Biomechanics Laboratory, The Pennsylvania State University, University Park, PA 16802-3408.
J Biomech Eng. 2021 May 1;143(5). doi: 10.1115/1.4049748.
To examine segment and joint attitudes when using image-based motion capture, it is necessary to determine the rigid body transformation parameters from an inertial reference frame to a reference frame fixed in a body segment. Determine the rigid body transformation parameters must account for errors in the coordinates measured in both reference frames, a total least-squares problem. This study presents a new derivation that shows that a singular value decomposition-based method provides a total least-squares estimate of rigid body transformation parameters. The total least-squares method was compared with an algebraic method for determining rigid body attitude (TRIAD method). Two cases were examined: case 1 where the positions of a marker cluster contained noise after the transformation, and case 2 where the positions of a marker cluster contained noise both before and after the transformation. The white noise added to position data had a standard deviation from zero to 0.002 m, with 101 noise levels examined. For each noise level, 10 000 criterion attitude matrices were generated. Errors in estimating rigid body attitude were quantified by computing the angle, error angle, required to align the estimated rigid body attitude with the actual rigid body attitude. For both methods and cases, as the noise level increased the error angle increased, with errors larger for case 2 compared with case 1. The singular value decomposition (SVD)-based method was superior to the TRIAD algorithm for all noise levels and both cases, and provided a total least-squares estimate of body attitude.
为了检查基于图像的运动捕捉中的节段和关节姿态，有必要从惯性参考系确定刚体变换参数到固定在体节中的参考系。确定刚体变换参数必须考虑到两个参考系中测量坐标的误差，这是一个总体最小二乘问题。本研究提出了一种新的推导方法，表明基于奇异值分解的方法为刚体变换参数提供了总体最小二乘估计。将总体最小二乘法与确定刚体姿态的代数方法（TRIAD 方法）进行了比较。检查了两种情况：情况 1 是在变换后标记簇的位置包含噪声，情况 2 是在变换前后标记簇的位置都包含噪声。添加到位置数据的白噪声的标准偏差从零到 0.002 m，检查了 101 个噪声水平。对于每个噪声水平，生成了 10000 个标准姿态矩阵。通过计算将估计的刚体姿态与实际刚体姿态对齐所需的角度、误差角来量化刚体姿态估计的误差。对于两种方法和两种情况，随着噪声水平的增加，误差角增大，与情况 1 相比，情况 2 的误差更大。对于所有噪声水平和两种情况，基于奇异值分解（SVD）的方法都优于 TRIAD 算法，并且提供了刚体姿态的总体最小二乘估计。
