Analytical characterization of self sustained nonlinear oscillators modelling human walking and bouncing

TL;DR

This paper introduces advanced mathematical techniques to analyze hybrid Van der Pol Rayleigh Duffing oscillators modeling human walking, providing new insights into their stability and parameter estimation methods.

Contribution

It applies the Krylov Bogolyubov perturbation, modified multiple scales, and describing function methods to improve analytical understanding of nonlinear oscillators for human motion modeling.

Findings

Analytical proof of limit cycle stability using Krylov Bogolyubov method

Modified multiple scales method for nonlinear effect approximation

Conditions for convergence between perturbation and describing function amplitudes

Abstract

Researchers have developed hybrid Van der Pol Rayleigh Duffing type oscillators to model human induced forces; however, their analytical framework has largely relied on the Lindstedt Poincare perturbation method, energy balance approaches, and harmonic balance techniques. This paper aims to apply new mathematical tools to these existing models and address potential research gaps. An analytical proof for the stability of the limit cycle has been formulated by using the Krylov Bogolyubov perturbation method. The multiple scales method has been modified to highlight an iterative algorithm for determining the order of approximation required to capture nonlinear effects. The describing function method is utilised to formulate an alternate amplitude. Comparisons between first order amplitudes obtained from perturbation analysis and the describing function formulations reveal conditions under which the two approaches converge. These conditions are exploited to formulate additional constraints for the estimation of model parameters, offering a systematic alternative to purely optimisation based approaches.