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.

************************************************** *********

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.

************************************************** *********

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

------------------------------------------------------------------------

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

------------------------------------------------------------------------

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

================================================== ===========

------------------------------------------------------------------------

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

------------------------------------------------------------------------

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

------------------------------------------------------------------------

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

***********SUMMARY OF RESPONSES (EDITED)********************

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.

************************************************** *********

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.

************************************************** *********

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

------------------------------------------------------------------------

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

------------------------------------------------------------------------

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

================================================== ===========

------------------------------------------------------------------------

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

------------------------------------------------------------------------

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

------------------------------------------------------------------------

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

***********SUMMARY OF RESPONSES (EDITED)********************