Two steps is enough: No need to plan far ahead for walking balance

Before designing a controller in detail, we ask this question: for a given robot state, and every possible control action, what is the minimum number of steps needed to get to a given target state (if it is possible to do so)? Our biped model is a 2D inverted pendulum with massless legs. We have two controls: (i) the magnitude of an impulsive push-off just before heel-strike and (ii) the step length (location of the next heel-strike). The maximum impulse and minimum step time are bounded to reflect limited motor strength. We compute the set of initial mid-stance velocities from which the biped can reach a given target mid-stance velocity in n or fewer steps. The result: for most target speeds and initial velocities, and with realistically strong actuators, it is possible to reach the target in two steps, if it is possible to reach it at all. This `two steps is enough' result expands on Koolen et al.'s [1] results for capturability of the linear inverted pendulum and is consistent with some human balance and visual guidance experiments.

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