Expressing angles relative to reference postures: A mathematical comparison of four approaches

Three-dimensional joint angles are most often quantified using Euler Angles. These measures are often easier to interpret if they are reported relative to a reference posture. However, since Euler Angles are not vectors, directly subtracting their values is difficult to justify mathematically. We compared four methods for subtracting a reference posture: the Subtraction Method (SM), directly subtracting the Euler Angles; two variants on the Relative Segment Method (RSM), one aligned to global (gRSM) and one aligned to the distal segment (dRSM), which considers the relative rotation of each segment to the reference posture; and the Relative Joint Method (RJM), which considers the relative rotation of the joint coordinate systems compared to that of the reference posture. One exemplar male subject (height: 175 cm; body mass: 90 kg; age: 27) performed three trials where they extended, laterally bent to the right, and extended while returning to a neutral posture between these movements. Two reference postures were compared: a standing neutral posture, and 90 degrees of flexion. All four methods showed strong agreement when the reference posture was a neutral one (lowest R²=0.971). However, when the reference posture was 90 degrees of flexion, both the RJM and gRSM swapped their lateral bend and axial twist measures. Additionally, when the reference posture was oriented 90 degrees from the global coordinate system, the gRSM swapped flexion and lateral bending. Therefore, the RJM, dRSM, and even the SM, are more robust than the gRSM. Either the RJM or dRSM are recommended as it is a compromise between mathematical validity and interpretability, however, the RJM seems to provide more readily interpretable angular velocities. The SM is only a viable approach under very strict conditions and should be avoided.