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Liu Wen
05-14-1995, 11:12 AM
Dear Biomechanists:
Thanks to all of you that replied to my question on the optimiztion of
3D motion measurement. Enclosed you will find my original request, followed
by a summary of responses.

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Dear Biomechanists:
I have some questions about the determination of rotation matrix R and
the translation vector V from noisy landmarker measurements. Suppose that the
measurement errors are independent and normally distributed with constant
standard deviation. Will I obtain higher accuracy of R and V if I use more
than 3 markers and use the least square algorithm (Veldpaus et al, 1988) than
that if I use only three markers and without using the least square algorithm?
If answer is yes, is there any one proved it, theoretically or numericaliy?
As always, I will post a summary of replies.

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With 4 or more markers, at the very least you will be able to calculate an
RMS value by back transforming your points through the calculated
least squares tranformation and determining the distances from the
original points. Use this method with more and more points to decide if the
RMS is getting lower. Note that while a low RMS is necessary for an accurate
transformation calculation, it is not sufficient. I think you will
probably find that the more points you use, the better.

Neil
--
N. Glossop, Ph.D.,
Toronto, Canada
neil@isgtec.com

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W. Liu asked:
> Will I obtain higher accuracy of R and V if I
> use more than 3 markers and use the least square algorithm
>(Veldpaus et al, 1988) than that if I use only three markers and without
>using the least square algorithm?

Will you obtain higher accuracy if you use more than two experimental points
to fit a straight line by least-squares?

Obviously, yes to both questions ...

You will even obtain better results for only three markers if you use
the Veldpaus et. al. (1988) algorithm or other least-squares methods
than if you do direct reconstruction of axes (e.g. two points form local z,
third point forms local x normal to local z).

Variance in estimating finite displacement motion is inversely
related to number of markers. Using 4 markers instead of 3 markers will
reduce standard deviation in displacement measurement by 13 percent.

REFERENCE (among many)
A. de Lange, R. Huiskes, and J.M.G. Kauer (1990) Measurement
errors in roentgen-stereophotogrammetric joint-motion analysis. J.
Biomechanics 23(3):259-269.

H.J. Sommer III, Professor of Mechanical Engineering, 327 Reber
Building The Pennsylvania State University, University Park, PA 16802 USA
(814)863-8997, FAX (814)863-4848, Internet HJSME@ENGR.PSU.EDU

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You may want to look at Soederkvist & Wedin: J. Biomech
26(12):1473-1477, which addresses issues relating to the configuration of
markers.


================================================== ===========
John F. Cummings (John.Cummings@UC.EDU)
Noyes-Giannestras Biomechanics Laboratories
University of Cincinnati, ML0048 V: (513) 556-4171
Cincinnati, OH 45221-0048 F: (513) 556-4162
================================================== ===========

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Dear Biomechanists:
You should take more than 3 points, say nbpoints, and then you
have 3 options:
problem: find the transformation matrix between frame A and frame B:
1) for all the combination of 3 points in nbpoints, calculate the matrix
as you usually do, and finally keep the matrix that gives the smallest RMS
error. This RMS error is calculated by applying this matrix to the points
in frame A and calculate the RMS difference between the transformed points
and the points in frame B.
2) You can calculate the transformation matrix for all the
combination of 3 points in nbpoints, transform the points from frame A to
frame B. At the end of the process, you will have nbpoints groups of
points. You can calculate the center of gravity of each group and reject
the points that are too far from that center, these points are probably
too noisy.
3) You can use the least square approach all the nbpoints together

I do not have precise refernces about this, I just know by experience
that it is better to take more points, but that if you take too much points,
you will increase the calculation time

Paule Brodeur
Ph.D. student at Ecole Polytechnique de Montreal:
brodeur@grbb.polymtl.ca
presently in France for a collaboration: paule@le-eva.univ-bpclermont.fr

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You should talk to Dr. Sorin Siegler in the Mechanical Engineering
Department there at Drexel. He has done some experiments with
regard to the effect of increasing the number of markers on optimally
determining 3D marker location.
--
Daniel P. Nicolella Phone: (216) 368 - 6446
Case Western Reserve University FAX: (216) 368 - 6445
Mechanical & Aerospace Engineering INTERNET:
dann@falstaff.mae.cwru.edu

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Liu,

Herman Woltring wrote at least one paper on this subject. It is
Woltring et al.,"Measurement Error Influence on Helical Axis Accuracy in the
Description of 3-D Finite Joint Movement in Biomechanics", Biomechanics 1983,
ASME, New York, NY 1983.

Marcus J.H., The Accuracy of Screw Axis Analysis Using Position Data
from Anatomical Motion Studies, Master's Thesis, Michigan State
University, 1980.

I have also performed studies showing the effects errors in
measurement data have on three dimensional motion studies. In my thesis
there are theoretical as well as numerical studies of error for several
popular algorithms.

Peterson S.W., Measurement and Analysis of Human Joint Motion,
Ph.D. Thesis, University of Minnesota, 1985.

I also have an as yet unpubliched paper which contains several good ideas
for performing motion studies using landmark coordinates. If you would
like a copy, please send me your regular mail address.

Peterson, S.W. and Erdman, A.G., A Survey of Algorithms for Computing Rigid
Body Motions from Landmark Data, submitted to the Journal of Biomechanical
Engineering.

Good luck,

Steve Peterson
fredrick@vuse.vanderbilt.edu

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