Codman's paradox of the arm rotations is not a paradox: mathematical validation

Abstract

Movement of a straight arm centred at the shoulder joint in three successive 90 degrees rotations, each around the respective orthogonal coordinate axis, leads to an apparently unrelated 90 degrees rotation around the longitudinal arm axis. This empirical fact is known as Codman's paradox, after a Bostonian surgeon who first reported it in 1934. However, by means of homogeneous coordinates, it is herein demonstrated that the phenomenon is just a mechanical property mathematically described by the equivalence between the matricial product of three orthogonal rotation matrices applied to a position vector and the matricial product of a single rotation matrix applied to the same vector. The latter rotation matrix corresponds to the middle one in the former group of three. When polar coordinates are used, the demonstration is even simpler, for the total shift vector clearly shows a single net effect on the longitudinal axis rotation. Thus, Codman's paradox is not a paradox. This property improves the muscle dynamics arm knowledge and might find applications in robotics.