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asiriani75
04-10-2009, 05:49 AM
Dear All,

We're presenting here our conclusion from the replies for: How to choose Euler angles sequences of rotation.

Thank you all for your emails, and thanks to Kevin McQuade at the University of Washington for his help with the translation.

Professor Anamaria Siriani de Oliveira
University of São Paulo – Brazil - Physical Therapy Division
Research Fellow, Department of Rehabilitation Medicine,
University of Washington. Seattle, USA

Summary of replies - How to choose Euler angles sequence of rotation?
In 12/03/2009 03:02
As as pointed out by Ton below, discussion of this issue goes back 15 yrs on Biomech-L !! It is unlikely to be definitively answered here as well. As I tell my students the real answer continues to be ... “it depends”.

As an introductory reference, Wikipedia text about Euler Angles was suggested more than once as a good material: http://en.wikipedia.org/wiki/Euler_angles

>From Adam Bartsch came this reading suggestion "Principles of Dynamics", 2nd Edition 1988, by Greenwood, Prentice Hall Publishers. pages 354-357.

And Bhushan Borotikar suggested the excellent webpage:
http://www.euclideanspace.com/maths/geometry/rotations/euler/index.htm
In general there was somewhat of a consensus that the choice of the Euler Angles Sequence should be guided by the specific requirements of individual experiments and the motions of interest. . While there are certainly alternatives to Euler/Cardan angles [ie Helical Axes, two-step rotations (Cheng J.Biomech (33),2000 ), etc.] this query was limited to a request for what people are using for Euler sequences.

Clark R. Andersen presented a practical example, of how to organize chooses. He suggested that selecting an Euler angle sequence in declining order of degree of motion should work well. For example, if you are measuring flexion of the wrist, and considering that the next largest motion would be radial-ulnar deviation, the sequence would be flexion axis, radial-ulnar axis, pronation-supination axis. Even when working with mixture of motions, as in a study where both radial-ulnar and flexion-extension motions are performed in relative isolation, again you should choose a fixed sequence in declining order of degree of motion. Additionally, Clark advises us to align our coordinate frame as well as possible, so that the coordinate axes align with the major rotation/anatomical axes.

This notion of picking primary motions of interest then choosing a sequence to both minimize gimbal lock, and have the primary motion of interest the first rotation was ecoed by several respondents, and nicely summarized in the presentation of Jim Richards – (the link provided in Ton van den Bogerts comment)
For the lower limb these decisions are relatively straight forward because the motions of the hip knee and ankle are more constrained that in the Scapula and Humerus thus Gimbal lock/singularity problems can be easily avoided in the lower limb where Grood & Suntay methods predominate but where other Cardan sequenced can also easily be applied.
We also have recommendations organized by the Standardization and Terminology Committee (STC) of the International Society of Biomechanics that covers this topic. These recommendations define for each joint; a standard for the local axis system in each articulating bone is generated.

Wu et al. ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion--part I: ankle, hip, and spine. International Society of Biomechanics. Journal of Biomechanics (2002) vol. 35 (4) pp. 543-8

Wu et al. ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion--Part II: shoulder, elbow, wrist and hand. Journal of Biomechanics. (2005) vol. 38 (5) pp. 981-992

>From this point of view, it is important keep on mind that following the STC-ISB suggestion for Euler angles sequence (if YZX, XYZ, ZYX, or whatever) should be preceded by the attention to the STC-ISB standardization for defining the local axis system and naming axes. The STC recommendations have the purpose of stimulating feedback and discussion, and facilitating further revisions. The recommendations are not so dogmatic so as to imply that they are definitive or the best for all conditions of 3-D motion analysis.

As Ton van den Bogert, wrote, using upper limb as example: “The shoulder standard that many people seem to be using now is one where the first axis is the inferior-superior axis of the thorax, the third axis is the long axis of the humerus, and the middle axis is perpendicular to those two. This has a singularity when the arm points straight down, but no singularity when abducted”. As long as one recognizes this caveat and defines and experiment where initial conditions are deterministic by other means , then this method is acceptable – If however if is crucial to have accurate horizontal plane information with the arm in a vertical position either 0 or 180 degrees ( north or south pole), then the choice o, f for example, a cardan sequence with singularity at 90 might be deemed the best solution.

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Some specific responses are attached below:
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De: Thomas Maack
Dear Dr. Siriani,

What joint are you working on?
I did research on the knee joint. Grood & Suntay describes the angular decomposition I used to analyze my data. They decompose rotations into, first a rotation about the x-axis (one that goes medial-laterally and is fixed in the femur), followed by a rotation about the y-axis (an axis that goes anterior-posterior and is mutually perpendicular to the x and z axes), finally followed by a rotation about the z-axis (one that goes proximal-distal and is fixed in the tibia).
I decomposed linear displacements according to a coordinate frame fixed in the femur.

Cheers,
Thomas Maack
M.S. Ohio State University
1. ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion
2. Grood & Suntay, 1983, A joint coordinate system for the clinical description of three-dimensional motions: application to the knee.
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De: Alberto Leardini

Dear Anamaria, you might find of help some of the thoughts in the following Review.
Gait and Posture 21 (2005) 186–196
Human movement analysis using stereophotogrammetry Part 1: theoretical background
Aurelio Cappozzo, Ugo Della Croce, Alberto Leardini, Lorenzo Chiari

Kind regards,
************************************************** ****************Alberto Leardini, DPhil
Movement Analysis Laboratory
Director Prof. Sandro Giannini
Centro di Ricerca Codivilla-Putti
Istituto Ortopedico Rizzoli
Via di Barbiano 1/10, 40136 Bologna ITALY
tel: +39 051 6366522
fax: +39 051 6366561
email: leardini@ior.it
http://www.ior.it/movlab/

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De: "Baker, Joshua"
Siriani,
Grood and Suntay have a great article discussing Euler angle rotation sequence (EARS) and the knee joint. I have found it to be very beneficial as they discuss their sequence with respect to clinical assessment. You will find the reference below:
Grood ES, Suntay WJ. A joint coordinate system for the clinical description of three-dimensional motions: application to the knee. J Biomech Eng. 1983,105(2):136-144.

In my experience this joint coordinate system (JCS) has been adopted by most for the entire lower extremity. There are a few different JCS out there for the ankle. The upper extremity is a whole different ball game. I believe the ISB released their standards for the shoulder joint (I have it somewhere, if you need it let me know and I will go digging). Good luck with everything,
Josh

Joshua Baker MPT
Department of Kinesiology
Office: HHS 2505 E
Phone: 530-4690
Fax: 530-2477

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De: "Crawford, Neil - SJHMC" Neil.Crawford@CHW.EDU
Dr. Siriani,
This method was written for the spine but is applicable to other joints - you just have to identify which plane is the plane of symmetry (e.g., sagittal plane in the spine) and which two rotations should be complementary (e.g., in the spine, lateral bending and anteroposterior bending should be complementary and different in nature to twisting).
Neil Crawford
Barrow Neurological Institute
Phoenix, Arizona

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Dear Dr. de Oliveira
I humbly suggest that you borrow a copy of "Principles of Dynamics", 2nd Edition 1988, by Greenwood, Prentice Hall Publishers.
Page 354-357 gives you the information you desire.

With kind regards,

Cleveland Clinic Spine Research Laboratory
Lutheran Hospital, 2-C
1730 West 25th Street
Cleveland, OH 44113
216.363.5749

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De: Cédric Schwartz
Hello,
If you are working on human kinematic, you may be interested by the recommandations of the International Society of Biomechanics :

G.Wu. ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion—part I: ankle, hip, and spine. Journal of Biomechanics, Volume 35, Issue 4, Page 543

and:
G.Wu, F.van der Helm, H.(DirkJan) Veeger, M.Makhsous, P.Van Roy, C.Anglin, J.Nagels, A.Karduna, K.McQuade, X.Wang. ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion—Part II: shoulder, elbow, wrist and hand. Journal of Biomechanics, Volume 38, Issue 5, Pages 981-992
Sincerly,

Cédric Schwartz, PhD student
Laboratoire de Traitement de l'Information Medicale (LaTIM)
INSERM U650 - Université de Bretagne Occidentale (UBO) - TELECOM Bretagne
IFR 148 ScInBioS - Science et Ingénierie en Biologie-Santé
CHU Morvan, Bâtiment 2Bis (I3S), 5 avenue Foch, 29609 Brest, France

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From: Mathew Yarossi
Hi,
I have attached a paper discussing rotation sequence in shoulder kinematics. Generally you want to choose your angle with the largest range of motion or the one of greatest interest to be the first rotation has the greatest influence over subsequent rotations. Best Regards. Mat.

Mathew Yarossi, B.S.
Study Engineer/Research Coordinator
Human Performance and Movement Analysis Laboratory (HPMAL)
Kessler Foundation Research Center
1199 Pleasant Valley Way
West Orange, NJ 07052
Tel: 973-243-6816
Fax: 973-243-6984
E-mail: myarossi@kesslerfoundation.net

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From: Florian Kugler
Dear Prof. Sirani de Oliveira,
Are you aware of the standardization attempt of the ISB in this matter?
Wu et al. ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion--part I: ankle, hip, and spine. International Society of Biomechanics. Journal of Biomechanics (2002) vol. 35 (4) pp. 543-8

Wu et al. ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion--Part II: shoulder, elbow, wrist and hand. Journal of Biomechanics. (2005) vol. 38 (5) pp. 981-992

Good luck and best regards,

Florian Kugler.
Humboldt-University Berlin
Department of Training and Movement Sciences
Philippstr. 13, Haus 11
10115 Berlin
Tel.: +49 (0)30 2093-46007
Fax: +49 (0)30 2093-46008
Mobile: +49 (0)151 14977567
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From: Søren Lewia
Hello Mr. Anamaria Siriani de Oliveira,
During my work of developing a kinetic three-dimensional full body model for a VICON motion capturing system I sticked to these papers. Here the ISB tries to establish standard coordinate systems to overcome the problems in interpretation of reported data you described. I hope that it helps. By the way, if you find an easy way to clearly explain the shoulder angles, please keep me informed ;-)

Kind regards,
Sören Lewis

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From: Jeffrey Bingham
Dear Siriani,

The particular sequence you work with will depend entirely on what information you want to explore.
If you are interested in joint coordinate systems you might consider using the method of Grood and Suntay (Grood and Suntay, 1983; Craig, 1986).
Keep in mind that euler angles are very sensitive to coordinate system definition. For example, in the knee the transepicondylar axis and cylindrical flexion axis will give significantly different results for measured flexion during gait for the exact same motion.
There are most certainly conventions for defining coordinate systems and euler angle sequences for most anatomical movement. I would encourage you to investigate these, but also consider defining a coordinate system and sequence that is intuitive to yourself as well.

Best,

Jeff Bingham
Neuroengineering Lab
Georgia Institute of Technology

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From: Jeremy Suggs
Anamaria,

A classic reference, at least for knees, is the following:
J Biomech Eng. 1983 May;105(2):136-44.
A joint coordinate system for the clinical description of three-dimensional motions: application to the knee.
Grood ES, Suntay WJ.

Hope this helps,
Jeremy

Jeremy F. Suggs, Sc.D.
Associate, Biomechanics Practice
Exponent, Inc.
3401 Market St., Suite 300
P: 215.594.8875
F: 215.594.8899

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From: "Andersen, Clark R."

It has been my experience that selecting an Euler sequence in declining order of degree of motion works best. For example, if you are measuring flexion of the wrist, and considering that the next largest motion would be radial-ulnar, the sequence would be Flexion, Radial-ulnar, Pronation-supination. When working with a mixture of motions, for example in a study where both radial-ulnar and flexion-extension motions are performed in relative isolation, again go with a fixed sequence in declining order of motion. Additionally, it helps a great deal to align your coordinate frame as well as possible so that the coordinate axes align with the major rotation axes. Sorry I don't have a reference for these comments, they are strictly empirical.

Clark Andersen
Division of Biomechanics and Bone Physiology Research
Department of Orthopaedic Surgery and Rehabilitation
The University of Texas Medical Branch
301 University Blvd.
Galveston, TX 77555-0174
Phone: (409) 747-3221
Fax: (409) 747-3240

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From: " Jaap Harlaar "
Hi there
basic reference is Grood & Suntay (1983 ?) :
fl/ext first, the AB/Adduction (floating axis) , finally endo/exo rotation.
Mathematically this is the cardanic decomposition
There s no general "best" really, in specific cases (near-) singularity should be avoided as much as possible, then picked the one that is most meaningful to the users
In some cases a different approach makes more sense, i.e Euler with the first en 3rd axis identical. (eg the shoulder , see Doorenbosch; Harlaar & Veeger, which follows up on Pearl )

good luck, Jaap Harlaar

Jaap Harlaar PhD
HumanMovement Laboratory - Dept. Rehabilitation Medicine
Research Institute MOVE - VU University Medical center
post: p.o.box 7057, 1007 MB Amsterdam
email: j.harlaar@vumc.nl tel.: +31 20 44 40 773 visit: VUmc, -1 Y 056
De Boelelaan 1117 1081 HV Amsterdam The Netherlands
www.vumc.nl/revalidatie / www.move.vu.nl / www.fbw.vu.nl / www.esmac.org
www.i-s-p-o.nl / www.smalll.nl / www.freemotion.tk / www.bodymech.nl

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From: "Scholz, M. (Melanie)"
Dear Prof. Siriani,

Attached is a document by the ISB, setting a standard for reporting kinematics, including angles.
If you record your kinematics using Northern Digital Inc.'s Optotrak and First Principles, you can use rigid bodies (defined according to those conventions) to track the 3D movement of the segments, and you get your Euler angles immediately, even in real time (http://www.ndigital.com/lifesciences/certus-motioncapturesystem.php).
Hope this helps,

Melanie

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From: "Cannella,Marco"
your question sounds to me in this way: what language will save me money on a phone call?

the reson is reported in wikipedia:
http://en.wikipedia.org/wiki/Rotation_matrix : The problem of singular alignment, the mathematical analog of physical gimbal lock , occurs when the middle rotation aligns the axes of the first and last rotations. It afflicts every axis order at either even or odd multiples of 90°, causing Euler angles to be abandoned for quaternions in many applications. Setting these unavoidable issues aside, angles for any order can be found using a concise common routine (Herter & Lott 1993 ; Shoemake 1994 ).

I find interesting looking at quaternion: http://en.wikipedia.org/wiki/Euler%27s_rotation_theorem

hope you may find my considerations helpfull

marco

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From: Bhushan Borotikar
It is very intriguing topic indeed. The selection of the sequence depends upon the joint or the model you are using. For example, for knee joint, you could use XYZ sequence or ZYX sequence depending on how you name your axes. For XYZ sequence, X becomes your flexion axis and for ZYX sequence, Z becomes your flexion axis. Basic details about Euler angles can be found at

http://www.euclideanspace.com/maths/geometry/rotations/euler/index.htm

Bhushan Borotikar

Department of Biomedical Engineering,
Cleveland Clinic, Cleveland, Ohio, USA

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From: "Macwilliams, Bruce"
In my opinion, when such things are known, such as during human motion, the largest motion (e.g. flexion/extension) should be the first angle, the next largest (e.g. internal/external rotation) should be the third. This is because the second angle occurs about a non-physical calculated axis.

Bruce MacWilliams, Ph.D.
Movement Analysis Lab - Shriners Hospital for Children
Fairfax Rd. @ Virginia St. Salt Lake City, Utah 84103-4399
Phone: 801-536-3800 Fax: 801-536-3782

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From: "van den Bogert, Ton"
This is a question with relevance to many subscribers, so I am posting an answer to the list.
The answer depends on which joint or body you are interested in. Euler/Cardanic angles have singularities at certain postures, and the trick is to pick a sequence that never reaches the singularity.
For the lower extremity, the Cardanic joint coordinate system (Grood & Suntay 1983) is well established. Standards can be found on www.isbweb.org, click on "information services" and "standards". Those sequences always use abduction as the middle axis, and those joints never reach the singularity at 90 degrees.
For the shoulder joint it is very difficult to avoid singularities, because of its large range of motion. A very nice presentation on this can be found here:
http://www.udel.edu/HNES/faculty/richards/BluePresentation_files/frame.htm
The shoulder standard that many people seem to be using now is one where the first axis is the inferior-superior axis of the thorax, the third axis is the long axis of the humerus, and the middle axis is perpendicular to those two. This has a singularity when the arm points straight down, but no singularity when abducted.
Helical angles, attitude vectors, Euler parameters, quaternions, Rodrigues parameters have all been used as alternatives that do not have the numerical singularity of Euler/Cardanic angles. But the problem with those is that there is no good connection to clinical terminology for joint movement. The debate on this goes back to the early days of Biomch-L, a compilation can be found here: http://www.biomch-l.org/files/ANGLES3D_TOPIC.txt.
Another useful reference is Woltring HJ, J Biomech 1994.

Ton van den Bogert
A.J. (Ton) van den Bogert, PhD
Department of Biomedical Engineering
Cleveland Clinic Foundation
http://www.lerner.ccf.org/bme/bogert/

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From: Joseph McIntyre
Prof. De Oliverira,
In my Euler angle analysis of human motion I choose the largest physiologic rotation as the first Euler angle. Then follow the Euler angle progression trying to fit the second Euler angle to second largest physiologic rotation. As an example if you are studying the motion of the arm at the shoulder the largest rotation is about an axis extending from the spine out through the joint so your first Euler angle would be about this axis. As Euler angle descriptions of motions are not unique, you should select a rotation set that best fits your data and research goals and it will be as accurate as any other set.

Sincerely,
Joseph McIntyre
Ph.D. Student
Auburn Univ. Mechanical Engineering Dept.
Motion Capture Laboratory
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From: Jiajie Guo
Hi Dr. Siriani,
Commonly used Euler angle sequence is ZYZ, but it is noted that some people may use another sequence.
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From: Paulo Roberto Garcia Lucareli
Cara Anamaria,
( you need to transcribe this to english
Normalmente usamos a seguinte sequência eixo (YXZ) que normalmente seque flexão-extensão, adução abdução e rotação.
Já trabalho com isso a algum tempo e só quando na marcha os valores de rotação pelvica são muito altos (>40º) usamos (ZXY) para corrigir a obliquidade.
No entanto todas as sequencias estão certas, mas a que eu te disse é aquela que mais se aproxima do visual e do que nós fisios estamos acostumados.
Qualquer dúvida me avise.
Abraços

Paulo Roberto Garcia Lucareli
Fisioterapeuta, Ms
Laboratório de Estudos do Movimento Einstein - LEME
Hospital Israelita Albert Einstein - São Paulo - Brasil
NAPAN - Nucleo de Apoio a Pesquisa em Análise do Movimento
São Paulo - Brasil

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To add to Ton's nice reply, I would recommend both Wikipedia and Eric
Weisstein's Mathworld as good quick references on this topic, which in
turn provide more detailed references at the end of the article.

http://en.wikipedia.org/wiki/Euler_angles
http://mathworld.wolfram.com/EulerAngles.html

Best,

Postdoctoral Fellow
Harvard School of Engineering and Applied Sciences
Phone : 607.339.6653

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From: Nasser Rezzoug

G.Wu, 2002, ISB recommendation on definitions of joint coordinate system of
various joints for the reporting of human joint motion—part I: ankle, hip,
and spine, Journal of Biomechanics, Volume 35, Issue 4, Page 543

G.Wu, F.van der Helm, H.(DirkJan) Veeger, M.Makhsous, P.Van Roy, C.Anglin,
J.Nagels, A.Karduna, K.McQuade, X.Wang, 2005, ISB recommendation on
definitions of joint coordinate systems of various joints for the reporting
of human joint motion—Part II: shoulder, elbow, wrist and hand
Journal of Biomechanics, Volume 38, Issue 5, Pages 981-992

Regards

Nasser Rezzoug, assistant professor
HandiBio EA 4322
Université du Sud Toulon Var
rezzoug@univ-tln.fr