Numerical nonlinear analysis for dynamic stability of an ankle-hip model of balance on a balance board

Study of human upright posture (UP) stability is of great relevance to fall prevention and rehabilitation, especially for those with balance deficits for whom a balance board (BB) is a widely used mechanism to improve balance. The stability of the human-BB system has been widely investigated from a dynamical system point of view. However, most studies assume small disturbances, which allow to linearize the nonlinear human-BB dynamical system, neglecting the effect of the nonlinear terms on the stability. Such assumption has been useful to simplify the system and use bifurcation analyses to determine local dynamic stability properties. However, dynamic stability analysis results through such linearization of the system have not been verified. Moreover, bifurcation analyses cannot provide insight on dynamical behaviors for different points within the stable and unstable regions. In this study, we numerically solve the nonlinear delay differential equation that describes the human-BB dynamics for a range of selected parameters (proprioceptive feedback and time-delays). The resulting solutions in time domain are used to verify the stability properties given by the bifurcation analyses and to compare different dynamical behaviors within the regions. Results show that the selected bifurcation parameters have significant impacts not only on UP stability but also on the amplitude, frequency, and increasing or decaying rate of the resulting trajectory solutions.