Kinematic Variability, Local, and Orbital Stability as Predictors of Falls in Passive Dynamic Walking

Abstract

Falls that occur during walking are a significant health care problem. However, there is as yet no generally accepted way to quantify walking stability. While clinicians often use variability as a proxy for stability, such measures do not quantify how systems respond to perturbations. Our goal was to determine how small-to-moderate perturbations affect both walking variability and stability. We applied random perturbations to a simple passive dynamic model of walking to simulate walking down a bumpy slope. Because the model’s global basin of attraction remained fixed, applying larger perturbations directly increased the risk of falling in the model. We generated 10 simulations of 300 consecutive strides of walking at each of 6 perturbation amplitudes up to the maximum level the model could tolerate without falling. We quantified “orbital stability” by calculating the maximum Floquet multipliers for the model. We quantified “local stability” by calculating local exponential rates of divergence of neighboring state space trajectories. As perturbation amplitudes increased, orbital stability and long-term local instability did not change. These measures reflected the fact that the model never actually “fell” during any of our simulations. Conversely, the mean variability of the walker’s kinematics increased exponentially and measures of short-term local instability increased linearly. These measures thus predicted the increased risk of falling exhibited by the model. For all simulated conditions, the walker remained orbitally stable, while exhibiting substantial local instability. This was because very small initial perturbations diverged away from the limit cycle, while larger initial perturbations converged toward the limit cycle. These results provide insight into how these different proposed measures of walking stability are related to each other and to risk of falling.