Dear colleagues,

About 10 days ago, I posted a request for algorithms commonly used in the functional method of determining the location of the hip joint center. Here is the original request:

> We are currently reviewing different methods of locating the hip joint
> center. It seems to me that most of the indirect methods
> basically fall into
> two categories: (1) methods based on the pelvic markers, in some cases in
> conjunction with anthropometric measurements, and (2) numerical
> methods s
> as the functional method originally reported by Cappozzo (1984).
> The methods that fall into group 1 include
> Tylkowsky et al. (1982)
> Andriacchi et al. (1980)
> Bell et al. (1990): hybrid of Tylkowsky and Andriacchi
> Davis et al. (1991)
> Studies based on the numerical method (rotational method) include
> Piazza et al. (2001)
> Leardini et al. (1999)
> Bell et al. (1990)
> Cappozzo (1984)
> I am particularly interested in the functio
nal method which is
> based on the
> least square approximation of
> L = sum[ ( (xi-xc)^2 + (yi-yc)^2 + (zi-zc)^2 )^0.5 - R ] ==> min
> where xi, yi, zi = the coordinates of a marker on the thigh (or
> centroid of
> several markers) at frame i described in the pelvis reference
> frame, xc, yc,
> zc = hip joint center described in the pelvis reference frame, and R =
> radius of the sphere formed by this marker about the hip joint center.
> Can anyone suggest a numerical alogori
thm that will perform the
> aforementioned least square approximation effectively. I could
> not find any
> specifics on this in the published papers. I am in the process of
> writing a
> program to do this and I'd like to compare different algorithms including
> mine. Any pointers or suggestions will be appreciated. As usual,
> I will post the summary on Biomch-L.

I am grateful to those who responded to my request: Sian Jenkins, Tomislav Pribanic, Mark Thompson, Xavier Savatier, Arnel Aguinald
o, Nels Madsen, Kjartan Halvorsen, Alberto Leardini, and James Shippen. Individual responses are compiled at the end of this summary.

Here are the studies recommended that deal with the algorithms:

* Gander, W., Golub, G. H., & Strebel, R. (1994). Fitting of circles and ellipses: least squares solution. TechReport Departement Informatik, ETH Zurich.

* Halvorsen, K., Lesser, M., & Lundberg, A. (1999). A new method for estimating the axis of rotation and th
e center of rotation. J Biomechanics 32, 1221-1227.

* Hiniduma Udugama Gamage, S.S., & Lasenby, J. (2002). New least squares solution for
estimating the average centre of rotation and the axis of rotation. Journal of Biomechanics, 35, 87-93.

Traditional algorithms recommended include Newton's method (Newton-Raphson), and Nelder-Mead downhill simplex method.

Some colleagues sent me MATLAB codes (Mark and Alberto), manuscript under review (Kjartan), and even a detailed rundown of the equations (Nel
s). Kjartan also offered his doctoral dissertation. I am grateful to you all.

I compiled details of the methods mentioned above and put them on a webpage. Follow the following links: > Theories > II. Theories and Practices in Motion Analysis > Joint Kinematics > Joint Center: Functional Method. Any feedback regarding this page would be appreciated.

I am in the process of writing a program for the functional method algorithms. I will report the comparison results soon.

With gratitu

Young-Hoo Kwon
- Young-Hoo Kwon, Ph.D.
- Biomechanics Lab, Texas Woman's University

Responses to my request:
Professor Kwon,

In addition to the literature you've already mentioned, I would suggest that you
the paper 'New least squares solutions for estimating the average ce
ntre of
rotation and the axis of rotation' by Sahan S. Hiniduma Udugama Gamage and Joan
Lasenby Journal of Biomechanics, 35 (1) : 87-93. 2002;
and also
Halvorsen, K., Lesser, M. and Lundberg, A., 1999. A new method for estimating
the axis of rotation and the center of rotation. Journal of Biomechanics 32 :

There are a number of groups currently working on functional techniques, and so
I expect that you will get an interesting set of responses.
I look forward to your summary,

Sian J
Sian E. M. Jenkins
Orthopaedic Engineering
University of Oxford

Oxford Gait Laboratory
Nuffield Orthopaedic Centre NHS Trust
tel/ (fax): +44 1865 227946 / (742348)

Dear Sir,

have you been lately, long time no hear from you?
I do not know much about finding joint centers and what you need it for (I
can only guess that your aim is inverse dynamic solutio
n values?).
Furthermore, I suspect that expression you are minimizing is of non-linear
nature. In that case I can recommend you Levenberg-Marquardt method. I have
used it and I found it very effective for camera calibration. Also, many
others think of it as standard for the non-linear lest square (i.e. data
modeling) problems. Here below is a site where you can find out more about
it and some other numerical methods too (along with codes in C for many of
the things). If I can help anything more do no
t hesitated to contact me.

Regards, Tomislav.

Tomislav Pribanic, M.Sc., EE
Department for Electronic Systems and Information Processing
Faculty of Electrical Engineering and Computing
3 Unska, 10000 Zagreb, Croatia
tel. ..385 1 612 98 67, fax. ..385 1 612 96 52
E-mail :

Dear Young-Hoo

I am also interested in the problem of locating the hip joint centre, but

have come across the problem of fitting spheres to point clouds in the
context of characterising the shape of the human acetabulum.

In the paper Thompson et al 2000 I used an algorithm from a technical
report from ETH Zurich, Gander et al 1994. I merely added a third dimension
to their circle fitting routine.

In fact the algorithm they use turns the minimisation problem to a root
finding problem using the Newton Raphson method. There is then the problem
of giving the algorithm a good initi
al estimate. I just use the mean of the
point cloud and the magnitude of its position vector, but Gander et al do
propose a more sophisticated method in the technical report.

The code is below.

I would also like to draw your attention to the paper by Gamage et al
(2002) which uses a slightly different cost function to the one you have
quoted. This allows for small relative motions between markers on the same
body segment. Also they present a much more general numerical solution to
the prob

I hope that this helps!

best wishes



S. S. H. U. Gamage and J. Lasenby. New Least Squares Solution for
Estimating the Average Centre of Rotation and the Axis of Rotation. Journal
of Biomechanics 35:87-93, 2002.

W. Gander, G. H. Golub, and R. Strebel. Fitting of Circles and Ellipses:
Least Squares Solution. TechReport Departement Informatik, ETH Zurich.

M. S. Thompson, T. Dawson, J-H. Kuiper, M. D. Northmore-B
all, and K. E.
Tanner. Acetabular Morphology and Resurfacing Design. Journal of
Biomechanics 33 (12):1645-1653, 2000.


I'm currently working on the same problem. I found a paper which describes a
least squares algorithm based on the fact that every markers on the thigh
lie on co-centric spheres. See :

S.S.H.U. Gamage et al., J. of Biomech, 35 (2002) 87-93

I tried to implement the algorithm of this article with Mathcad but
unsuccessfully at thi
s time. If you find some equivalent solutions, I will
very pleased if you could share them with me !

I hope this will help you.
Best regards,

Research Engineer
Mont St Aignan - FRANCE

Best congratulations for your web site !

Hi Young-Hoo:

It's great to hear from you again.

We're actually finishing up a study on the functional method for hip joint
center location using MRI-estimated hjc locations as validation.

t of the four articles you mentioned in group 2, only Piazza et al. (2001)
described in relatively greater detail the algorithm (mixed quadratic/cubic)
used to minimize the error function. What's unique about their study was
that the initial values for the minimization process were randomly located
within a cube centered on the true hjc location (albeit a ball-n-socket
model not a real hip). Additionally, they increased the max # of iterations
and function evaluations to 1000 and 10000, respectively,
but I believe
these max values simply prevent the minimization from crashing out before an
adequate fit is achieved.

Shea et al. (1997) presented a study on current software (Orthotrak) that
specifies hjc locations as % of pelvic width (ie, Bell's model) using the
functional method as their validation. The algorithm they used to minimize
the error function was a downhill simplex method but was not detailed in the
abstract, but I imagine this was an optimization tool coded in Matlab.

The algorit
hm we are employing is based on the Nelder-Meed simplex method,
which is available in Matlab's function toolbox (use fminsearch function for
more than one variable). What I am testing out is a process in which the
minimization results from the trajectory a thigh fixed marker (or centroid)
are used as initial values for the error minimization based on the
trajectory of a virtual knee center. The idea is similar to Piazza's study
in that the initial guesses will "drive" the minimization to a relatively

better fit (at least that's the idea). The biggest difference with this case
is that we are evaluating hip joints in-vivo. The optimal motion being
analyzed is the standard combination of hip flex/ext, abd/add, and
circumduction. The estimated hip joint locations are then compared to MRI
based coordinates for validation.

You can find a very detailed description of the Nelder-Meed algorithm in
Matlab's function (funfun) toolbox. I can send it to you if you don't have

I would be interested in
testing out your algorithm should it become
available. Like others in our field, we wish to estimate the hjc location
using only the hip motion of walking rather than that of a fully 3D range of
motion. This would be advantageous for patients with very limited hip ROM.
The challenge, of course, is estimating this point from what ultimately is a
predominantly planar trajectory during walking.

Hope this helps!


Arnel Aguinaldo
Biomechanical Engineer
Motion Analysis Laboratory
ren's Hospital San Diego
San Diego, California USA

Attached is something I worked up in response to your question (Word file).
Let me know if you have any difficulty accessing what I have prepared.
Let me know if you have any questions or concerns about what I have prepared.
As I didn't have much time to spend on it, I apologize in advance for any
spelling, grammatical, or typographical errors.
Let me know if you have any interest in pursuing the proposed id

Enjoy, Nels.

Nels Madsen
Associate Dean for Assessment and Special Programs
Samuel Ginn College of Engineering
Ramsay Hall
Auburn University, AL 36849
334 844 3329
334 844 3307 FAX

Dear Dr. Kwon,

I would like to express my appreciation for your biomechanics web pages.
They cover a nice collection of important methods in biomechanics.

To your question.
Look up the recent paper
Gamage & Lasenby "New least squares sol
utions for estimating the average
center of rotation and the axis of rotation". J Biomechanics 35: 87-93,

Gamage & Lasenby provides a closed form solution to the least squares
criterion you set up in your message.
An improvement to the method of Gamage & Lasenby has recently been
submitted to J Biomechanics:
Halvorsen "A bias compensated least squares estimate of the center of
I may provide you with a preprint upon request. For a more in depth
treatment of functional method
s for determing the joint center you may
want to have a look at my PhD thesis
"Model-based methods in Motion Capture"
which I will defend the coming friday, October 4. I will be glad to send
you a copy.

Yours sincerely,

Kjartan Halvorsen
PhD student

The Department of Systems and Control
Uppsala University
+ 46 18 471 3150

Dear Young-Hoo, my original work is based on a very simple procedure I
enclose (in Matlab). It is
based on the sphere equation. You don't need
numerical algorithms in a first approximation.

Do not forget the following in those methods based on pelvic 'geometry'
(rather than
Seidel, G.K., Marchinda, D.M., Dijkers, M., Soutas-Little, R.W., 1995. Hip
joint center location from palpable bony landmarks - a cadaver study.
Journal of Biomechanics 28, 995-998.

Bell et al. (1990) compared in fact his own hybrid (from Tylkowsky and
Andriacchi), based on regression equations on the pelvic
'geometry', with
the functional of Cappozzo: I would not include this Bell et al. (1990) in
the 'functional' list.

By the way, you may find a useful review in
Wu G, Siegler S, Allard P, Kirtley C, Leardini A, Rosenbaum D, Whittle M,
D'Lima DD, Cristofolini L, Witte H, Schmid O, Stokes I.
ISB recommendation on definitions of joint coordinate system of various
joints for the reporting of human joint motion-part I: ankle, hip, and
spine. J Biomech. 2002 Apr;35(4):543-8


************************************************** ********
Alberto Leardini, DPhil
Movement Analysis Laboratory
Centro di Ricerca Codivilla-Putti
Istituti Ortopedici Rizzoli
Via di Barbiano 1/10, 40136 Bologna ITALY
tel: +39 051 6366522
fax: +39 051 6366561

"Where is the Life we have lost in living,
Where is the wisdom we have lost in knowledge,
Where is the knowledge we have lost in information."
Thomas Stearns Eliot, Choruses from ''The Rock'
' (1934)
************************************************** ************************


If you are (or know) a Matlab user I recommend the function FMINSEARCH which uses the Nelder-Mead simplex (direct search) method. This will reaily eat your problem.


James Shippen

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