On the use of a Euclidean norm function for the estimation of local dynamic stability from 3D kinematics using time-delayed Lyapunov analyses. Issue 10 (October 2016) - Beaudette, Shawn M.
Highlights: To estimate local dynamic stability (LDS) state-space reconstruction is necessary. The Euclidean norm function ( N ) can be used to collate the dynamics of a 3D system. With N, non-linear transformations can occur, skewing Lyapunov exponents (λmax ). λmax estimate error can be reduced by eliminating zero-crossings in kinematic data. 3D components should be shifted completely into positive space prior to applying N . Abstract: Several different state-space reconstruction methods have been employed to assess the local dynamic stability (LDS) of a 3D kinematic system. One common method is to use a Euclidean norm ( N ) transformation of three orthogonal x, y, and z time-series' followed by the calculation of the maximum finite-time Lyapunov exponent ( λ max ) from the resultant N waveform (using a time-delayed state space reconstruction technique). By essentially acting as a weighted average, N has been suggested to account for simultaneous expansion and contraction along separate degrees of freedom within a 3D system (e.g. the coupling of dynamic movements between orthogonal planes). However, when estimating LDS using N, non-linear transformations inherent within the calculation of N should be accounted for. Results demonstrate that the use of N on 3D time-series data with arbitrary magnitudes of relative bias and zero-crossings cause the introduction of error in estimates of λ max obtained through N . To develop a standard for the analysis of 3D dynamic kinematic waveforms, we suggest that all dimensions of a 3D signal be independently shifted to avoid the incidence of zero-crossings prior to the calculation of N and subsequent estimation of LDS through the use of λ max .