We present a control method for a simple limit-cycle bipedal walker that uses adaptive frequency oscillators (AFOs) to generate stable gaits. Existence of stable limit cycles is demonstrated with an inverted-pendulum model. This model predicts a proportional relationship between hip torque amplitude and stride frequency. The closed-loop walking control incorporates adaptive Fourier analysis to generate a uniform oscillator phase. Gait solutions (fixed points) are predicted via linearization of the walker model, and employed as initial conditions to generate exact solutions via simulation. Global stability is determined via a recursive algorithm that generates the approximate basin of attraction of a fixed point. We also present an initial study on the implementation of AFO-based control on a bipedal walker with realistic mass distribution and articulated knee joints.

This paper presents a modified passive dynamic walking model with hip friction. We add Coulomb friction to the hip joint of a two-dimensional straight-legged passive dynamic walker. The walking map is divided into two parts – the swing phase and the impact phase. Coulomb friction and impact make the model's dynamic equations nonlinear and non-smooth, and a numerical algorithm is given to deal with this model. We study the effects of hip friction on gait and obtain basins of attraction of different coefficients of friction.

We study a particular example of a recursive distributional equation (RDE) on the unit interval. We identify all invariant distributions, the corresponding ‘basins of attraction’, and address the issue of endogeny for the associated tree-indexed problem, making use of an extension of a recent result of Warren.