Theoretical justification for distal foot power equation

Abstract

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.

Introduction

Rigid-body, link-segment models are commonly used in experimental studies of human movement. The rigid-body assumption should be better for some segments with a primary long bone, such as the shank and thigh, than others with many articulating bones, such as the foot and trunk. Over two decades ago, Siegel et al. (1996) introduced an equation for approximating the power due to the non-rigid nature of the foot that has come to be known as “distal foot power.” In recent years, numerous studies have used and expanded upon this concept of distal foot power (Farinelli et al., 2019, McGibbon and Krebs, 1998, Takahashi et al., 2012, Takahashi and Stanhope, 2013, Zelik et al., 2015, Zelik and Honert, 2018, Zhao et al., 2020). Despite the increasing use of this approach for estimating foot power, no rigorous derivation of distal foot power has been provided in the literature. Siegel et al. (1996) briefly described how to calculate distal foot power, but their justification for the formulation is insufficient and perhaps problematic, as is discussed later. The purpose of this communication is to provide a more thorough justification for the distal foot power equation. A proper understanding of the theoretical basis of the distal foot power equation, as well as its underlying assumptions, should help researchers as they interpret results obtained using this formula.