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Steve Piazza
07-18-2007, 06:59 AM
Ton asked what axes we would find if we performed kinematic fitting using the two marker sets in our animation -- i.e., would we get the different U-joints that were used to generate the motions? We think the answer is yes. While we didn't run these simulated data through the algorithm, we did collect data using an anthropomorphic linkage in different configurations representing very different TC and ST axis orientations (these experiments are described in our 2006 paper) and we were always able to locate both axes to within a few degrees. Such tests would presumably be subject to stereophotogrammetric errors but there was very little play in these revolute joints and obviously no skin movement.

When we tested cadaver specimens with bone-mounted clusters, however, we got poor results. As with the mechanical linkage, we had stereophotogrammetric errors with no skin movement errors. The main difference was that the biological joints deviated more from ideal revolute behavior. This could be observed in the helical axes we computed from talocalcaneal and tibiotalar motions, and this deviation was especially evident in the latter case. To truly understand why 2-axis fitting is not robust (if it isn't), we need to better determine whether marker locations are affected by deviations from revolute behavior in ways that are correlated with joint angle or whether additonal degrees of freedom have a substantial influence on the shape of the cost function and the location of its minimum.

Below we have also appended (with permission) a recent post on this topic from Mike Schwartz.

Steve Piazza
Greg Lewis
Penn State University

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On 7/17/07, van den Bogert, Ton wrote:
Steve made a very nice animation to illustrate of the ill-posedness of
the problem
(http://www.biomechanics.psu.edu/research/spiazza/universal_joints.avi).
This shows that you only need to perturb the marker data by a few mm to
obtain a drastically different estimate of the axes of rotation.

However, you need perfectly correlated perturbations (between markers,
between markers and joint angle) to get that result. Measuring errors
are not like that. Don't understimate the power of combining a large
number of noisy measurements into an accurate estimate of a single model
parameter. This is well known in statistics, of course. If you average
N values, each having a random error E, the error in the average is only
E/sqrt(N). Now, in foot motion data the errors are not all random, some
of it is correlated to joint motion (as we discussed before in relation
to skin movement error). But there is probably sufficient lack of
correlation (between markers, and between markers and joint angles) that
the error can be reduced significantly by the parameter estimation
procedure.

Steve, did you try estimating the joint axes from the simulated marker
motions in that animation? Was the answer correct, and was it sensitive
to random or systematic errors in marker data? My results suggested
that one parameter (the medial deviation of the subtalar joint axis) was
poorly estimated, but the others were not. But I did not do extensive
sensitivity analyses.

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On 7/13/07, Michael Schwartz wrote:Steve has shown quite clearly (albeit using animation rather than
mathematics) that the optimization (a.k.a. kinematic fitting) problem is
ill-posed for the TC-ST axis.

Recognizing this, we (Krista Evans and I) tried a few "gimmicks", such as
starting with a careful, anatomically-based initial guess. That is, we used
markers (physical and virtual) placed on palpated landmarks to estimate the
position of the TC and ST axes. We then used this orientations as initial
guesses to the solution. Furthermore, we restricted the search space to
cones around these axes (i.e. only allow the engine to search within a
certain angular range of the initial guess).

Even with all these aides...the optimizer still often found "bizarre"
solutions.

Of course, in retrospect, this isn't too surprising. After all, why did we
think it was called a "universal" joint?

I know that Steve and Greg have done some work trying to isolate the
individual axes (TC and ST) by holding the foot/ankle in specific positions,
then using *functional* methods to find the still-free joint. While I
understand the idea of this (i.e. "locking" the ST axis by inverting the
foot, or "locking" the TC axis by dorsiflexing the foot), I really wonder
how reliable/accurate it is. How stationary does the "locked" axis actually
remain?

I guess that I have been (slowly) coming to the conclusion that the
ill-posedness of the kinematic fitting problem is such a big issue that, at
least until we have a better understanding of the details, I have started
shying away from the idea. For what it's worth...when I first read Lu and
O'Connor I was convinced that the approach was going to be a huge step
forward for the field. I just don't think (based on my own work and that of
others) that we are much closer than we were 5 years ago.

I am guessing that Richard Baker must be at the beach with his family (or
else down in the salt mines trying to earn some extra money to pay for the
palatial expansion of his home). I am quite keen to hear his take on all
this, as I know he has been doing a lot of work in the area lately.

-Mike-
_______________________
Michael H. Schwartz, Ph.D.

Director of Bioengineering Research
Gillette Children's Specialty Healthcare

Associate Professor - Orthopaedic Surgery
University of Minnesota

Graduate Faculty - Biomedical Engineering
University of Minnesota