Chaos Detection Using Artificial Neural Network-Based Copula Entropy Representations for a 6-DOF Mobile Robot

Defining intricate movement in high-dimensional robotic systems necessitates techniques that differentiate signal anomalies, statistical interrelations, reconstruction discrepancies, and sensitivity to beginning conditions, without considering any singular metric as definitive evidence of deterministic chaos. This research introduces a cohesive framework for examining six coordinated end-effector motion elements—X, Y, Z, roll, pitch, and yaw—assessed experimentally with an optical motion-capture system. Signal complexity was assessed by Approximate Entropy, Sample Entropy, Permutation Entropy, Fuzzy Entropy, Multiscale Entropy, Composite Multiscale Entropy, as well as bivariate and dynamic bivariate entropy. Spectral analyses were employed to delineate frequency-content distribution, whereas correlation and copula frameworks assessed both linear and nonlinear interdependence. Phase-space reconstruction and the Rosenstein greatest Lyapunov exponent technique was further utilized to investigate the divergence of reconstructed states. Anomaly scores based on reconstruction were produced utilizing a single-hidden-layer artificial neural network and a deep autoencoder. The findings indicate heterogeneous complexity across the six motion elements. Roll and pitch demonstrate the most robust linear correlation, with a value of 0.86, but the rotational signals often exhibit more multiscale complexity compared to the translational signals. The superficial ANN reconstructs the synchronized responses with greater precision than the deep autoencoder, with an RMSE of around 0.014 and an MAE of about 0.010, in contrast to 0.212 and 0.151 for the deep model, respectively. Copula analysis indicates that Permutation Entropy exhibits the most robust rank correlation with the ANN anomaly score (Kendall’s τ = 0.38), succeeded by Fuzzy Entropy (τ ≈ 0.27), while Approximate Entropy and Sample Entropy have weak or insignificant correlations. Gaussian, Clayton, and Gumbel copulas offer different representations of symmetric, lower-tail, and upper-tail dependence, although the choice of model depends on the entropy measure and selection criterion. Estimates of the greatest positive Lyapunov exponent suggest varying levels of divergence in reconstructed trajectories; nonetheless, their susceptibility to fitting intervals, noise, and reconstruction parameters constrains a conclusive chaotic interpretation. The suggested approach offers additional insights into signal intricacy, multiscale organization, reconstruction dynamics, and interdependence in robotic movement. The results indicate the existence of oscillatory, transitory, nonstationary, and locally erratic activity, although they do not definitively confirm the presence of persistent deterministic chaos. Dependable validation necessitates consistent Lyapunov assessments, constrained non-periodic dynamics, resilient phase-space reconstruction, and surrogate-data evaluation.