  Stability of running on rough terrain depends on the propagation of
perturbations due to the ground. We consider stability within the sagittal
plane and model the dynamics of running as a two-dimensional body with an
alternating aerial and stance phase. Stance is modeled as a passive, impulsive
collision followed by an active, impulsive push-off that compensates for
collisional losses. Such a runner has infinitely many strategies to maintain
periodic gaits on flat ground. However, these strategies differ in how
perturbations due to terrain unevenness are propagated. Instabilities manifest
as tumbling (orientational instability) or failing to maintain a steady speed
(translational instability). We find that open-loop strategies that avoid
sensory feedback are sufficient to maintain stability on step-like terrains
with piecewise flat surfaces that randomly vary in height. However, these
open-loop runners lose orientational stability on rough terrains whose slope
and height vary randomly. Only by avoiding tangential collisions is
orientational stability recovered. Tangential collisions may be avoided through
leg-retraction to match foot and ground speed at touch down. By analyzing the
propagation of perturbations, we derive a single dimensionless parameter that
governs stability and guides the design and control of both biological and
robotic runners.
