Parametrization of cyclic motion and transversal sections

Abstract

Cyclic motion is at the core of many sports such as running, swimming, or cycling. The study of corresponding kinematic variables is fundamental for the evaluation of training routines and the assessment of performance. Conventional kinematic analysis of human cyclic locomotion derives characteristic features from a few single cycles (e.g., average peak, average difference between peaks, as in Harris and Smith, 1996). However, most current methods ignore the full dynamics of the motion that may reveal important additional insight into patterns of motor control (Schablowski-Trautmann & Gerner, 2006).An approach to the analysis of multi-dimensional kinematic variables is to view them in their state space and to intersect the cyclic orbit with a co-dimension 1 hyperplane, called transversal section (Vieten, Sehle & Jensen, 2013). This corresponds to so-called Poincaré sections, a tool for the analysis of dynamical systems nearby a periodic solution. Features can be extracted from the intersection points on a single transversal section in order to characterize the overall motion. E.g., the variance may indicate the degree of regularity of cyclic motion. It may also be of interest to compute a representative average cycle and to study how the kinematic variables change with the phase angle of the cyclic motion.The timing of muscular, neurological, and respiratory systems varies according to the environmental, bio-mechanical, and morphological constraints. Therefore, individual cycles in cyclic motion differ in duration and local speed. It follows that the calculation of an average cycle of a set of cycles requires a phase alignment.Our contributions are: (1) An algorithm for calculating an average periodic cycle based on dynamic time warping (DTW) and modifications of DTW barycentric averaging (DBA). Then cycles are aligned with the average cycle yielding their parametrization by phase. (2) A definition of the quality of cycle intersections with Poincaré sections, providing a criterion for the right choice of the section.