Dear fellow biomechanicians,

Some of you will (I hope) answer Herman Woltring's call, and give

comments on practical use of net forces/moments/powers in functional

movement analysis. I want to reply to some technical statements made

by Herman. Most differences of opinion seem to originate from different

areas of application; my interest is in estimation of 'internal forces',

which need not be part of 'functional movement analysis'.

(1) Are joint powers physical quantities? In case of only monoarticular

muscles, yes. In that case one joint power equals the sum of several

muscle powers. When there are polyarticular muscles, the only equality

is: sum of all joint powers = sum of all muscle powers. A joint power

is then just another mathematical transformation of measured variables,

without physical meaning. It may have practical value though, but

others should comment on that.

An example. If you say that ankle power is very large just before take-

off in a vertical jump, you are restating what was observed: a large

distance between GRF and joint center, simultaneous with a large

extension velocity. The information was there already. But again, the

information might be more useful (for pattern recognition?) in the

transformed form; you get one variable instead of the original three:

GRF vector and point of application, and joint angular velocity.

Mechanical interpretation of joint power is dangerous: if you assume

that all ankle power is generated by ankle muscles, you are wrong

because the gastrocnemius muscle (even when it does not contract) can

transmit power from the knee extensors to the ankle joint.

Herman is very right that interpretation of physical joint forces is

also not straightforward. Very little is known on how cartilage and

bone react to specific loading patterns. But, you can not use

differentiation of force to get the 'jerk' if you do not estimate the

force in the first place.

(2) I agree with Herman, that the ICR (or 3D IHA) should ideally be part

of a biomechanical analysis. But using the ICR as moment reference

point requires high-quality kinematics and processing. It would be a

pity if a cheap and simple analysis would be considered below standard.

Especially if there is no good reason to prefer this difficult

transformation of measured variables over another.

(3) I apologize for using confusing terminology. It is indeed logical

to reserve the term 'net joint force' for the force obtained from 'net

kinetic analysis', i.e. models with one force and one moment transmitted

by each joint. In other types of analysis, which include estimation of

muscle forces, the use of just 'joint force' seems more appropriate.

When a joint is a complex (powerless) kinematic connection, this 'joint

force' can be the resultant vector of several forces, e.g. contact and

ligament forces. In that case, one might be tempted to add 'net'. To

avoid confusion, the term 'constraint force' or 'constraint reaction

force' from theoretical mechanics could be an alternative.

(4) My discussion on muscle moment arms was, admittedly, not properly

generalized to 3D. I had only joints in mind with one degree of freedom

(DOF). In that case, the IHA (instantaneous helical axis) depends only

on the joint angle, and muscles cannot change it. Within this limi-

tation the moment arm is still d(length)/d(angle). When for example,

the knee joint is part of a model, one must decide on the number of DOF.

If it is simplified to one DOF, the above theory applies. If the laxity

is an essential part of the analysis, more DOF are required. The

removal of kinematic constraints means that the corresponding constraint

forces are also lost (see below). This produces incorrect muscle

forces, because only the muscles are assumed to be responsible for the

observed movement, unless the actual physical constraints (the joint

ligaments) are added to the model. This shows that it is best to

reduce the degrees of freedom as much as possible in a dynamic analysis.

In a truly general 3D-theory, the concept of generalized coordinates

is convenient. If a joint has N degrees of freedom, you need N

variables (generalized coordinates) to specify the position of body 2

relative to body 1. There are many ways to define such variables, as

was shown by the '3D joint angles' debate some time ago on Biomch-L, but

the theory always applies. Each of the N generalized coordinates is

associated with a generalized force (the 'moments', if the coordinates

are angles). The relationship between physical forces and generalized

forces is linear, and the coefficients (the 'moment arms') can be found

using the principle of virtual work:

SUM(F.dr) = Q.dq (for all dq).

Where dq is a small change in the N-vector q of generalized coordinates, dr

is the resulting (3D-vector) change in position of the point of

application of each force vector F. Q is the N-vector of generalized

forces. Actually, the principle of virtual work defines Q. From this,

we find for each component Qi of Q:

Qi = SUM(F.(dr/dqi)) (the d's mean partial derivative here)

For a muscle, the direction of the vector F is exactly opposite to the

direction of lengthening, so F.(dr/dqi) = -|F|dL/dqi. For a ground

reaction force, the full vector equation must be used.

The conditions for static equilibrium are now simply: Q=0. Dynamic

equations of motion can also be formulated in generalized coordinates:

Q = M(q).q"

I am not familiar with the method to find the inertia matrix M, which

may depend in a complex way on q. My dynamics software (DADS) does not

use generalized coordinates but 'cartesian' coordinates, which are more

suitable for general-purpose software.

Finally, there are 6-N (in 3D) constraint force variables (the 'joint

force'). Examples: A ball-and-socket joint (the hip) has N=3, and the

constraint force is a 3D force vector. A universal joint (in Dutch:

"kruiskoppeling"), as used in machines, has N=2 because it does not

allow internal/external rotation. The 4 constraint reaction

forces are one 3D force vector, plus one torque. Note that joints can

also have translational degrees of freedom ('slider' joints), where the

corresponding qi is best measured in meters, not in radians. The

generalized theory still applies.

Enough of theoretical mechanics now, let's get back to more practical

matters!

-- Ton van den Bogert

University of Utrecht, Netherlands.

Some of you will (I hope) answer Herman Woltring's call, and give

comments on practical use of net forces/moments/powers in functional

movement analysis. I want to reply to some technical statements made

by Herman. Most differences of opinion seem to originate from different

areas of application; my interest is in estimation of 'internal forces',

which need not be part of 'functional movement analysis'.

(1) Are joint powers physical quantities? In case of only monoarticular

muscles, yes. In that case one joint power equals the sum of several

muscle powers. When there are polyarticular muscles, the only equality

is: sum of all joint powers = sum of all muscle powers. A joint power

is then just another mathematical transformation of measured variables,

without physical meaning. It may have practical value though, but

others should comment on that.

An example. If you say that ankle power is very large just before take-

off in a vertical jump, you are restating what was observed: a large

distance between GRF and joint center, simultaneous with a large

extension velocity. The information was there already. But again, the

information might be more useful (for pattern recognition?) in the

transformed form; you get one variable instead of the original three:

GRF vector and point of application, and joint angular velocity.

Mechanical interpretation of joint power is dangerous: if you assume

that all ankle power is generated by ankle muscles, you are wrong

because the gastrocnemius muscle (even when it does not contract) can

transmit power from the knee extensors to the ankle joint.

Herman is very right that interpretation of physical joint forces is

also not straightforward. Very little is known on how cartilage and

bone react to specific loading patterns. But, you can not use

differentiation of force to get the 'jerk' if you do not estimate the

force in the first place.

(2) I agree with Herman, that the ICR (or 3D IHA) should ideally be part

of a biomechanical analysis. But using the ICR as moment reference

point requires high-quality kinematics and processing. It would be a

pity if a cheap and simple analysis would be considered below standard.

Especially if there is no good reason to prefer this difficult

transformation of measured variables over another.

(3) I apologize for using confusing terminology. It is indeed logical

to reserve the term 'net joint force' for the force obtained from 'net

kinetic analysis', i.e. models with one force and one moment transmitted

by each joint. In other types of analysis, which include estimation of

muscle forces, the use of just 'joint force' seems more appropriate.

When a joint is a complex (powerless) kinematic connection, this 'joint

force' can be the resultant vector of several forces, e.g. contact and

ligament forces. In that case, one might be tempted to add 'net'. To

avoid confusion, the term 'constraint force' or 'constraint reaction

force' from theoretical mechanics could be an alternative.

(4) My discussion on muscle moment arms was, admittedly, not properly

generalized to 3D. I had only joints in mind with one degree of freedom

(DOF). In that case, the IHA (instantaneous helical axis) depends only

on the joint angle, and muscles cannot change it. Within this limi-

tation the moment arm is still d(length)/d(angle). When for example,

the knee joint is part of a model, one must decide on the number of DOF.

If it is simplified to one DOF, the above theory applies. If the laxity

is an essential part of the analysis, more DOF are required. The

removal of kinematic constraints means that the corresponding constraint

forces are also lost (see below). This produces incorrect muscle

forces, because only the muscles are assumed to be responsible for the

observed movement, unless the actual physical constraints (the joint

ligaments) are added to the model. This shows that it is best to

reduce the degrees of freedom as much as possible in a dynamic analysis.

In a truly general 3D-theory, the concept of generalized coordinates

is convenient. If a joint has N degrees of freedom, you need N

variables (generalized coordinates) to specify the position of body 2

relative to body 1. There are many ways to define such variables, as

was shown by the '3D joint angles' debate some time ago on Biomch-L, but

the theory always applies. Each of the N generalized coordinates is

associated with a generalized force (the 'moments', if the coordinates

are angles). The relationship between physical forces and generalized

forces is linear, and the coefficients (the 'moment arms') can be found

using the principle of virtual work:

SUM(F.dr) = Q.dq (for all dq).

Where dq is a small change in the N-vector q of generalized coordinates, dr

is the resulting (3D-vector) change in position of the point of

application of each force vector F. Q is the N-vector of generalized

forces. Actually, the principle of virtual work defines Q. From this,

we find for each component Qi of Q:

Qi = SUM(F.(dr/dqi)) (the d's mean partial derivative here)

For a muscle, the direction of the vector F is exactly opposite to the

direction of lengthening, so F.(dr/dqi) = -|F|dL/dqi. For a ground

reaction force, the full vector equation must be used.

The conditions for static equilibrium are now simply: Q=0. Dynamic

equations of motion can also be formulated in generalized coordinates:

Q = M(q).q"

I am not familiar with the method to find the inertia matrix M, which

may depend in a complex way on q. My dynamics software (DADS) does not

use generalized coordinates but 'cartesian' coordinates, which are more

suitable for general-purpose software.

Finally, there are 6-N (in 3D) constraint force variables (the 'joint

force'). Examples: A ball-and-socket joint (the hip) has N=3, and the

constraint force is a 3D force vector. A universal joint (in Dutch:

"kruiskoppeling"), as used in machines, has N=2 because it does not

allow internal/external rotation. The 4 constraint reaction

forces are one 3D force vector, plus one torque. Note that joints can

also have translational degrees of freedom ('slider' joints), where the

corresponding qi is best measured in meters, not in radians. The

generalized theory still applies.

Enough of theoretical mechanics now, let's get back to more practical

matters!

-- Ton van den Bogert

University of Utrecht, Netherlands.