Centers and angles of rotation of body joints: a study of errors and optimization.

Abstract Large variations are generally reported in the locations of centers of rotation (CR) for each of various joints in the human body. Some of these reports present conflicting results. This paper shows that this may be due in part to suboptimal experimental design as well as the phenomenon of error magnification. An algorithm is presented for computing the coordinates of the CR and the angle of rotation from the x, y coordinate measurements of two point markers on a moving body in two different positions. Error analysis is performed using a mathematical model that introduces systematically a positive or a negative error into each of the 8 x, y coordinates in all possible combinations, resulting in 256 CR locations. CR error zones are computed and graphed. Parametric analysis of the experimental set-up leads to optimization of the set-up. A typical case is analyzed and its errors computed. It is shown that small errors present in the measurements of the x, y coordinates of the markers are magnified to relatively large errors in the CR coordinates. In a suboptimal case, this magnification may be 30–50 times or more. The results show that, besides the magnitude of x, y coordinate errors, other factors responsible for determining the magnitude of errors in the location of the CR are: the magnitude of angle of rotation, the orientation of the markers with respect to the CR and their distances from the CR. In conjunction with the CR, the angle of rotation is also analyzed. Guidelines for optimal experimental set-up for minimizing the output errors are presented.