## Re: Kinematic Fitting vs. Functional Method

Mike Schwartz and I went a little deeper into the problems of kinematic
fitting. He wrote (with permission to repost to Biomch-L):

> What is the justification for assuming
>
> 1. minimum cost is a *sufficient* condition for the solution
> 2. the errors are symmetrically (randomly) distributed about
> the solution
>
> A bit more on #1. To me, it's clear that the minimum cost is
> a *necessary* solution, but as anyone who has tried this
> method has found, there are multiple solutions. These are
> perhaps "false" minima, but possibly "real"
> minima. It's not clear that insisting on marker tracking has
> a unique solution of the actual joint parameters. I have
> tried (and failed) to prove this to be the case. Have you

I can think of poorly defined optimizations where joint parameters are
undefined. For instance, you can't find axes of a gimbal (3-DOF) joint
by optimization. Any set of three non-coinciding axes will be able to
exactly reproduce your observed marker trajectories. But when the
problem is well defined, I think the minimum is unique (but perhaps hard
to find).

> A bit more on #2. I think this one is a bit more problematic.
> This condition is clearly necessary for the least-squares
> approaches to work. The cost can accumulate from several
> sources: i) mis-location/mis-orientation of the joint
> parameters, ii) soft tissue artifact, and iii) marker
> reconstruction error. I think we can safely say that the
> errors due to marker reconstruction are random (and small).
> However, the nature of the error term due to mis-location and
> soft tissue are clearly not symmetrically distributed about
> the actual solution for the joint parameters. Like #1, I've
> spent some time trying to get a grip on this (i.e. whether
> it's really a problem, how big a problem, etc...). However, I
> became so discouraged with the optimization approach that I
> have taken a sabbatical from it.

Absolutely, and I agree that non-random errors can be "problematic".
Skin movement error is typically correlated to joint motion. I did a
sensitivity analysis in the 1994 paper, by simulating a skin movement
error in the lateral malleolus marker. I perturbed the Z of that marker
proportionally to the subtalar angle (1 cm per 35 deg rotation). This
caused 6 degrees error in the subtalar joint axis orientation even
though the fit was still excellent (0.26 mm!) with absolutely no hint
that there was a 1 cm soft tissue error. This demonstrates that a good
fit does not always imply a good model. In the Discussion I said: "The
influence of systematic errors, such as soft tissue movement, is more
problematic [than random errors], because the curvature of the cost
function is very small in certain directions. Therefore, the optimum is
easily shifted by small artifacts caused by systematic errors which
distort the cost function."

Finding two axes in the ankle joint complex is probably much harder
(i.e. flatter cost function) than finding a hip center or a knee flexion
axis. But even in those applications it is probably a good idea to use
simulations to estimate the sensitivity to skin movement error.

--

Ton van den Bogert

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