### What are the shoulder angle calculations in Plug-in Gait?

The first step in the shoulder modelling process is the definition of the shoulder, elbow and wrist centres and the Thorax, Clavicle and Humerus segments. The shoulder angle calculations are then based on YXZ Euler angle rotations between the Thorax and the Humerus Segments as follows:

• LShoulderAngles 1 Flexion Anti-clockwise about Thorax Y, 2 Abduction Anti-clockwise about Thorax X, 3 Internal Rotation Anti-clockwise about Thorax Z
• RShoulderAngles 1 Flexion Anti-clockwise about Thorax Y, 2 Abduction Clockwise about Thorax X, 3 Internal Rotation Clockwise about Thorax Z

The explanation for the sometimes strange angles seen when using the above method for determining shoulder motion is the occurrence of 'Gimbal Lock' and the quirk in clinical descriptions of motion known as 'Codman's Paradox'. 'Gimbal Lock'. Gimbal Lock occurs when using Euler angles and any of the rotation angles becomes close to 90 degrees, for example lifting the arm to point directly sideways or in front (shoulder abduction about an anterior axis or shoulder flexion about a lateral axis respectively).

In either of these positions the other two axes of rotation become aligned with one another, making it impossible to distinguish them from one another, a singularity occurs and the solution to the calculation of angles becomes unobtainable. For example, assume that the humerus is being rotated in relation to the thorax in the order Y,X,Z and that the rotation about the X-axis is 90 degrees. In such a situation, rotation in the Y-axis is performed first and correctly. The X-axis rotation also occurs correctly BUT rotates the Z axis onto the Y axis. Thus, any rotation in the Y-axis can also be interpreted as a rotation about the Z-axis.

True gimbal lock is rare, arising only when two axes are close to perfectly aligned. 'Codman's Paradox': The second issue however, is that in each non-singular case there are two possible angular solutions, giving rise to the phenomenon of “Codman’s Paradox” in anatomy (Codman, E.A. (1934). The Shoulder. Rupture of the Supraspinatus Tendon and other Lesions in or about the Subacromial Bursa. Boston: Thomas Todd Company), where different combinations of numerical values of the three angles produce similar physical orientations of the segment. This is not actually a paradox, but a consequence of the non-commutative nature of three-dimensional rotations and can be mathematically explained through the properties of rotation matrices (Politti, J.C., Goroso, G., Valentinuzzi, M.E., & Bravo, O. (1998).

Codman's Paradox of the Arm Rotations is Not a Paradox: Mathematical Validation. Medical Engineering & Physics, 20, 257-260). Codman proposed that the completely elevated humerus could be shown to be in either extreme external rotation or in extreme internal rotation by lowering it either in the coronal or sagittal plane respectively, without allowing any rotation about the humeral longitudinal axis.

For a demonstration of this, follow the sequence below:

1. Place the arm at the side, elbow flexed to 90 degrees and the forearm internally rotated across the stomach.
2. Elevate the arm 180 degrees in the sagittal plane.
3. Lower the arm 180 degrees to the side in the coronal plane.
4. Note that the forearm now points 180 degrees externally rotated from its original position with no rotation about the humeral longitudinal axis actually having occurred.
5. Appreciate the difficulty then in describing whether the fully elevated humerus was internally or externally rotated.

This ambiguity can cause switching between one solution and the other, resulting in sudden discontinuities. A combination of 'Gimbal Lock' and 'Codman's Paradox' can lead to unexpected results when joint modelling is carried out.

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