The largest Lyapunov exponent (LyE) is an accepted method to quantify gait stability in young and old adults. However, a range of LyE values has been reported in the literature for healthy young and elderly adults in normal walking. Therefore, it has been impractical to use the LyE as a clinical measure of gait stability. The aims of this systematic review were to summarize different methodological approaches of quantifying LyE, as well as to classify LyE values of different body segments and joints in young and elderly individuals during normal walking. The Pubmed, Ovid Medline, Scopus and ISI Web of Knowledge databases were searched using keywords related to gait, stability, variability, and LyE. Only English language articles using the Lyapunov exponent to quantify the stability of healthy normal young and old subjects walking on a level surface were considered. 102 papers were included for full-text review and data extraction. Data associated with the walking surface, data recording method, sampling rate, walking speed, body segments and joints, number of strides/steps, variable type, filtering, time-normalizing, state space dimension, time delay, LyE algorithm, and the LyE values were extracted. The disparity in implementation and calculation of the LyE was from, (i) experiment design, (ii) data pre-processing, and (iii) LyE calculation method. For practical implementation of LyE as a measure of gait stability in clinical settings, a standard and universally accepted approach of calculating LyE is required. Therefore, future studies should look for a standard and generalized procedure to apply and calculate LyE.
The development of methods that can identify athlete-specific optimum sports techniques-arguably the holy grail of sports biomechanics-is one of the greatest challenges for researchers in the field. This 'perspectives article' critically examines, from a dynamical systems theoretical standpoint, the claim that athlete-specific optimum sports techniques can be identified through biomechanical optimisation modelling. To identify athlete-specific optimum sports techniques, dynamical systems theory suggests that a representative set of organismic constraints, along with their non-linear characteristics, needs to be identified and incorporated into the mathematical model of the athlete. However, whether the athlete will be able to adopt, and reliably reproduce, his/her predicted optimum technique will largely be dependent on his/her intrinsic dynamics. If the attractor valley corresponding to the existing technique is deep, or if the attractor valleys corresponding to the existing technique and the predicted optimum technique are in different topographical regions of the dynamic landscape, technical modifications may be challenging or impossible to reliably implement even after extended practice. The attractor layout defining the intrinsic dynamics of the athlete, therefore, needs to be determined to establish the likelihood of the predicted optimum technique being reliably attainable by the athlete. Given the limited set of organismic constraints typically used in mathematical models of athletes, combined with the methodological challenges associated with mapping the attractor layout of an athlete, it seems unlikely that athlete-specific optimum sports techniques will be identifiable through biomechanical optimisation modelling for the majority of sports skills in the near future.
The aims of this study were to demonstrate "order error" in the calculation of continuous relative phase (CRP) and to suggest two alternative methods-(i) constructing phase-plane portraits by plotting position over velocity; and (ii), the Hilbert transform-to rectify it. Order error is the change of CRP order between two degrees of freedom (e.g., body segments) when using the conventional method of constructing phase-plane portraits (i.e., velocity over position). Both sinusoidal and non-sinusoidal simulated signals as well as signals from human movement kinematics were used to investigate order error and the performance of the two alternative methods. Both methods have been shown to lead to correct results for simulated sinusoidal and non-sinusoidal signals. For human movement data, however, the Hilbert transform is superior for calculating CRP.
This paper evaluates the effectiveness of, and highlights issues with, conventional paradigms in applied sports biomechanics research and comments on their capacity to optimise techniques of individual athletes. In empirical studies, group-based analyses often mask variability between athletes and only permit probabilistic 'in general' or 'on average' statements that may not be applicable to specific athletes. In individual-based analyses, performance parameters typically exhibit a small range and a flat response over iterative performance trials, making establishing associations between performance parameters and the performance criterion problematic. In theoretical studies, computer simulation modelling putatively enables athlete-specific optimum techniques to be identified, but given each athlete's unique intrinsic dynamics, it is far from certain that these optimum techniques will be attainable, particularly under the often intense psychological pressures of competition, irrespective of the volume of practice undertaken. Sports biomechanists and coaching practitioners are advised to be more circumspect with regard to interpreting the results of applied sports biomechanics research and have greater awareness of their assumptions and limitations, as inappropriate interpretation of results may have adverse consequences for performance and injury.
Whether higher variability in older adults' walking is an indication of increased instability has been challenged recently. We performed a computer simulation to investigate the effect of sensorimotor noise on the kinematic variability and stability in a biped walking model. Stochastic differential equations of the system with additive Gaussian white noise was constructed and solved. Sensorimotor noise mainly resulted in higher kinematic variability but its influence on gait stability is minimal. This implies that kinematic variability evident in walking gaits of older adults could be the result of internal sensorimotor noise and not an indication of instability.
This study aimed to determine the effect of added noise, filtering and time series length on the largest Lyapunov exponent (LyE) value calculated for time series obtained from a passive dynamic walker. The simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series. The generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples. The LyE was calculated as the exponential rate of divergence of neighboring trajectories of the state space using Rosenstein's algorithm. To determine the effect of noise on LyE values, seven levels of Gaussian white noise (SNR=55-25dB with 5dB steps) were added to the time series. In addition, the filtering was performed using a range of cutoff frequencies from 3Hz to 19Hz with 2Hz steps. The LyE was calculated for both noise-free and noisy time series with different lengths of 6, 50, 100 and 150 strides. Results demonstrated a high percent error in the presence of noise for LyE. Therefore, these observations suggest that Rosenstein's algorithm might not perform well in the presence of added experimental noise. Furthermore, findings indicated that at least 50 walking strides are required to calculate LyE to account for the effect of noise. Finally, observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating LyE.
Global adoption of wind energy continues to increase, while improving the efficiency of turbine settings requires reliable wind speed (WS) models. The latest models rely on artificial intelligence (AI) optimizations which constructs tests on a range of novel hybrid models to examine the reliability. Gradient Boosting (GB), Random Forest (RF), and Long Short-Term Memory (LSTM) are used in new combinations for data pre-processing. A Time Varying Filter-based Empirical Mode Decomposition (TVFEMD) model is coupled with the GB and LSTM standalone models, to create TVFEMD-GB and TVFEMD-LSTM hybrids, which are run in competition with each other. Eventually, a preferred hybrid form is established, simultaneous hybridization of TVFEMD with GB and LSTM. This study is the first to hybridize these fundamental systems, and create a TVFEMD-GB-LSTM model that can forecast WS. This study finds that the novel hybrid models exhibit superior performance to standalone GB and LSTM models, opening the pathway to alternative WS prediction techniques.