A comparison of methods for identifying the Jacobian for uncontrolled manifold variance analysis

Abstract

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.

Figure 1

Time-series of V_UCM and V_ORT results for a representative participant (A) and V_UCM (B) and V_ORT (C) results averaged-across participants obtained from each method within four periods of the reaching trajectory. The asterisks represent statistical significant differences with a level of significance set at p<0.05.

Figure 2

Regressions of V_UCM and V_ORT between each pair of methods for estimating the Jacobian based on values for all participants (white circles, M1–M2; black circles, M1–M3 and white squares, M2–M3).