Theoretical justification for distal foot power equation

The distal foot power equation is a simple yet powerful tool for estimating the power dissipation or generation within the foot even while modeling it as a rigid body. It was introduced over two decades ago, but has seen a resurgence of use in recent years. Nevertheless, the theoretical justification for this formula has thus far been limited. It is difficult to properly use any equation and interpret the results from analyses using it without a solid understanding of how it is derived as well as its underlying assumptions. In this communication, a thorough derivation of the distal foot power equation is provided first for the case where the foot is interacting with a rigid ground without sliding and then second generalized for situations when the foot may slide relative to a deformable ground surface. For the first case, the derivation makes clear that distal foot power represents the power due to the deviation of the foot from a rigid body state for the portion of the foot between its mass center (or other point of reference) and the center of pressure. For the second case, distal foot power represents not only the internal deformation power of the foot, but also the power due to sliding of the foot on the ground and the power due to deformation of the ground near the point of contact.