Imagine playing basketball on an uneven ground. Small variations in the slope of the ground would make dribbling the ball significantly harder. Much like bouncing a ball, running is also the repeated motion of ballistic flight punctuated by a stance phase where the runner redirects their motion into the next flight phase. To understand how running dynamics is affected by terrain slope, we mathematise this caricature of running as a bouncing gait by abstracting the runner to a circular disc in the sagittal plane and specifying simple collision laws of the disc with the ground. Our model expands on canonical simple models of running by incorporating angular dynamics of the runner during flight, since the runner has no control over their angular momentum when they are not in contact with the ground. This can cause the runner to tumble.

Using Monte Carlo methods, dynamical systems theory and probability theory, we find a critical role for reducing tangential collisions with the ground (prevent scuffing collisions at landing) for maintaining body orientation. Foot placement strategies whereby runners land on flat regions of the terrain further aid in reducing instabilities. While no strategy in the absence of feedback control can maintain stability, anticipatory strategies suffice for preventing tumbling falls. Further, we find scaling relations that can be used to assess running stability of animals and robots of varying sizes and morphologies.

For more information, see the published paper: Dynamics and stability of running on rough terrain.