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Charnwood Dynamics Ltd. Coda cx1 User Guide – Advanced Topics III - 3

CX1 USER GUIDE - COMPLETE.doc 26/04/04

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SEGMENT ROTATIONS - The Mathematics Of Euler Angles

The reporting of angular coupling patterns for limb segments is now routine in gait

analysis. Standardisation throughout the clinical environment and within CAMARC

has

tended towards one particular set of angular couplings, namely Euler Angles, which are

considered by many to deliver the best compromise in the representation of complex

clinical rotations in three dimensions.

By their very nature, however, Euler Angles present difficulties of interpretation for the

clinician who must decide upon their relevance to clinical movements. Not least amongst

these difficulties is the visualisation of complex rotations in 3D, without which we can not

attribute meanings to the individual Euler Angles. In addition, one must attempt to digest

the vector algebra and trigonometry by which sets of marker positions within a Cartesian

frame are processed into rotation angles.

It is assumed the reader is familiar with the usual notions of co-ordinate frames with

orthogonal axes. Basic concepts of vectors, matrices and trigonometry will suffice to

appreciate the mathematics described below. Readers are urged to form their own

pictures (mental or otherwise) of the geometries descibed; the author’s own diagrams of

3D events on 2D page would only serve to confuse.

The formulae for Euler Angles are, in fact, rather simple (at least in vector notation), but

before visiting those it will be worthwhile exploring behind the scenes.

Background

A rigid body, free to move in space, is said to have six degrees of freedom, three of which

may be associated with translational movements, the other three with rotations. In clinical

movement analysis such a body is represented by a minimum of three strategically placed

markers whose measured positions are sufficient data to allow definition of an embedded

co-ordinate frame or ‘vector basis’ (EVB).

The details of constructing an EVB with the Gram-Schmidt Orthogonalisation Process are

given in another user document - 3D Segmental Analysis

- but we should note here that

the embedded axes are always aligned to be anatomically meaningful; in particular, the

longitudinal axis of a limb segment usually becomes the local ‘Z’ axis, the medio-lateral

axis ‘Y’, and antero-posterior ‘X’, all mutually orthogonal.

Here we are concerned only with the orientation of a segment and its EVB.

In 1748 the Swiss mathematician Leonhard Euler noted that the orientation of a rigid body

could be described, relative to some neutral position, by a succession of three rotation

angles about a particular set of axes.

Nowadays, the term ‘ Euler Angles ’ is used for

any equivalent ordered set of angles corresponding to rotations about given axes, usually

orthogonal axes.

The meaning and validity of the derived clinical angles are determined by the choice of

axes and rotation sequence; there is ample scope for confusion here. In order for limb-

segment angles to be clinically relevant we attempt to define the orientation of the distal

segment relative to the proximal segment by comparing the ‘attitudes’ of corresponding

axes of the segment-embedded co-ordinate frames. There are many ways of doing this.

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