## 1 Introduction
### ANN Applications and Nonlinear Measures in Gait Analysis  
Our study outlines an innovative approach to understanding the intricacies of gait dynamics and their metabolic implications through the lens of ANNs and nonlinear analysis. Nonlinear measures quantify human locomotion's complex and dynamic nature and have the potential to offer insights that would be invaluable for customizing assistive device settings to improve walking. Six nonlinear metrics relevant to the examination of gait variability and complexity are studied: the Lyapunov Exponent using the [PERSON] algorithm (LyE\({}_{\text{R}}\)), Detrended Fluctuation Analysis (DFA), the Approximate Entropy (ApEn), the correlation dimension (CD), the Sample Entropy (SpEn), and the Lyapunov Exponent using the [PERSON] algorithm (LyE\({}_{\text{W}}\)).  
These measures provide a nuanced view of human locomotion, highlighting the unpredictable and complex nature of gait patterns that traditional linear analyses might overlook. For example, the Lyapunov Exponent (LyE) is critical in analyzing dynamic systems, including the stability and variability of walking [15]. The LyE measures the local divergence from an attractor, such as a steady state condition, thereby quantifying the motor system's capacity to dampen minor disturbances [16]. The algorithms of [PERSON] et al. (LyE\({}_{\text{W}}\)) [16; 17; 18; 19] and [PERSON] et al. (LyE\({}_{\text{R}}\)) [15; 20] each apply unique normalization processes to the time series data [15]. The DFA is used to analyze time series data to identify long-range correlations [21; 22; 23; 24; 25; 26]. It employs the fractal scaling index, alpha. In gait analysis, an alpha of 0.5 to 1.0 indicates statistical persistence in the intervals between steps, meaning these intervals are consistent over a long distance, with minor fluctuations corrected in subsequent strides. Conversely, an alpha value below 0.5 denotes anti-persistence, where intervals vary and appear randomly over a long range, with immediate adjustments for deviations.  
The ApEn [27; 28; 29; 30] quantifies regularity and unpredictability in time series data. The ApEn is particularly useful in analyzing complex systems where data patterns are not immediately apparent, such as subtle changes in the inherent repeatability of cyclic gait patterns, which may reflect underlying pathology. High ApEn values suggest greater system complexity and less predictability, whereas lower values indicate more regular and predictable patterns. The CD, which has been used to study the complexity of gait patterns between different populations [31], elucidates how data points from a dynamical system, such as a kinematic time series from walking, are arranged within a state space. Small correlation dimension values suggest a limited number of available degrees of freedom, while values in the range of 1.5 to 2.5 are typically associated with deterministic data. In contrast, large values (e.g., 4.0 to 6.0) indicate entirely random data. The SpEn, the final nonlinear measure studied, is less dependent on the data series length and displays a greater capacity to distinguish between similar and dissimilar patterns in a dataset than the ApEn (e.g., single- and dual-task conditions) [32; 33].