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 shortterm 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 threedimensional, which enables to construct a state space based on a uni or multivariate time series. Furthermore, two variants of the shortterm LLE are present in the literature which differ in length of a time span, over which the shortterm LLE is computed. Both a state space structure and the consistency of the observations based on values of both shortterm LLE variants were analyzed using time series representing the joint angles at ankle, knee, and hip joints. The shortterm 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 shortterm 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.
Stability means the ability to return to a stable state after having been subjected to some form of perturbation. Focusing on gait, if infinitesimally small perturbations, naturally occurring tiny variations in the walking surface and/or natural noise in the neuromuscular system, are concerned, then the ability of the locomotor system to keep the gait smooth by attenuating them is called local dynamic stability (LDS) [
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. The method is based on a trajectory in a state space which is reconstructed from time series generated by a dynamical system. The 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.
The authors intended to investigate how a state space structure affects the LDS. The input data for constructing a state space were time series describing the movement at a single joint. The authors created state spaces on the basis of one or threedimensional time series for hip, knee, and ankle joints separately and used the reconstructed trajectory for the LDS assessment according to the approach briefly described in the next section.
A symptom of extreme sensitivity to initial conditions is the exponential rate of divergence of trajectories from their starting points which are located in a state space very close to each other. This rate, which is called the largest Lyapunov exponent (LLE), is defined as follows:
A state space structure is an important aspect of the LDS assessment. The reconstruction procedure is based on two parameters:
For a multivariate time series, which is composed of
Concentrating on the importance of analyzing LDS and utilizing its results, some other pieces of research deserve a mention, e.g., Terrier et al. [
The discussion of state space structures in the context of motion data was initiated by Gates and Dingwell [
It should also be pointed out that the range of applications of the LLE as a measure of sensitivity to infinitesimal changes in initial conditions goes beyond the gait analysis. For instance, Jagrič et al. [
As mentioned above, the examined state space structures were constructed on the basis of time series built of joint angles at hip, knee, and ankle joints. Three time series, which are related to the given joint, represent specific types of movement in sagittal, frontal, and transverse planes. For instance, movements at hip joint are called flexion/extension, abduction/adduction, and internal/external rotation, respectively. However, analysis of human gait focuses often on the sagittal plane to which the vast majority of the work during gait is assigned (ca. 74%, 85%, and 93% in case of hip, knee, and ankle joints, respectively) [
Finally, three state space structures, which were constructed for each joint separately, were used to verify if the differences in LLE values between the opposed scenarios are significant for individual joints.
The authors also present a modification of the LDS computation method, i.e., an algorithm for estimation of one of its crucial parameters, embedding dimension, for the case of a multivariate gait time series, in which the parameter is not estimated for each of the component time series separately, but holistically.
The following research questions are addressed in the paper:
Are there any significant differences in the local dynamic stability between compared gait variants, which can be revealed using the individual state spaces?
Is the predominant role of sagittal plane preserved in a state space which is based on a multivariate time series?
Does the length of the time span, over which the shortterm LLE is computed, influence the difference in the local dynamic stability between compared gait variants?
The research procedure was composed of the following steps:
Data acquisition and preprocessing.
Estimation of the reconstruction parameters.
Trajectory reconstruction.
Estimation of the shortterm LLE, taking into consideration both the aforementioned time span variants.
Inspired by reports in the literature, the authors decided to incorporate three different state space structures into research. The
The dimensionality of
This research is a part of an extensive project carried out in cooperation with the University of the Third Age (U3A). The project focuses on elderly people that would like to remain active over the age of 65. Table
Participants’ characteristics.
Age (years)  Height (m)  Weight (kg)  BMI (kg/m^{2})  

Median  71  1.65  70  25.86 
Mean ± SD  70.64 ± 3.52  1.66 ± 0.07  76.12 ± 15.57  27.66 ± 5.18 
The authors state that the study has been approved by the Ethical Committee and all the subjects gave informed written consent to participate in the research after they were briefly introduced to the research protocol.
The CAREN extended system, which was used as the research environment, guaranteed not only fully immersive virtual scenery and 6 DOF motion platform but also safety and comfort of the participants. Besides, during the experiments, the walkers were under constant medical supervision. It is worth mentioning that the U3A students willingly took part in the experiments, especially when the research environment turned out to be so immersive, attractive, and safe at the same time, as the CAREN extended system is.
The participants performed six scenarios of selfpaced or fixed speed treadmill walking on level ground or on inclined platform, which are briefly presented in Table
Scenarios of experiments.
Scenario  Type of walking  Platform slope 


Selfpaced  0 

Selfpaced  0 

At fixed treadmill speed 
0 

At fixed treadmill speed 
0 

Selfpaced 


Selfpaced 

In each scenario, the subjects walked through a virtual forest. The CAREN treadmill’s selfpaced mode enables the subject to initiate gait and walk at her/his own pace which determines the instant walking speed. The treadmill adjusts then its speed to adapt to the subjects’ pace. The selfpaced mode was used in
The participants practiced each scenario until they were able to walk comfortably. Next, three trials were recorded using the integrated Vicon motion capture system at the frequency of 100 Hz giving together 18 gait sequences for every subject (several exceptions were caused by fatigue). Every time series, which is analyzed by means of the LLE, should include the equal number of strides as well as the equal number of data points [
The recorded data were initially filtered and optionally repaired (e.g., in view of occluded markers) using the Vicon software. The beginning of each stride was demarcated based on precisely marked occurrence of the “heelstrike” event. A stride interval varied not only across subjects but also across experiments’ scenarios. Mean and standard deviation values of the stride interval for different scenarios are included in Table
Mean stride interval values for six scenarios (s).








Mean  1.0924  1.0853  0.9590  1.4226  1.2316  1.0079 
SD  0.1085  0.1063  0.0998  0.2746  0.1537  0.1222 
Where necessary, the time series were cropped to 50 strides. Next, every stride was separately normalized using linear interpolation to contain 100 points. Subsequently, the time series were subject to estimation of reconstruction parameters. It deserves a mention that the most frequently occurring values for the dimensionality of the
of a length equal to 50 which is equivalent to one step, i.e., a half of a stride (the shortterm LLE is then labeled by
of a length equal to 100 which is equivalent to one stride (
The mean period parameter as the threshold for temporal separation of the nearest neighbors on two different segments of the reconstructed state space trajectory, which repeatedly imitate two initially neighboring trajectories, was estimated as the reciprocal of the mean frequency of the power spectrum [
Finally, the results were aggregated across all the six scenarios, three state space structures, and individual joints. All the computations were performed using MATLAB and MySQL DBMS. The statistical analysis of the results focused on investigating if differences between the shortterm LLE values in three pairs of compared scenarios (
Examples of times series representing a movement in the sagittal plane, recorded for a 75yearold woman performing the
(a) Time series representing a movement in sagittal plane (red line: left ankle joint, blue line: left knee joint, and green line: left hip joint); (b–d) 3D projections of corresponding reconstructed trajectories for: (b) left ankle joint; (c) left knee joint; (d) left hip joint. Each of the examined joints is characterized by a specific shape of the trajectory.
The final results are presented as box plots. On each box, the boundary between the areas of different colors indicates the median, the × symbol denotes the mean, the edges of the box are the 25^{th} and 75^{th} percentiles, the “whiskers” indicate the most extreme values which are not outliers, i.e., the smallest value that is larger than or equal to
Box plots for
Box plots for
Box plots for
Box plots for
Box plots for
Box plots for
Some general remarks, which were formulated at the earlier stage of research [
The main goal of the statistical analysis is to investigate if the differences between the opposed scenarios for individual joints are significant, taking into consideration that the shortterm LLE values were computed in two variants using different state space structures.
For a given joint
The expected conclusions are placed in the alternative hypotheses, according to which observations in one group tend to be greater than observations in the other group, e.g.,
The selection of an appropriate statistical test for verification of hypotheses should be preceded by the analysis of distribution for both measures
The outcome of the Shapiro–Wilk test of normality at significance level of 5% turned out to be negative in 74 from 216 cases (ca. 34%). Thus, the verification of the hypotheses requires a nonparametric test without any assumptions related to the distribution of scores.
With regard to the outcomes of the Shapiro–Wilk test, the hypotheses were verified for both measures independently using the nonparametric Mann–Whitney–Wilcoxon test at significance level of 5%. Because the number of null hypotheses is large (3 joints
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The total number of rejected null hypotheses for individual pairs of scenarios is as follows:
As far as two other pairs of scenarios are concerned, the high number of rejected null hypotheses indicates that the sensitivity to tiny local perturbations in both compared scenarios is different. The lower values of the shortterm LLE for the
Independently of the LDS measure and the applied state space, the significant differences between
A similar remark refers to a hip joint as regards the
The lower values for hip joints for
The verification outcomes for
Inspired by the close connection between
Therefore, although each examined state space reveals locally unstable behavior during gait as well as its attenuation in the current stride, a direct comparison of the shortterm Lyapunov exponents computed for different state spaces bears the risk of a wrong conclusion.
A similar remark refers to another test in which the accepted hypotheses based on
It should also be noted that the structure of a state space affects the computation time which comprises the following operations on a uni or multivariate time series: interpolation, estimation of time delay(s) and embedding dimension(s), reconstruction of the state space trajectory, and computation of the shortterm LLE. The mean computation time values for individual state spaces are presented in Table
Mean computation time values for individual state spaces (s).




30.96  80.89  28.30 
As expected, the mean computation time is maximal for the
It is worth mentioning that some other studies which require the LLE computation and are based on a univariate time series could benefit from adding one or more time series to input data used for a state space construction. For instance, dynamic features of eye movement [
The method of computation of the local dynamic stability requires a proper reconstruction of a state space trajectory. The authors intended to investigate how a state space structure affects the shortterm LLE as the measure related to the local dynamic stability. One of the state spaces was constructed using an algorithm which estimates the embedding dimension holistically in the case of a multivariate gait time series while taking into account the quota of work done during gait in individual motion planes. It should also be mentioned here that a direct comparison of the shortterm LLE computed for different state spaces is burdened with the risk of a wrong conclusion. Due attention should also be paid to the length of the time span, over which the shortterm LLE was estimated.
The improved method of LDS assessment will be used in experiments focused on finding easytomeasure, objective biomarkers that could classify PD (Parkinson’s disease) patients in early (preclinical) stages of the disease. Identification of the first deviation from the norm in patient’s physical movement like walking, which is often unobservable to a neurologist, might help follow disease progression, make more adjusted treatment, and lead to modification of disease course.
The motion capture data (joint angles) used to support the findings of this study are available from the corresponding author upon request.
The data used in this project were obtained from the Centre for Research and Development of the PolishJapanese Academy of Information Technology (PJAIT) (
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Statutory Research Funds of Institute of Informatics, Silesian University of Technology, Gliwice, Poland (BK/204/RAU2/2019).