Small slope implies low speed for McGeer's passive walking machines

We consider passive dynamic walking machines of the kind originally studied by McGeer. For passive walking on arbitrarily small slopes, we show that any existing gaits must be correspondingly slow. The argument is first presented for nonsingular mass distributions, where it is shown that small slopes preclude long steps and that small steps imply low speeds. The argument is then extended to singular walkers (viewed as physically meaningful limiting cases of nonsingular walkers). A design for a different passive machine that might walk on flat ground is discussed briefly. The discussion in this paper lends insight into biped walking theory and may help to inspire designs for efficient bipedal robots.