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This paper examines the passive dynamics of straight-ahead level ground quadrupedal running and explores its use in formulating design guidelines that would: a) reduce steady-state roll and b) self-stabilize the rolling motion, thus making the control of the robot more straightforward. To study the effect of mechanical design in the rolling motion, a simple bounding-in-place (BIP) template is introduced as a candidate frontal plane model that captures the targeted steady-state behavior of a straight-ahead level ground running quadruped robot. This model is parametrically analyzed and local stability analysis shows that the dynamics of the open loop passive system alone can confer stability of the motion! These results might explain the success of simple, open loop running controllers on existing experimental robots and can be further used in developing control methodologies for legged robots that take advantage of the mechanical system.