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.

Ground Reaction Force (GRF) is an essential gait parameter. GRF analysis provides important information regarding various aspects of gait. GRF has been traditionally measured using bulky force plates within lab environments. There exist portable force sensing units, but their accuracy is wanting. Estimation of GRF has applications in remote wearable systems for rehabilitation, to measure performance in athletes, etc. This article explores a novel method for GRF estimation using the Lyapunov-Floquet (LF) and invariant manifold theory. We assume human gait to be a periodic motion without external forcing. Using time delayed embedding, a reduced order system can be reconstructed from the vertical GRF data. LF theory can be applied to perform system identification via Floquet Transition Matrix and the Lyapunov Exponents. A Conformal Map was generated using the Lyapunov Floquet Transformation that maps the original time periodic system on a linear Single Degree of Freedom (SDoF) oscillator. The response of the oscillator system can be calculated numerically and then remapped back to the original domain to get GRF time evolution. As an example, the GRF data from an optical motion capture system for two subjects was used to construct the reduced order model and system identification. A comparison between the original system and its reduced order approximation showed good correspondence.

Many researchers have proved the potential of autoparametric system in controlling stability and parametric resonance. In this paper, two different designs for auto-parametrically excited mass-spring-damper systems were studied: one system was controlled by rotational motion of the spring, and the other system was controlled by sliding motion of the spring. The theoretical models were developed to predict the behavior of the systems and also generated stability charts to analyze the systems. For each system, the numerical results from both the nonlinear equation and linearized equation were analyzed and compared. Simulation models were constructed to validate the analytical results. The error between simulation and theoretical results was within 2%. Both theoretical and simulation results displayed that the implementation of autoparametric system could help reduce the resonance by up to 33% and amplify the resonance by up to 34%.

In this work, the authors first study different designs of prosthetic ankles. Then, a new design is proposed, and its dynamics are discussed. Various experiments are conducted to verify the concept. The results of the experiments are discussed, and a conclusion is drawn based on the discussion. An OpenSim simulation is run to emulate the effects of the prosthesis on an amputee’s residual leg. Further iterations of the same design are then discussed and presented. Finally, a conclusion is drawn on the usability of the ankle along with some suggestions for future research.