15.1.2 How singularities are resolved

As explained above, the plane-based calculation becomes unstable when the upper arm approaches alignment with the trunk (a gimbal-lock singularity). A regularization strategy is applied to ensure continuity and prevent numerical jumps:

Detection of proximity to the singularity. The degree of alignment between the humeral axis and the trunk axis is evaluated. The more parallel—or antiparallel—they are, the greater the likelihood that the method based on the trunk–upper arm plane will become unstable.
Use of an alternative estimate of upper-arm “twist.” Near the singularity, rather than relying exclusively on the dihedral angle between planes—which can change abruptly in response to minimal variations—an alternative measure of rotation about the upper-arm axis is used because it is inherently more stable in this configuration. Conceptually, this alternative measure is based on the segment's own orientation relative to a trunk-fixed reference and behaves more reliably when the trunk–upper arm plane becomes ambiguous.
Progressive interpolation between the two estimates. The final humeral-rotation value is not replaced abruptly; instead, it is obtained by blending the two estimates according to proximity to the singularity:

far from the singularity, the dihedral angle between planes dominates because it provides more biomechanical information;
near the singularity, the alternative measure dominates because it is more stable;
in the intermediate region, a smooth, continuous transition is applied to prevent discontinuities.

Collinear shoulder–elbow–wrist case. If the shoulder–elbow–wrist plane is degenerate because the points are collinear, the system cannot obtain a reliable dihedral angle and directly uses the alternative estimate, which does not depend on this plane.

In summary, humeral rotation is calculated using a geometric formulation based on the trunk–upper arm reference and the upper-arm–forearm plane. It is stabilized by a controlled transition to an alternative measure of rotation about the upper-arm axis when the joint configuration approaches conditions under which the plane-based definition loses uniqueness or becomes numerically fragile.