CODMANS PARADOX OF THE ARM ROTATIONS IS NOT A PARADOX - MATHEMATICAL VALIDATION

Movement of a straight arm centred at the shoulder joint in three succ essive 90 degrees rotations, each around the respective orthogonal coo rdinate axis, leads to an apparently unrelated 90 degrees rotation aro und the longitudinal arm axis. This empirical fact is known as Codman' s paradox, after a Bostonian surgeon who first reported it in 1934. Ho wever, by means of homogeneous coordinates, it is herein demonstrated that the phenomenon is just a mechanical property mathematically descr ibed by the equivalence between the matricial product of three orthogo nal rotation mat;ices applied to a position vector and the matricial p roduct of a single rotation matrix applied to the same vector. The lat ter rotation matrix corresponds to the middle one in the former group of three. When polar coordinates are used, the demonstration is even s impler, for the total shift vector clearly shows a single net effect o n the longitudinal axis rotation. Thus, Codman's paradox is not a para dox. This property improves the muscle dynamics arm knowledge and migh t find applications in robotics. (C) 1998 IPEM. Published by Elsevier Science Ltd. All rights reserved.