View Full Version : Moments of forces about C of R - Is it a good idea?

Ian Stokes
05-28-1991, 02:47 PM
To: Fabio Catani, Herman Woltring, Ed Chao and Biomch-l readers.

During an email correspondence with our encyclopaedic moderator (HJW),
he encouraged me to review the Biomch-l archive for November 1990 (prior to my
membership in the list) in order to read the discussion initiated by Fabio
Catani, and joined by Herman Woltring and Ed Chao concerning the use of
center/axis of rotation as a reference point for forces and moments at joints.
The following borrows heavily from a poster I presented at the Combined
Canadian and American Societies of Biomechanics meeting, Montreal, Quebec, in
August 1986. The abstracts were not published that year, other than in the
Conference Proceedings. The poster was motivated by my desire to clarify my
own mind about the assumptions we make about joint moment equilibrium and
lines of actions of joint forces.

Fabio Catani's original question was about how to find the center of
rotation of the knee and hip joints in order to calculate the joint moments.
He got excellent advise about how to do this, but I question why he should use
the center of rotation, rather than a more easily located reference. There is
moment equilibrium about any and all points, so why worry about where to place
the reference point? Usually because we are interested in going from
measurements of external moments and inertial forces to estimation of internal
forces in anatomic structures - joints and muscles. Estimation of joint
forces is usually solved by writing six equilibrium equations (for force, and
torques in a 3 axis system). This method can be applied both to the complete
kinematic analysis, and to the quasi-static case when it is assumed that
acceleration terms are negligible. Usually there are more than six unknowns
in the six equations, so assumptions have to be made so that possible
solutions can be obtained. These assumptions can be divided into two groups:
(a) assumptions about the line of action of the joint force and (b)
assumptions about the distribution of muscle and ligament forces around the
joint. The first assumption enables equations of moment equilibrium to be
simplified. Since the unknown joint force has no moment about a point on its
own line of action it can be eliminated from the equation of moments about
such a point. If the direction of the joint force is also known, then a
constraint equation can be written. People want to measure moments about the
center of rotation, presumably, because they are looking for a point or axis
about which the joint force has no moment.

For my poster I reviewed 1483 papers published in the Journal of
Biomechanics between Jan 1969 and Dec 1986. 56 were concerned with estimation
or evaluation of articular forces and/or moments of muscles about joints.
Among these 56 papers, the assumption about the line of action of the joint
force was classified into one of 4 groups:

(1) Contact Point (9 papers):
Nine used observations of the area of contact between joint surfaces to
define a point on the line of action of the joint force.

(2) Axis of Rotation (17 papers):
Fourteen assumed that the joint force passed through a point on the axis
of rotation (or center of rotation for planar motion) and used this as a point
about which to consider moment equilibrium. Two papers used this point and
the further constraint of having the joint force pass through the joint
contact area. One of these papers considered errors introduced by uncertainty
about these points. One considered that both the center of curvature and the
center of rotation are co-linear with the contact point, and therefore the
joint force should pass through both of these points.

(3) Center of Curvature (16 papers):
Eleven papers assumed that the joint force passed through the center of
curvature of an articular surface. Five papers made this assumption and also
constrained the line of action of the joint force to pass through the contact

(4) Miscellaneous (14 papers):
One aligned the tibiofemoral forces in the knee with points half way
across the condyles and perpendicular to the articular surfaces. Two papers
used the center of curvature of the hip and the center of contact of the knee.
One analyzed joint forces in the fingers and assumed that they acted along the
long axes of phalanges, transecting the joint contact. One used previously
published data for moment arms of the triceps surae. In three cases no
assumption was stated. Two papers showed how the principle of virtual work
could be used to find moment arms of muscles by measuring tendon length/joint
angle relationships. One determined experimentally where there was a region
of zero moment due to external forces and muscle forces, and related it to the
anatomic features of the knee joint. One study of the shoulder investigated
whether the joint center of rotation lay on the line of the joint force. Two
papers addressed the question of what differences in the calculated joint
forces would be produced by adopting either the joint center of curvature or
the experimentally determined axis of rotation of the temporomandibular joint
(TMJ) as a point about which to consider moment equilibrium. In the TMJ joint
(which is relatively unconstrained by ligaments or by its articular surfaces)
the axis of rotation could become quite remote from the joint itself.

Thus, many different assumptions have been used both singly and in
combination. The three major ones are that the joint force passes through the
center of contact, the center of curvature, or the center of rotation.
Clearly, an inter-articular force must pass through the point (or region) of
contact between surfaces. In the absence of friction, it must also be aligned
with the mutual perpendicular between the surfaces, so it would pass through
the centers of curvature of the surfaces. But is it valid to assume that the
joint force has zero moment about the center of rotation? On the face of it
the center of rotation is a kinematic property of the joint. Why should it
have any special significance for force and moment equilibrium?

What are the constraints on the center of rotation? The relative motion
at the cartilage surface can be considered to have three components: rolling,
gliding, and spin. At any instant, there is a plane passing through the
articular contact in which rolling and gliding take place. The center of
rotation is the point where the axis of rotation cuts this plane. Spin is an
rotation between the two components of the joint, about an axis which is the
mutual perpendicular through the joint contact. It can be shown that the
center of rotation for rolling and gliding lies on the perpendicular to the
common tangent plane. Its position is at the articular contact for pure
rolling, but increasingly distant from it as the amount of gliding is

Since the joint contact, the center of curvature and the center of
rotation are co-linear under normal conditions, it is therefore correct to
assume that the joint force has no torque about the center of rotation. This
should be true under both static and dynamic conditions, providing the center
of rotation is controlled by contact at joint surfaces with inelastic
cartilage and there is zero friction.

The axis or center of rotation is, however, notoriously difficult to
measure experimentally. Joint laxity can make its position quite variable, so
it is not certain that the center of rotation found in one experiment can be
applied directly to estimating forces under different loading conditions. The
same also applies to the contact area or center of curvature, but there are
bounds on their positions, and they are not subject to large measurement error
sensitivity. Instantaneous center of rotation is impossible to measure in
practice, since the errors tend to infinity as the increment of motion tends
to zero.

All joint force calculations (especially those in which inertial forces
are significant) would be simpler if the reference point were constant.
Indeed, the three papers recommended by Ed Chao use a fixed 'joint center'
based on anatomic reference points. For most real joints, especially those
with minimal constraints on motion, neither the center of curvature, center of
rotation nor the contact point can be expected to be fixed to either part of
the joint. Therefore, as biomechanicians we must always be critical and
imaginative in our analyses. There is no simple rule which can be applied in
all cases. However, a rigid adherence to using the center of rotation of a
joint is unwise. Neither biomechanical theory, nor practical considerations
support it.

Ian Stokes
Burlington, VT, USA.