PERIODIC MOTIONS OF A HOPPING ROBOT WITH VERTICAL AND FORWARD MOTION

This article analyzes the global dynamical behavior of simplified hopp ing robot models that are analogous to Raibert's experimental machines . We first review a one-dimensional vertical hopping model that captur es both the vertical hopping dynamics and nonlinear control algorithm. Second, we present a more complicated two-dimensional model that incl udes both forward and vertical hopping dynamics and a foot placement a lgorithm. These systems are analyzed using a Poincare return map. In t his approach, issues of stability and global dynamical behavior are re duced to the study of the fixed points of this map. For the one-dimens ional model, a closed-form return map is obtained. For the two-dimensi onal model, we derive an exact return map based on the first integrals of motion. Because this map can only be constructed numerically, we a lso derive an analytical approximation to the return map based on pert urbation methods. The approximate return map is shown to closely predi ct the behavior of the exact map for small forward running velocities. In addition. the approximate return map can be used to quantitatively explore the coupling of vertical and lateral dynamics and to determin e the effect of the foot placement algorithm on dynamical behavior. Th e bifurcation diagrams, which capture variations in dynamical behavior with respect to the variations in system and control parameters, are also constructed. The bifurcation diagrams exhibit a period-doubling c ascade. In other words, for certain system parameter values, Raibert's control algorithm can lead to an anomalous nonuniform, but stable, ho pping behavior Using the vertical model results as a guide, we interpr et the interesting dynamical behavior of this system.