorbs no power at any time. Picture one body (bone) as stationary

while the other is moving. All points on the line of action of the joint force

must have velocities perpendicular to this force (power is the dot product

of force and velocity). In a moving rigid body, every line on which all

velocities have the same direction *must* pass through the ICR. Please take

a few seconds to verify this statement...

So, the joint force also passes through the ICR. Incidentally, this also

proves that the ICR of the knee joint during the swing phase coincides with

the intersection of the cruciate ligaments. The joint force is in that case

the resultant of ligament forces only.

Note that this only applies to true kinematic connections, without frictional

losses or energy storage in elastic cartilage or joint ligaments. Neglecting

these small amounts of energy is probably allowed. Also note that in this

definition, 'joint force' is taken to mean the total 'constraint reaction

force' in mechanical terms, sometimes called 'net joint force'. If you

only want the contact force, without ligaments, the ligament forces become

additional unknowns in the equilibrium equations and that is not what you

want.

A joint, defined as a kinematic connection between two bodies, is more than

just the bone-to-bone contact surfaces. It also includes the structures

that guide the movement without exchanging energy with the system. I.e.

ligaments that can be considered inextensible for the purpose of dynamic

analysis.

Remains the problem that the ICR has (in general) no fixed position on either

bone, and that the ICR is not easily determined during actual movements.

That is exactly why the ICR is taken as the reference point in the moment

equation. That way you do not have to know it! Of course, this implies that

all other moments must alse be calculated about the ICR. For muscular

forces this is no problem: the moment arm with respect to the ICR is the

partial derivative of origin-insertion length with respect to the joint

angle (also to be proved by the principle of virtual work). Using this

definition, moment arms of muscles are easily determined from

cadaver measurements or a rigid-body model incorporating the line of action.

Only for calculation of external (ground reaction force) moments is an

estimated location of the ICR required. Hopefully, moment arms of ground

reaction forces are large enough (or the moments small enough) to be

insensitive to errors in the ICR.

So, my opinion is that moments should be calculated about the ICR. I would

like to hear Ian's reply, or other opinions.

-- Ton van den Bogert

University of Utrecht, Netherlands.

The posting on joint centers of rotation by Ian Stokes is interesting;

a (literal) shift of perspective from the center of rotation can be useful.

I tend to disagree however, with his final conclusion that "...neither

biomechanical theory, nor practical considerations support it (the use

of joint centers as reference points)". If you do not make assumptions

about the line of action of the joint force (JF), the JF must be described

by 3 variables (2D) instead of 2: two components of force, and one for

the line of action. This introduces one extra unknown variable into the moment

equilibrium equation, and typically there are too many unknowns already.

So you must make an assumption about the line of action of the JF. But there

is only one thing a priori known about this: the JF goes through the

instantaneous center of rotation (ICR).

This can be proved using the principle of virtual work. A joint is defined

as a 'kinematic' connection, i.e. the force associated with this connection

generates or absorbs no power at any time. Picture one body (bone) as stationary

while the other is moving. All points on the line of action of the joint force

must have velocities perpendicular to this force (power is the dot product

of force and velocity). In a moving rigid body, every line on which all

velocities have the same direction *must* pass through the ICR. Please take

a few seconds to verify this statement...

So, the joint force also passes through the ICR. Incidentally, this also

proves that the ICR of the knee joint during the swing phase coincides with

the intersection of the cruciate ligaments. The joint force is in that case

the resultant of ligament forces only.

Note that this only applies to true kinematic connections, without frictional

losses or energy storage in elastic cartilage or joint ligaments. Neglecting

these small amounts of energy is probably allowed. Also note that in this

definition, 'joint force' is taken to mean the total 'constraint reaction

force' in mechanical terms, sometimes called 'net joint force'. If you

only want the contact force, without ligaments, the ligament forces become

additional unknowns in the equilibrium equations and that is not what you

want.

A joint, defined as a kinematic connection between two bodies, is more than

just the bone-to-bone contact surfaces. It also includes the structures

that guide the movement without exchanging energy with the system. I.e.

ligaments that can be considered inextensible for the purpose of dynamic

analysis.

Remains the problem that the ICR has (in general) no fixed position on either

bone, and that the ICR is not easily determined during actual movements.

That is exactly why the ICR is taken as the reference point in the moment

equation. That way you do not have to know it! Of course, this implies that

all other moments must alse be calculated about the ICR. For muscular

forces this is no problem: the moment arm with respect to the ICR is the

partial derivative of origin-insertion length with respect to the joint

angle (also to be proved by the principle of virtual work). Using this

definition, moment arms of muscles are easily determined from

cadaver measurements or a rigid-body model incorporating the line of action.

Only for calculation of external (ground reaction force) moments is an

estimated location of the ICR required. Hopefully, moment arms of ground

reaction forces are large enough (or the moments small enough) to be

insensitive to errors in the ICR.

So, my opinion is that moments should be calculated about the ICR. I would

like to hear Ian's reply, or other opinions.

-- Ton van den Bogert

University of Utrecht, Netherlands.