Segment Rotations- The Mathematics of Euler Angles


The relative motion of a segment compared to another can be described from many different ways (quaternion, Euler angles and Cardan angles, rotation matrix or projections angles). However, most of these present difficulties of interpretation for the user who must decide upon their relevance to movements. The Euler/Cardan angles are considered by many to deliver the best compromise in the representation of complex rotations in three dimensions.

The ODIN segrot1.pngwith the help of other associates such as the International Shoulder Group (ISB) for example) took great care to choose the rotation orders so that the angles remain as close as possible to the clinical definitions of joint and segment motions.

Whatever the Euler sequence and irrespective of the order of rotation, the angles are defined as follows:

  • The α angle describes a rotation around the Z-axis
  • The γ angle describes a rotation around the Y-axis
  • The β angle describes a rotation around the X-axis

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 described. 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.


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 later 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. The ISB published recommendations for the definition of the body EVBs.

The Euler / Cardan angles only concern the orientation of a segment and its EVB. In 1748 the Swiss mathematician Leonhard Euler noted that the orientation of a rigid body relative to some another could be described 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 Euler / Cardan angles only concern the orientation of a segment and it’s EVB. Euler angles are readily calculated (requiring no joint centre model), and correspond to quite relevant axes which are generally orthogonal and therefore kinetically useful.

Eulerian System

Consider a distal limb segment initially in the ‘neutral’ position, such that its EVB axes coincide exactly with those of the proximal segment. Clearly the Euler Angles are all zero for this situation but if, at some other instant, the distal segment is re-positioned elsewhere the new set of Euler Angles ought to quantify the necessary rotations (about the proximal axes) by which it arrives there.

A single rotation through a given angleabout a given (proximal) axis may be represented by a rotation matrix:


If  is the rotation matrix composed of three successive rotations.

There are two main types of conventions called “proper” Euler angles and Cardan angles. There are six possibilities of choosing the proper Euler angles and also six Cardan angle combinations. All the combinations are summarised in the following table:

Sequence 1 2 3 4 5 6

NOTE: the Cardan sequence XYZ means that  and is the result of a rotation about the (proximal) Z axis, followed by rotation about the Y axis, and finally about the X axis.

Choosing the Axis Sequence

For any re-orientation of a distal segment (with EVB)

and is the result of a rotation about the (proximal) Z axis, followed by rotation about the Y axis, and finally about the X axis.

7.4 Choosing the Axis Sequence

For any re-orientation of a distal segment (with EVB ODIN P1.png) there are twelve Euler / Cardan decompositions available, some more appropriate than others. Ideally, and to avoid confusion, everyone should adopt the same convention, using the ‘best’ axis sequence for the job, if such exists.

The criteria for this choice are many: clinical relevance; joint, robustness; intuitive ease of use; correspondence with alternative schemes; generality of application etc. The ISB recommendations took the specificities of each major joint into account and chose the rotation orders so that the angles remain as close as possible to the clinical definitions of joint and segment motions. The sequence orders for every joint are detailed in the guide: “ODIN – human analysis protocol”.

Even if the joint doesn’t have three degrees of freedom, the Euler / Cardan sequence has to be the result of three successive rotations. This allows, for example, a user to determine the angle of varus / valgus of the forearm with respect to the upper-arm or the varus / valgus and the axial rotation of the tibia compare to the femur.

NOTE: the glenohumeral and thoracofumeral joints are the only joints based on an Euler sequence. All the other are based on a Cardan sequence. This is due to the fact that the floating axis notion defined by Grood and Suntay (5) cannot be used. The clinical terms flexion – extension and adduction – abduction cannot be used because flexion followed by adduction give radically different results than given by adduction followed by flexion Anglin and Wyss (6)

Yaw, Pitch and Roll (but not necessarily in that order)

Seafaring terminology often finds its way into the discussion thanks to the generality of the terms ‘yaw’, ‘pitch’, ‘roll’ in respect of a ship’s motion. This is all very well but for the potential confusion over frames of reference. The inference from these terms is of rotations measured against the ship’s own co-ordinate frame immediately prior to its re-orientation. This is analogous to describing re-orientation of the distal segment in the absence of the proximal segment, which is perfectly acceptable – some texts on biomechanics proceed exactly in this fashion on the subject of Euler Angles.

It is vitally important to realise that a sequence of Euler angles given with respect to the moving co-ordinate frame of a (distal) segment is different to the sequence which describes the same re-orientation with respect to the proximal segment. Fortunately it is only the sequence which differs: the magnitudes of the angles and the types of rotations they measure are the same, but the order is reversed (7)

The chosen ‘ZXY’ sequence in respect of proximal segment axes would be reversed to Y-X-Z with respect to (moving) distal segment axes, corresponding to the nautical sequence: pitch, followed by yaw, followed by roll (for most segments)

(5) Grood, E. S., and W. J. Suntay. “A joint coordinate system for the clinical description of three-dimensional motions: application to the knee.” J Biomech Eng 105, no. 2 (1983): 136-144.

(6) Anglin, C., and U. P. Wyss. “Review of arm motion analyses.” Proc Inst Mech Eng H 214, no. 5 (2000): 541-555.

(7) Paul, R. P., 1982. Robot Manipulators, pp. 45 – 71. Cambridge: The MIT Press

(In either case, the sequence described is that by which the segment arrives at its orientation. Anyone wishing to ‘undo’ the rotations to return to the neutrally aligned orientation must reverse carefully!)

Mathematical decomposition

Having settled for the ‘ZXY’ sequence we must calculate the compound rotation matrix:

Conveniently, the 2nd element of the 3rd column of our matrix happens to be a sinus.

and we solve first, therefore, the X axis rotation angle β as

Having found β, its value may be plugged back into the equations for a γ and α:

we obtain γ from the 1st element of the 3rd column wherein

so that the rotation angle about the Y axis is given by

Likewise, from the 2nd element of the 1st column:

so that the rotation angle about the Z axis is

Note that the order in which the decomposition is solved is NOT the same as the Euler sequence of rotations. In fact the mid-sequence angle is solved first.

Closer scrutiny of the equations for γ and α reveals that these angles would remain undefined in the event cos β = 0 (division by zero isn’t allowed). This only happens when β= ∓ 90°, i.e. when ad/abduction reaches 90°, a condition known as ‘gimbal-lock’ and is unlikely in the context of lower limb movements. This may, of course, be taken as one of the constraints upon the choice of Euler sequence – the chosen scheme must avoid a mid-sequence rotation of 90°.

Neutral alignments

The Euler angles described here are taken to be measured relative to a hypothetical position of neutral alignment, i.e. all distal segment axes aligned exactly with proximal axes. In clinical applications these hypothetical alignments may not even be possible, let alone considered neutral. In normal stance, for example, one would expect to be able to claim neutral positions for the segments but the alignments are clearly never perfectly square and will therefore register non-zero Euler angles. Thus, the extent to which ‘real’ neutral positions are offset from hypothetical alignment is a matter for the clinician to judge. Unreasonably large offsets between segments in neutral position may lead to slight crosstalk in the angular coupling patterns observed.