Abstract
The ability of the locomotor system to maintain continuous walking despite very small external or internal disturbances is called local dynamic stability (LDS). The importance of the LDS requires constantly working on different aspects of its assessment method which is based on the short-term largest Lyapunov exponent (LLE). A state space structure is a vital aspect of the LDS assessment because the algorithm of the LLE computation for experimental data requires a reconstruction of a state space trajectory. The gait kinematic data are usually one- or three-dimensional, which enables to construct a state space based on a uni- or multivariate time series. Furthermore, two variants of the short-term LLE are present in the literature which differ in length of a time span, over which the short-term LLE is computed. Both a state space structure and the consistency of the observations based on values of both short-term LLE variants were analyzed using time series representing the joint angles at ankle, knee, and hip joints. The short-term LLE was computed for individual joints in three state spaces constructed on the basis of either univariate or multivariate time series. Each state space revealed walkers' locally unstable behavior as well as its attenuation in the current stride. The corresponding conclusions made on the basis of both short-term LLE variants were consistent in ca. 59% of cases determined by a joint and a state space. Moreover, the authors present an algorithm for estimation of the embedding dimension in the case of a multivariate gait time series.
Highlights
Stability means the ability to return to a stable state after having been subjected to some form of perturbation
It requires constantly working on different aspects of local dynamic stability (LDS) assessment method, which is derived from the dynamical systems theory. e method is based on a trajectory in a state space which is reconstructed from time series generated by a dynamical system. e dynamical properties of a system in the true state space are preserved under the reconstruction process, which enables to analyze the system’s behavior using the reconstructed trajectory, with particular emphasis on system’s sensitivity to initial conditions
Examples of times series representing a movement in the sagittal plane, recorded for a 75-year-old woman performing the Normal scenario are presented in Figure 1(a)
Summary
Stability means the ability to return to a stable state after having been subjected to some form of perturbation. Gait stability is of great importance for older people who are considered prone to falls. It requires constantly working on different aspects of LDS assessment method, which is derived from the dynamical systems theory. E method is based on a trajectory in a state space which is reconstructed from time series generated by a dynamical system. E dynamical properties of a system in the true state space are preserved under the reconstruction process, which enables to analyze the system’s behavior using the reconstructed trajectory, with particular emphasis on system’s sensitivity to initial conditions. E authors created state spaces on the basis of one- or three-dimensional time series for hip, knee, and ankle joints separately and used the reconstructed trajectory. Computational and Mathematical Methods in Medicine for the LDS assessment according to the approach briefly described
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