title: On the use of a Euclidean norm function for the estimation of local dynamic stability from 3D kinematics using time-delayed Lyapunov analyses.
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 (λmax) 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 λmax 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 λmax.
authors.affiliation: Department of Human Health and Nutritional Sciences, University of Guelph, Guelph, ON, Canada. Electronic address: sbeaudet@uoguelph.ca.
authors.forename: Shawn M
authors.initials: SM
authors.lastname: Beaudette
authors.affiliation: Department of Graduate Education and Research Programs, Canadian Memorial Chiropractic College, Toronto, ON, Canada. Electronic address: showarth@cmcc.ca.
authors.forename: Samuel J
authors.initials: SJ
authors.lastname: Howarth
authors.affiliation: School of Human Kinetics, University of Ottawa, Ottawa, ON, Canada. Electronic address: Ryan.Graham@uottawa.ca.
authors.forename: Ryan B
authors.initials: RB
authors.lastname: Graham
authors.affiliation: Department of Human Health and Nutritional Sciences, University of Guelph, Guelph, ON, Canada. Electronic address: shmbrown@uoguelph.ca.
authors.forename: Stephen H M
authors.initials: SH
authors.lastname: Brown
mesh_terms: D001696:Biomechanical Phenomena; D055595:Mechanical Phenomena; D013223:Statistics as Topic / Q000379:methods*
keywords: Euclidean norm; Kinematics; Local dynamic stability; Lyapunov exponent; New method; Non-linear dynamics