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Author(s) Quintana, J.
Title Non-linear dynamic invariants based on embedding reconstruction of systems for pedaling motion
Abstract Recent empirical studies of human motion (e.g. walking) have revealed complex dynamical structures, even under constant environmental conditions. These structures, also known as variability, have been used to determine disease severity, medication utility, and fall risk. On one hand, variability on human motion is attributed to the body's ability to find the most stable solution coordinating all physiological systems over different timescales, whose behaviors are both highly variable and strongly dependent on each other. On the other hand, the term variability has been associated with different mathematical definitions related to chaotic systems, which are known as dynamic invariants. They are quantities describing the dynamical behavior of a system with the special property that the value of that quantity does not depend on the coordinate system and it can be obtained either directly from the original state space or from the reconstructed embedding space (See next page) obtained from time series data. The classical embedding theory, based on Takens’ theorem assumes that there is a rule governing the dynamics of a continuous system with n variables whose actual values depend on first order differential equations. This has been used before to describe motion and other physiological signals but this doesn't take into account a feedback in one or more variables (represented by delay differential equations) as has been suggested to model the human body. This can explain why they are some contradictory results in previous works about standard values of invariants. Thus, it must be questioned whether the conclusions in previous works based on Takens’ theorem, assuming only first order differential equations, are complete and correct. There is a lack of studies about the description of the dynamic structure of pedaling motion and the potential benefits of this to distinguish subtle differences between pedaling motion patterns. As an example for application, we tested this idea for fatigue detection. We propose to use dynamic invariants with an extension of the classical embeddings using two embedding windows instead of only one. Two dynamic invariants based on embedding space (Maximal Lyapunov Exponent and Recurrence Period Density Entropy) were evaluated using pedaling motion data with low, medium and high workloads. Evidence of better classification based on invariants with this extended embedding was found assuming that they will show changes due to fatigue. The benefit of this work is a new method for analysis and classification of the differences between quasi periodical movements.
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