Stability of Human Balance With Reflex Delays Using Galerkin Approximations

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation constrained by a boundary condition, then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. Numerical examples demonstrate that proportional-derivative-acceleration feedback results in larger stability margins than proportional-derivative feedback in the presence of reflex delays. The study of balance is critical for guiding development of diagnostic criteria, assistive devices, and rehabilitation strategies for the growing elderly population.