View Full Version : Re: Kinematic Fitting vs. Functional Method

Van Den Bogert, Ton
07-10-2007, 04:37 AM
Mike Schwartz wrote:

> Kinematic fitting modifies some/all joint parameters and
> joint angles, often in a nested manner, to minimize a cost
> function (e.g. cost = sum of squared distances between
> measured and predicted marker trajectories). Further
> explanation can be found in Lu T-W and O'Connor JJ, J
> Biomech, 32:129-134, Charleton et al, Gait Posture,
> 20:213-221, or Rienbolt et al, J Biomech, 38:621-626.
> As a personal note -- I have had **no end of grief** trying
> to use optimization approaches. I believe that this is due to

This is my personal experience also. It is easy to get carried away
with this concept because mathematically it is perfectly valid, but in
my experience you need to be very careful not have too many unknown
model parameters.

Let's say you have N unknown joint parameters p1...pN. And you have a
model with M degrees of freedom q1...qM, and you capture Nf frames of
motion data. Then the number of unknowns is N + M*Nf, since the model
parameters do not change during movement. The number of forward
kinematic equations is Nf times the number of markers times 3 (for 3D).
For large enough number of frames, and when using more than M/3 markers,
this will always exceed the number of unknowns. Then we can solve this
with a least squares method and a cost function which is the sum of
squared residuals as defined by Mike above.

The nested optimization (guess p1...pN in the outer loop, and guess
q1...qM in each frame in an inner loop) is just a way to partition the
problem but it still should find the same solution.

If you test this with simulated motion data, it always works. But in
the real world there are errors in model and data which can easily
produce false minima in the cost function. We got this optimization to
work in our two-axis ankle model (Smith et al, J Biomech 1994) but
discovered the following limitations:

(1) Motion data must span sufficient range of motion in all joints.
(2) Optimization must start from many initial guesses to ensure finding
the global optimum.
(3) Orientation of the subtalar joint in the horizontal plane was
sensitive to measuring error.
(4) With data collected during weightbearing, the method failed,
probably because there was too much foot deformation which violated the
ideal two-axis model.

I expect that these problems get worse when trying to do this for
multiple joints simultaneously. It may be possible if you carefully
select a low number of joint parameters to estimate, leaving others
fixed. For instance, the axial rotation axis of the knee would not be
estimated from kinematics, but be anatomically defined along the line
from knee center to ankle center. I would be interested in hearing
Richard Baker's observations and opinions on this.

Some other related comments.

The kinematic fitting method is a "global optimization" approach (using
the term coined by Lu & O'Connor), which assumes certain kinematic
connections between body segments (hinge, ball, or even coupled
rotation/translation as in the SIMM knee model). This contrasts with
the 6-DOF method mentioned by Frank Buczek, which makes no such
assumptions. The 6-DOF method does not suffer from potential modeling
errors (since there are no joint models) but the global optimization
approach has some nice advantages:

(1) Fewer than 3 markers per segment are needed, so you can sometimes
avoid using markers on "wobbly" sites.
(2) Less sensitive to skin motion artifacts.
(3) Degrees of freedom can be made compatible with whole body dynamics
(4) Degrees of freedom can be made compatible with graphics (animation)

Depending on the scientific question, subject population, and the
movement being studied, we must carefully balance the effects of model
error (in global optimization methods) against the effects of data error
(greater in 6-DOF methods). In our lab we use global optimization
methods, but in one of our projects we model each joint as three slider
joints and three hinge joints, i.e. six degrees of freedom. This made
the results compatible with historical data which used 6-DOF analysis.
It also allowed the joints to "absorb" marker wobble at impact which
would otherwise lead to (brief) overestimation of knee valgus and
flexion. This was a problem because the movement was jump landing, and
the subject population included some with the potential for substantial
marker wobble.

Incidentically, this example shows that the software tools for global
optimization can be used for 6-DOF analysis. You can even model some
joints with 6 DOF and other joints with 1 DOF in the same model. The
6-DOF software tools probably do not have this versatility.

The philosophical problem with 6-DOF analysis is that we have large skin
marker errors, causing errors in translational motion in the joints
which are usually larger than the translational motion itself.
Therefore assuming zero translation is a logical approach, leading to
global optimization with many practical advantages. But as the above
example illustrates, modeling the translational motion can sometimes
make the rotational motions more accurate. Even if the translational
motion itself is poorly measured and of no interest.

I agree with Dan Benoit that we must be very careful when presenting
data on non-sagittal knee rotations. Apart from the skin movement
problem, it is well known that these results are sensitive to the
orientation of the joint coordinate system (Ramakrishnan & Kadaba,
1991). This is the main motivation for using coordinate systems with
functional axes, which are hopefully more reproducible than axes based
on anatomy.

I would like to propose that for joints or degrees of freedom that are
very "stiff" with limited range of motion, the joint moment is
potentially much more reliable than the joint motion. The joint moment
can be quite large even when there is almost zero motion in the joint.
Obviously the joint angle results would be totally overwhelmed by errors
in that case.

Generally I think there is no single correct model or marker set. It
always depends on purpose of the study, movements, subjects, etc. It's
best not to be dogmatic about these things. That is easy enough to say
in basic research, but the clinical labs need standardized methods. I
think it was Mike Schwarts who mentioned that he routinely uses standard
and non-standard models and markersets simultaneously. To me this to be
a very sensible way to gain insight into how much the results are
affected by these choices. If not much, we gain some confidence in the
results of our analysis.

This was much more than I intended to write. This is a useful
discussion and it is great to have these contributions in the Biomch-L


Ton van den Bogert
Biomch-L co-moderator


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