Basin of Attraction and Limit Cycle Oscillation Amplitude of an Ankle-Hip Model of Balance on a Balance Board

Abstract

The study of upright posture (UP) stability is of relevance to estimating risk of falls, especially among people with neuromuscular deficits. Several studies have addressed this problem from a system dynamic approach based on parameter bifurcation analyses, which provide the region of stability (RoS) and the delimiting bifurcation curves (usually Hopf and pitchfork) in some parameter-spaces. In contrast, our goal is to determine the effect of parameter changes on the size of the basin of attraction (BoA) of the UP equilibrium and the amplitude of the limit cycle oscillations (LCOs) emerging from the Hopf bifurcations (HBs). The BoA is an indicator of the ability of the UP to maintain balance without falling, while LCOs may explain the sway motion commonly observed during balancing. In this study, a three degree-of-freedom model for a human balancing on a balance board (BB) was developed. Analysis of the model revealed the BoAs and the amplitude of the LCOs. Results show that physical parameters (time-delays and feedback control gains) have a large impact on the size of the BoA and the amplitude of the LCOs. Particularly, the size of the BoA increases when balancing on a rigid surface and decreases when either proprioceptive or combined visual and vestibular (V&V) feedback gain is too high. With respect to the LCOs, it is shown that they emerge from both the subcritical and supercritical HBs and increase their amplitudes as some parameters vary.