Investigation of the chaotic dynamics of the vertical strance of a human body on the model of an inverted pendulum

Abstract

Different stances of human body are studied in medicine and biology for quantitative estimation and clinical diagnostics of  impairments and diseases of the musculoskeletal, nervous, vestibular systems and functions. Human body is composed of ~200 bones and ~600 muscles, and its upright position is unstable due to high complexity of the system and its control mechanisms. Among different techniques of the body sway recording the stabilography is one of the most simple and cheap unit. It is composed by a force platform that can measure the reaction forces over the contact areas between two feet and the platform. The former is portable and can be connected to any laptop via USB port. In this study the functions controlling the vertical stance of a person are studied accounting for the nonlinear dynamics of oscillations of the projection (X_C,Y_C) of center of mass (CM) of the body on the horizontal plane. The time series {X_C(t),Y_C(t)} have been measured on 28 healthy volunteers (age 21-42, height 156-182 cm, body mass 48-84.8 kg). The volunteers were asked to keep a quiet stance on two feet, similar stances with body mass shifted onto the left and then onto the right leg. Each stance has been repeated during 30 s with open and then with closed eyes. After a short break a test with balancing on the left and then on the right leg has been perfrmed. For each case, based on the mathematical model of the inverted pendulum, the calculated control functions u(t) in the form u(t)=k₁(r(t)-r₀)+ k₂(r^/(t)-r^/₀), where r(t) is the radius-vector of the CM, r₀ is its averaged value over time, (.)^/ means the time derivative. Using statistical analysis, the absence of correlations between the control functions for both different subjects and for different positions of the body of the same volunteer was shown. Based on the calculations of the Lyapunov exponent, the individuals have been classified into groups with stable, weakly and highly unstable control of the vertical position of the body. The modeling of such systems in the framework of nondeterministic chaos models with nonlinear control is discussed.