Dynamic Stability of Passive Dynamic Walking on an Irregular Surface

Falls that occur during walking are a significant health problem. One of the greatest impediments to solve this problem is that there is no single obviously “correct” way to quantify walking stability. While many people use variability as a proxy for stability, measures of variability do not quantify how the locomotor system responds to perturbations. The purpose of this study was to determine how changes in walking surface variability affect changes in both locomotor variability and stability. We modified an irreducibly simple model of walking to apply random perturbations that simulated walking over an irregular surface. Because the model’s global basin of attraction remained fixed, increasing the amplitude of the applied perturbations directly increased the risk of falling in the model. We generated ten simulations of 300 consecutive strides of walking at each of six perturbation amplitudes ranging from zero (i.e., a smooth continuous surface) up to the maximum level the model could tolerate without falling over. Orbital stability defines how a system responds to small (i.e., “local”) perturbations from one cycle to the next and was quantified by calculating the maximum Floquet multipliers for the model. Local stability defines how a system responds to similar perturbations in real time and was quantified by calculating short-term and long-term local exponential rates of divergence for the model. As perturbation amplitudes increased, no changes were seen in orbital stability (⁠$r2=2.43%$⁠; $p=0.280$⁠) or long-term local instability (⁠$r2=1.0%$⁠; $p=0.441$⁠). These measures essentially reflected the fact that the model never actually “fell” during any of our simulations. Conversely, the variability of the walker’s kinematics increased exponentially (⁠$r2⩾99.6%$⁠; $p<0.001$⁠) and short-term local instability increased linearly (⁠$r2=88.1%$⁠; $p<0.001$⁠). 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.