Periodic Motions of a Hopping Robot With Vertical and Forward Motion

Abstract

This article analyzes the global dynamical behavior of sim plified hopping robot models that are analogous to Raibert's experimental machines. We first review a one-dimensional ver tical hopping model that captures both the vertical hopping dynamics and nonlinear control algorithm. Second, we present a more complicated two-dimensional model that includes both forward and vertical hopping dynamics and a foot placement algorithm. These systems are analyzed using a Poincare return map. In this approach, issues of stability and global dynamical behavior are reduced to the study of the fixed points of this map. For the one-dimensional model, a closed-form return map is obtained. For the two-dimensional model, we derive an exact return map based on the first integrals of motion. Because this map can only be constructed numerically, we also derive an analytical approximation to the return map based on perturba tion methods. The approximate return map is shown to closely predict the behavior of the exact map for small forward run ning velocities. In addition, the approximate return map can be used to quantitatively explore the coupling of vertical and lat eral dynamics and to determine the effect of the foot placement algorithm on dynamical behavior The bifurcation diagrams, which capture variations in dy namical behavior with respect to the variations in system and control parameters, are also constructed. The bifurcation dia grams exhibit a period-doubling cascade. In other words, for certain system parameter values, Raibert's control algorithm can lead to an anomalous nonuniform, but stable, hopping be havior. Using the vertical model results as a guide, we interpret the interesting dynamical behavior of this system.