A New Approach to the Control of the Planar One-Legged Hopper

Abstract

The problem of energy-efficient control of running legged mecha nisms is addressed via the case study of the planar one-legged hopper. Owing to its mechanical simplicity, the planar one-legged hopper is generally considered as the basic prototype of mechanisms capable of ballistic running gaits. What makes this system particularly inter esting is that when properly endowed with energy-storing springs, it can be controlled without spending much energy for actuation. This possibility has already been investigated in a few studies, but has not yet been sufficiently explored. The present study is an at tempt in this direction. Being primarily motivated by control-design and analysis aspects, we have tried to set the basis of a methodol ogy for the derivation of a new class of simple controllers capable of stabilizing passive periodic motions. Elements of this class are derived by making specific novel choices for the model used for con trol design and the control actuation setting. Impulsive feedback inputs applied at discrete time instants and equivalent piecewise- constant feedback inputs, calculated at the beginning of each hop, are considered, instead of more conventional piecewise-continuous feedback controls. Stability analysis is performed on a simplified model of the system, and the soundness of the approach is illustrated via simulation. The study also unveils new results and intriguing questions, some of which could be central to a better understanding of the potentials and limitations of legged locomotion.