15.1 APPENDIX 1: CALCULATION OF UPPER-ARM AND THIGH ROTATIONS

Calculating humeral and femoral rotation is conceptually more complex than calculating other joint angles because it is defined not only by the direction of the segment, but also by its orientation “around” its own longitudinal axis.

Focusing on humeral rotation as an example, without loss of generality, rotation is commonly estimated by defining a reference plane associated with the trunk and upper arm and comparing it with the functional plane of the upper arm and forearm. This approach is robust in most postures, but it has an inherent problem: certain configurations cause the reference plane to become poorly defined or extremely sensitive to small postural variations.

The typical case occurs when the upper arm is almost aligned with the trunk axis—for example, when it is nearly vertical alongside the body. In this situation, the humeral axis is almost parallel to the trunk axis, and the “trunk–upper arm” plane loses geometric stability: small abductions or minimal deviations can produce large changes in the plane's orientation and therefore abrupt jumps in the estimated humeral rotation. This phenomenon is an example of a kinematic singularity, equivalent to what is known as gimbal lock in Euler-angle representation systems: one effective degree of freedom is lost, and the angular decomposition is no longer unique or becomes numerically unstable.

There is also another degenerate case: when the shoulder, elbow, and wrist are almost aligned. Under this condition, the plane defined by these three points—which is used to describe the orientation of the upper-arm–forearm assembly—also becomes indeterminate, making it impossible to calculate a reliable rotation angle.