Energy plane analysis for passive dynamic walking

In this paper, we introduce a novel concept to explore passive dynamic walking. When a planar bipedal robot walks down a shallow slope without any actuations, the total energy of the system moves dynamically because of interactions between energy loss and gain at impact. Our approach is to plot energy trajectories in a two-dimensional space spanned by the total energy and the rate change of it. We define new terms for energy plane analysis and then compare energy plane analysis with phase plane analysis to show the effectiveness of our approach. One of the main advantages is that the dynamic behavior of passive walking can be seen in a two-dimensional plane regardless of the dimension of the system. Our results help us to understand some aspects of passive dynamic walking such as its stability, the basin of attraction, and bifurcations.