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This article offers some analytical results concerning sim plified 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 ren dering of what we intuitively mean by a "correct" strat egy. The motivation for this work is the hope that it will facilitate the development of general design principles for "dynamically dexterous" robots.
Koditschek et al. (Sun,) studied this question.