Invariant Manifolds in Human Joint Angle Analysis During Walking Gait

The complex dynamics of human gait is yet to be completely understood. Researchers have quantified stability of walking gait using Floquet multipliers as well as Lyapunov exponents. In this article, we utilize the techniques and tools from dynamical system theory and invariant manifolds to map the gait data onto a time invariant representation of a dynamical system. As an example, the complex behavior of the joint angle during walking was studied using a conformal mapping approach that transformed the time periodic system into a time invariant linear system. Time-delay embedding was used to reconstruct the dynamics of the original gait system with time series kinematic data. This minimal realization of the system was used to construct a Single Degree of Freedom (SDOF) oscillator. The time evolution of the linear oscillatory system was mapped back using the conformal mapping derived using Lyapunov-Floquet Theory. This algorithm was verified for walking gait kinematics data for two healthy human subjects. A comparison was drawn between the phase space behavior of the original time periodic system and the remapped time invariant system. The two systems showed good correlation. The algorithm resulted in a well correlated phase space representation.