| 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. |