Politti J C, Goroso G, Valentinuzzi M E, Bravo O
Instituto de Bioingeniería, Facultad de Ciencias Exactas y Tecnología, Universidad Nacional de Tucumán, Argentina.
Med Eng Phys. 1998 Jun;20(4):257-60. doi: 10.1016/s1350-4533(98)00020-4.
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.
以肩关节为中心的直臂在三个连续的90度旋转中运动，每次围绕各自的正交坐标轴旋转，会导致围绕手臂纵轴出现明显不相关的90度旋转。这一经验事实被称为科德曼悖论（Codman's paradox），以一位1934年首次报道此现象的波士顿外科医生命名。然而，通过齐次坐标，本文证明该现象只是一种机械特性，从数学角度来看，它可由应用于位置向量的三个正交旋转矩阵的矩阵乘积与应用于同一向量的单个旋转矩阵的矩阵乘积之间的等价性来描述。后一个旋转矩阵对应于前一组三个矩阵中间的那个。当使用极坐标时，证明甚至更简单，因为总位移向量清楚地显示了对纵轴旋转的单一净效应。因此，科德曼悖论并非悖论。这一特性增进了对手臂肌肉动力学的认识，可能在机器人技术中找到应用。
