Abstract

The five-link biped is a simple, planar model of human-like walking in which scuffing can be avoided. In this paper, we focus on controller design and stability analysis for the important case of non-steady walking, toward such goals as avoiding obstacles on terrain or meeting specific requirements in speed or energetics. To achieve such tasks as new sensor information about upcoming terrain becomes available, control must be adjusted on-the-fly, preferably using a continuous family of controllers. Here, we present an illustrative case using only two, discrete sets of controllers and investigate the effect of switching between them on a stochastically rough terrain. Of note, we find that the tenth-order system dynamics of unsteady walking can be accurately represented as a Markov process, using only a sparse, quasi-2D mesh of discrete states. This transition matrix approach is then used to determine bounded limits on terrain noise for which guarantees of stability (i.e., never falling) may be given for a particular controller and for arbitrary switching between the controllers, as well as to estimate fall rates for cases where these bounds are exceeded. Our results also allow us to quantify the increase in stability gained by a simple policy of switching based on a noisy, single-step lookahead on terrain. This illustrative example, using two controllers that behave differently and allow for arbitrary switching, provides a framework for future work where tasks or requirements for biped walking are clearly defined and can only be achieved by a wider set of, or ideally a continuous range of, controllers.