Department of Physical Therapy, 307 McKinly Laboratory, University of Delaware, Newark, DE 19716, USA.
J Biomech. 2010 Mar 3;43(4):775-7. doi: 10.1016/j.jbiomech.2009.10.033. Epub 2009 Dec 14.
Uncontrolled Manifold (UCM) analysis has been used to identify a component of joint variance leading to pointer-tip position variability and a component representing motor abundant joint combinations corresponding to an equivalent pointer-tip position. A Jacobian is required for UCM analysis, typically derived from an analytic model relating joint postures to pointer-tip position. Derivation of the Jacobian is often non-trivial, however, because of the complexity of the system being studied. In this article, we compared the effect of different methods of deriving the Jacobian on results of UCM analyses during reaching. Jacobian matrices were determined at each percentage of the reach across trials using one of three methods: (M1) partial derivatives of the geometric model relating ten joint postures, segment lengths and pointer length to the position of a hand-mounted pointer tip; or (M2-M3) as the coefficients of linear regression between the ten joint postures and either (M2) the pointer tip position measured directly from motion capture or (M3) the pointer-tip position estimated from the geometric model. For all methods, motor abundant joint variance (V(UCM)) was larger than joint variance leading to a variable pointer-tip position (V(ORT)). Results did not differ among methods prior to the time of peak velocity. Thereafter, M2 yielded lower V(ORT) and slightly higher V(UCM) compared to M1. Method M3 was used to disambiguate the possible effect of estimating model parameters for the geometric model on the M1-M2 comparison. The advantages of the use of linear regression method in the UCM approach are discussed.
未受控制的流形 (UCM) 分析已被用于识别导致指针尖端位置变化的关节方差分量和代表与等效指针尖端位置对应的大量关节组合的分量。UCM 分析需要雅可比矩阵，通常由将关节姿势与指针尖端位置相关联的解析模型得出。然而，由于所研究系统的复杂性，雅可比矩阵的推导通常并不简单。在本文中，我们比较了在到达过程中，不同的雅可比矩阵推导方法对 UCM 分析结果的影响。在试验过程中，在到达的每个百分比处，使用以下三种方法之一确定雅可比矩阵：（M1）将十个关节姿势、节段长度和指针长度与手部安装的指针尖端位置相关联的几何模型的偏导数；或者（M2-M3）作为十个关节姿势与（M2）直接从运动捕捉测量的指针尖端位置或（M3）从几何模型估计的指针尖端位置之间的线性回归系数。对于所有方法，导致指针尖端位置变化的关节方差（V(ORT)）均大于大量运动的关节方差（V(UCM)）。在达到峰值速度之前，结果在方法之间没有差异。此后，与 M1 相比，M2 产生的 V(ORT) 更低，V(UCM) 略高。方法 M3 用于消除对几何模型参数进行估计对 M1-M2 比较的可能影响。讨论了在 UCM 方法中使用线性回归方法的优势。
