Stable Orbits for a Simple Passive walker Experiencing Foot Slip

We present a method for calculating the derivative of the Poincaré return map for hybrid arcs and apply this technique to a simple model of bipedal walking with foot slip. It turns out that there is a trade off in terms of stability. The smaller the steps taken, the more robust the stability of the periodic orbit becomes. However, as the steps become larger the eigenvalues of the (derivative of the) Poincaré map become smaller, i.e. faster exponential stability to the periodic orbit. In addition to determining which parameters (one of which is the stride angle) result in the existence of a periodic orbit, we explore which initial conditions for a fixed set of parameters lead to periodic motion, that is we find the basin of attraction for the periodic orbits. We end this paper with a way to approximate the boundary of this set.