We demonstrate that an irreducibly simple, uncontrolled, two-dimensional, two-link model, vaguely resembling human legs, can walk down a shallow slope, powered only by gravity. This model is the simplest special case of the passive-dynamic models pioneered by McGeer (1990a). It has two rigid massless legs hinged at the hip, a point-mass at the hip, and infinitesimal point-masses at the feet. The feet have plastic (no-slip, no-bounce) collisions with the slope surface, except during forward swinging, when geometric interference (foot scuffing) is ignored. After nondimensionalizing the governing equations, the model has only one free parameter, the ramp slope gamma. This model shows stable walking modes similar to more elaborate models, but allows some use of analytic methods to study its dynamics. The analytic calculations find initial conditions and stability estimates for period-one gait limit cycles. The model exhibits two period-one gait cycles, one of which is stable when 0 < gamma < 0.015 rad. With increasing gamma, stable cycles of higher periods appear, and the walking-like motions apparently become chaotic through a sequence of period doublings. Scaling laws for the model predict that walking speed is proportional to stance angle, stance angle is proportional to gamma 1/3, and that the gravitational power used is proportional to v4 where v is the velocity along the slope.

The fastest gait and speed of the largest theropod (carnivorous) dinosaurs, such as Tyrannosaurus, is controversial. Some studies contend that Tyrannosaurus was limited to walking, or at best an 11 m s(-1) top speed, whereas others argue for at least 20 m s(-1) running speeds. We demonstrate a method of gauging running ability by estimating the minimum mass of extensor (supportive) muscle needed for fast running. The model's predictions are validated for living alligators and chickens. Applying the method to small dinosaurs corroborates other studies by showing that they could have been competent runners. However, models show that in order to run quickly, an adult Tyrannosaurus would have needed an unreasonably large mass of extensor muscle, even with generous assumptions. Therefore, it is doubtful that Tyrannosaurus and other huge dinosaurs (approximately 6,000 kg) were capable runners or could reach high speeds.

Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species.