Analysis of A Simplified Hopping Robot

This article offers some analytical results concerning simplified models of Raibert's hopper. We represent the task of achieving a recurring hopping height for an actuated "ball" robot as a stability problem in a nonlinear discrete dynamical control system. We model the properties of Raibert's control scheme in a simplified fashion and argue that his strategy leads to closed-loop dynamics governed by a well-known class of functions, the unimodal maps. The rich mathematical literature on this subject greatly advances our ability to determine the presence of an essentially globally attracting fixed point-the formal rendering of what we intuitively mean by a "correct" strategy. The motivation for this work is the hope that it will facilitate the development of general design principles for "dynamically dexterous" robots.