Stability

We determine stability for the hybrid system of a bipedal walker by the method of Poincaré. A hybrid solution for stability is a fixed point, where after every impact the hybrid dynamics return to the same non-holonomic configuration. Searching for these fixed points is a formidable challenge, since the continuous dynamics are non-linear and the foot impact transitions are discontinuous. We use a combination of exhaustive search, or "shooting forward," along with Newton-Raphson's algorithm, described below:

The Newton-Raphson Method and its Application to Fixed Points