View Full Version : Info...Calculating Euler/Cardan angles from spatial coordinates

Marc Odorico
10-24-1999, 09:37 AM
Hi everyone!

The other day I posted a question regarding the calculation of
Euler/Cardan angles from spatial coordinates and received an overwelming

Following is a list of most of the responses that I hope may help others
seeking similar information.

Thanks to everyone who contributed!

Kind Regards,


Posted Message
>I'm in the process of trying to calculate the Euler/Cardan angles (or
>rotation matrix) for a system of ball-socket joints given the 3D
>spatial coordinates (in any reference frame) of the links attached to the
>joints and the order of rotation.


To:Marc Odorico

Hi Marc Odorico,

take a look at:

Riener, R., Straube, A (1997) Increased sensitivity in motion analysis
applying inverse dynamics: arm tracking movements in cerebellar
Journal of Neuroscience Methods, 72, p. 87-96.

Good luck,
Robert Riener


From: Gideon Ariel
To: Marc Odorico

Hi Marc:
When you get the answer, could you please send me the information
also. So far the best infomration on Euler/Cardan angles described in
Keith Vaughan "GaitCD" published by Keith. I have the CD and it is

This is one of the best reference:
Grood, E.S., & Suntay, W.J. (1983). A joint coordinate system for the
clinical description of three-dimensional motions: Application to the
knee. Journal of Biomechanical
Engineering, 105,136-144.

Gideon Ariel


From: Manolo Salmeron
To: Marc Odorico

Some useful references:

Spoor, Veldpaus;'Rigid body motion calculated from spatial
co-ordinates markers', J. Biomechanics, vol. 13 pp. 391-393,1980

Veldpaus, Woltring, Dortmans; 'A least-square algorithm for the
equiform transformation from spatial marker coordinates'. J.
Biomechanics, vol.
21, pp. 45-54, 1998


Manolo Salmeron


From: Michael Damsgaard
To: Marc Odorico

Hi Marc,

I would have a look on litterature in multibody systems dynamics, e.g.:

P. Nikravesh: "Computer-Aided Analusis of Mechanical Systems",
Prentise-Hall, 1988 or books by E.J.Haug, A.A.Shabana etc. They may be
difficult to buy.

Otherwise you may send me a more precise problem-statement, incl. a
drawing showing the coordinates you use and I 'll try to help you.

Good luck,


* Michael Damsgaard
* Institute of Mechanical Engineering, Aalborg University
* Pontoppidanstraede 101, DK-9220 Aalborg East, DENMARK
* phone: +45 96358080, +45 96359310 (direct)
fax: +45 98151675, email: md@ime.auc.dk


From: Christophe Degueurce
To: Marc Odorico

Dear colleague,

We had exactly the same difficulty as we tried to perfom such analysis
on the digital joints of the horse. The precise way to calculate such
are not always given in the publications. The most interesting
were provided by :

Fioretti S., Cappozzo A., Lucchetti L., Joint kinematics, in:
Three-dimensional analysis of human locomotion, Wilay & Sons Ed,
Chichester, 1997, 173-189. (very well explained!!!!)

Grood E.S., Suntay W.J., Comment on « justification of triaxial
goniometer for the measurement of joint rotation », J. Biomech. 14 (9)

Grood E.S., Suntay W.J., A joint coordinate system for the clinical
description of three-dimensional motions: application to the knee, J.
Biomech. Eng. 105 (1983) 136-144.

Selvik G., Roentgen stereogrammetry: a method for the study of the
kinematics of the skeleton system, Acta Orthop. Scand. suppl. 232
(1989) 1-51.

Woltring H.J., Representation and calculation of 3-D joint movement,
Hum.Mov. Sci. 10 (1991) 603-616.

Wu G., Cavanagh P.R., ISB recommendations for standardization in the
reporting of kinematic data, J. Biomech. 28 (1995) 1257-1261.

Hopping it will help you!

Dr Christophe Degueurce, DVM, PhD

Maître de conférences en Anatomie
UMR Biomécanique du Cheval

Conservateur du Musée Fragonard
Ecole nationale Vétérinaire d'Alfort
7, av du Général de Gaulle
94704 Maisons-Alfort cedex
Tel: 00 33 1 43 96 70 52
Fax: 00 33 1 43 96 31 62
Email: degueurc@vet-alfort.fr


From: Tomislav Pribanic
To: Marc Odorico

Dear Marc,

try site http://www.cs.bsu.edu/~ykwon/kwon3d/. I remember seeing
also some papers on the similar issue. I will let you know if it comes
back to me.

Best regards, Tomislav.

Tomislav Pribanic
Department for Electronic Systems and Information Processing
Faculty of Electrical Engineering and Computing
3 Unska, 10000 Zagreb, Croatia
E-mail : tomislav.pribanic@zesoi.fer.hr


From: Antonio Perillo-Marcone
To: Marc Odorico

I used to work with that some years ago and you are right, there is
plenty of literature out there. Perhaps you should try robotics books,
ussually have very detailed aproaches in this area (ie, Euler angles,
rotation matrices, etc). Since I worked on this such a long time ago I
cannot recall the names of the books I used, but I promise I will check
and get back to you within this week. There is an author's name I
though, why don't you try and have a look at this book:

Personal Author: Fu, K.S.
Title: Robotics : control, sensing, vision and intelligence /
K.S. Fu, R.C. Gonzalez, C.S.G. Lee
Imprint: New York : McGraw-Hill, c1987
Series Title: (CAD/CAM, robotics and computer vision)
Subject (LCSH): Robots
Subject index: Robots - Mechanical engineering
Added author: Gonzalez, Raphael C.
Added author: Lee, C.S.G.

I am not sure if that is one of the books I used, I'll let you know




From: Kjartan Halvorsen
To: Marc Odorico

Hello Marc,

The best (in the sense that the squares of the residuals are smallest
(lest squares, then)) way to compute the rotation matrix from 3d
coordinates is by the algorithm proposed by Söderkvist and Wedin:

Söderkvist I, Wedin P (1993) ``Determining the movements of the
skeleton using well-configured markers''. J. Biomechanics, 26

Now, the rotation matrix is definately not invariant to the choice of
frame of reference, so you have to be clear what your choice is (local
frame fixed to the links, or global frame).

The algorithm by Söderkvist and Wedin assumes that you have measured
the position of a set of markers before and after the rotation (or rigid
body transformation). The rotation matrix will depend on the frame of
reference that the coordinates of the markers are expressed in. The
paper discuss this.

I understood from your posting that computing the euler/cardan angles
once you have the rotation matrix is not a problem. If I misunderstood,
and you need a reference for this as well, try:

Angeles J. (1988) Rational Kinematics, Springer Verlag, Berlin.

Yours sincerely,
Kjartan Halvorsen

The Department of Systems and Control
Uppsala University
+ 46 18 471 7846


From: Boris Prilutsky
To: Marc Odorico

Dear Marc:

You can find an excellent overview of the topic and a great number of
examples from human biomechanics in the following book.

V. M. Zatsiorsky (1998). Kinematics of Human Motion. Human Kinetics.

Good luck!


From: Ettore Pennestri
To: Marc Odorico

I suggest you to use Euler parameters instead of
Euler angles. It is well known that Euler angles,
in some situations, do not uniquely determine
the finite motion. An excellent test on Euler parameters is the
one authored by E.J. Haug. Computer-Aided
Kinematics and Dynamics of Mechanical Systems,
Allyn and Bacon. You will find there the answers to your question.


Ettore Pennestri'


From: Ian Fisher
To: Marc Odorico

Hi Marc,

I remember those days! The best source I found for practical advice

Vaughan, Davis & O'Connor `Dynamics of Human Gait' (Human Kinetics,

They give clearly set out examples of finding Euler angles from gait
marker data. The book was out of print but there were plans to republish
it on CD -
I'm not sure if that has happened yet, you might try Human Kinetics
The theory of Euler angles is covered by Goldstein:

Goldstein,H `Classic Mechanics' (Addison Wesley)

of which most University Libraries have hundreds of copies.



From: Angela Tate
To: Marc.Odorico@ENG.MONASH.EDU.AU


E.S. Gorrd and WJ Suntay Transactions of hte ASME vol 015 may 1983
pg 136-144


kinzel hall and hill berry Biomechanics 1972 vol5 pp93-105
or Tupling a& pierrynowski in Med and Biol. Eng Comput 1987, 25, 527-532

have a great day

From: "a.l.hof"
Organization: faculty of medical sciences (RuG)
To: Marc.Odorico@ENG.MONASH.EDU.AU

I had the same problem. I found these in:

Woltring H One hundred years of photogrammetry in biolocomotion. In
Cappozzo et al Biolocomotion: A century of research using moving
pictures. ISB series Vol 1 Roma, Promograph 1992.

A rather unlikely source, but quite useful. Woltring did not give
very extensive references and no derivations, however.
For the initiated it seems all common knowledge, but I would very
much like to see a good introductory textbook on this subject.
Best wishes,

At Hof
Department of Medical Physiology &
Laboratory of Human Movement Analysis AZG
University of Groningen
Bloemsingel 10
Tel: (31) 50 3632645
Fax: (31) 50 3632751
e-mail: a.l.hof@med.rug.nl


From: fregly@aero.ufl.edu
To: Marc.Odorico@ENG.MONASH.EDU.AU


Here is an excellent reference for calculating the rotation matrix and
translation vector form the 3D marker coordinates of each body:

Soderkvist, Inge
Wedin, Per-Ake

Determining the movements of the skeleton using well-configured markers
Journal of Biomechanics v 26 n 12 Dec 1993. p 1473-1477
Publ. year:

Once you have the rotation matrix, you can calucuate Euler or Cardan
for any desired sequence of rotations using information in the

Woltring, Herman J., Human Movement Science, 1990? (I unfortunately do
have the precise reference for this, but it is an excellent overview
on the application of kinematics to human movement).

Kane, T.R., Likins, P.W., and Levinson, D.A. (1983) Spacecraft
McGraw-Hill Book Company, New York. (currently out of print

Hope this information is helpful.

B.J. Fregly
Assistant Professor
Department of Aerospace Engineering,
Mechanics, and Engineering Science
University of Florida
Tel: (352) 392-8157
Fax: (352) 392-7303
E-mail: fregly@aero.ufl.edu
Home page: www.aero.ufl.edu/~fregly


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