Michaud Benjamin, Jackson Monique I, Prince François, Begon Mickaël S
a Laboratoire de Simulation et Modélisation du Mouvement, Département de Kinésiologie , Université de Montréal , Québec , Canada.
Comput Methods Biomech Biomed Engin. 2014 Apr;17(5):507-15. doi: 10.1080/10255842.2012.696104. Epub 2012 Jul 3.
To determine the range of motion of a joint between an initial orientation and a final orientation, it is convenient to subtract initial joint angles from final joint angles, a method referred to as the vectorial approach. However, for three-dimensional movements, the vectorial approach is not mathematically correct. To determine the joint range of motion, the rotation matrix between the two orientations should be calculated, and angles describing the range of motion should be extracted from this matrix, a method referred to as the matrical approach. As the matrical approach is less straightforward to implement, it is of interest to identify situations in which the vectorial approach leads to insubstantial errors. In this study, the vectorial approach was compared to the matrical approach, and theoretical justification was given for situations in which the vectorial approach can reasonably be used. The main findings are that the vectorial approach can be used if (1) the motion is planar (Woltring HJ. 1994. 3-D attitude representation of human joints: a standardization proposal. J Biomech 27(12): 1399-1414), (2) the angles between the final and the initial orientation are small (Woltring HJ. 1991. Representation and calculation of 3-D joint movement. Hum Mov Sci 10(5): 603-616), (3) the angles between the initial orientation of the distal segment and the proximal segment are small and finally (4) when only one large angle occurs between the initial orientation of the distal segment and the proximal segment and the angle sequence is chosen in such a way that this large angle occurs on the first axis of rotation. These findings provide specific criteria to consider when choosing the angle sequence to use for movement analysis.
为了确定关节在初始方位和最终方位之间的运动范围，从最终关节角度中减去初始关节角度是很方便的，这种方法称为矢量法。然而，对于三维运动，矢量法在数学上并不正确。为了确定关节运动范围，应该计算两个方位之间的旋转矩阵，并从该矩阵中提取描述运动范围的角度，这种方法称为矩阵法。由于矩阵法实施起来不太直接，因此确定矢量法会导致微不足道误差的情况很有意义。在本研究中，将矢量法与矩阵法进行了比较，并给出了矢量法可合理使用情况的理论依据。主要发现是，如果满足以下条件，矢量法就可以使用：（1）运动是平面的（Woltring HJ. 1994. 人体关节的三维姿态表示：一项标准化建议。《生物力学杂志》27(12): 1399 - 1414），（2）最终方位与初始方位之间的角度较小（Woltring HJ. 1991. 三维关节运动的表示与计算。《人类运动科学》10(5): 603 - 616），（3）远端节段与近端节段初始方位之间的角度较小，最后（4）当远端节段与近端节段的初始方位之间仅出现一个大角度，并且角度序列的选择使得这个大角度出现在第一个旋转轴上。这些发现为在选择用于运动分析的角度序列时提供了需要考虑的具体标准。
