Abstract
Nonlinear dynamical analysis is a powerful approach to understanding biological systems. One of the most used metrics of system complexities is the Kolmogorov entropy. Long input signals without noise are required for the calculation, which are very hard to obtain in real situations. Techniques allowing the estimation of entropy directly from time signals are statistics like approximate and sample entropy. Based on that, the new measurement for quaternion signal is introduced. This work presents an example of application of a nonlinear time series analysis by using the new quaternion, approximate entropy to analyse human gait kinematic data. The quaternion entropy was applied to analyse the quaternion signal which represents the segments orientations in time during the human gait. The research was aimed at the assessment of the influence of both walking speed and ground slope on the gait control during treadmill walking. Gait data was obtained by the optical motion capture system.
Highlights
The parameters associated with chaos are measures of dimension, rate of information and the Lyapunov determinant
Kolmogorov entropy K is known as a chaos metrics and the value of entropy can be used for the classification of underling dynamic systems [1]
The research was aimed at the assessment of the influence of both walking speed and ground slope on the proposed quaternion entropy during treadmill walking
Summary
The parameters associated with chaos are measures of dimension, rate of information (entropy) and the Lyapunov determinant. Kolmogorov entropy K is known as a chaos metrics and the value of entropy can be used for the classification of underling dynamic systems [1]. The connection of Kolmogorov entropy and Lyapunov determinants of the system is defined by Pessin’s theorem. Sensitive dependence on initial conditions is a distinguishing feature of chaotic behavior. Arbitrarily close points in the phase space produce significantly different trajectories. Trajectories in chaotic systems diverge exponentially and Lyapunov exponents (LLE) were proved to be a good quantitative measure for the average rate of exponential divergence of two trajectories. Positive LLE is indicative of unpredictable behavior. K is basically equal to the sum of the positive LLE of the system
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