View Full Version : uniqueness and continuity of Euler angles

Richard Baker
01-12-2003, 01:33 PM
Dear all,

I've been away on my summer (sic) holidays so have come back to read the
recent correspondence on Cardan angles as a whole. My main concern with the
recent exchanges is that it focuses on just one of the six possible
sequences of Cardan angles (flexion,adduction,internal rotation for the hip
or knee, corresponding terms for ankle, upper limb, or axial skeleton). I
feel the biomechanics community has been a little guilty of assuming,
because this sequence is almost certainly the best for the hip and knee,
that it is therefore the most appropriate for all joints. Both the sequence
and the reference position to which measurements are related should be
carefully chosen for each joint investigated.

As most of you are aware all six possible sequences are mathematically
equivalent. No amount of consideration of the maths alone will allow us to
decide which sequence to choose. However some sequences give better
agreement with the conventional usage of clinical terminology than others.
For example, in order that internal/external rotation occurs about the long
axis of the distal segment (the conventional definition) this must be the
last angle of the Cardan sequence.

The sequence used by Ton (flexion,adduction,int. rot.) almost certainly
gives the best agreement between maths and conventional clinical usage for
the hip and knee. Care is needed at the ankle where it is the long axis of
the foot which must be the third axis. For the pelvis I have made the case
(Gait and Posture 13:1-6, 2001) that the sequence rotation, obliquity, tilt
gives better agreement with clinical usage than tilt, obliquity, rotation
(which is the equivalent of flexion, adduction, int. rot. and which is
almost universally used in gait analysis). I'd now go further and suggest
that this is preferable for all components of the axial skeleton.

I've been thinking about the shoulder for some time and have reached the
conclusion that there is no sequence of Cardan angles which is consistent
with clinical terminology. This is deeply rooted in the clinical
terminology which was never intended to be mathematically consistent. A
fundamental problem with clinical definitions of shoulder angle is that
they choose the anatomical position (arms by the subject's side) as the
reference position. Because of the restriction on the middle Cardan angle
lying in the range (-90 to +90) only one of flexion (sagittal plane) or
abduction (coronal plane) can be greater than 90 whereas conventional
clinical usage assumes that either can be greater than 90. The other
problem here is that if the flexion, adduction, int. rot. sequence is used
then the gimbal lock position occurs at 90 abduction which is an important
position for many shoulder functions.

There are ways around this and choices can be guided by an understanding of
the biomechanics (the most obvious is to choose a different reference
position). A wider debate is what CAN the biomechanics community do about
the shoulder? The bottom line is that conventional clinical usage of terms
cannot be used in a way that is mathematically consistent. I'd like to
think that the biomechanics community could give some guidance to the wider
clinical community in redefining nomenclature for shoulder movements. I
think this would be clinically beneficial in allowing for clearer
communication about shoulder problems as well as resolving our own
technical problems. But do sufficient people within that wider
community listen to what the biomechanics community is saying to make this
a viable proposition?


Richard Baker PhD CEng
Gait Analysis Service Manager and Director of Research
Hugh Williamson Gait Laboratory, Royal Children's Hospital, Victoria 3052,
Tel: +61(0)3 9345 5354, Fax: +61(0)3 9345 5447

Adjunct Associate Professor, La Trobe University
Honorary Senior Fellow, University of Melbourne

To unsubscribe send SIGNOFF BIOMCH-L to LISTSERV@nic.surfnet.nl
For information and archives: http://isb.ri.ccf.org/biomch-l