Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 407156, 6 pages

http://dx.doi.org/10.1155/2015/407156

## Parametric Modeling of Human Gradient Walking for Predicting Minimum Energy Expenditure

Departament de Biologia Animal, Universitat de Barcelona, 08028 Barcelona, Spain

Received 17 December 2014; Revised 2 February 2015; Accepted 3 February 2015

Academic Editor: Zhonghua Sun

Copyright © 2015 Gerard Saborit and Adrià Casinos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A mathematical model to predict the optimum gradient for a minimum energetic cost is proposed, based on previous results that showed a minimum energetic cost when gradient is −10%. The model focuses on the variation in mechanical energy during gradient walking. It is shown that kinetic energy plays a marginal role in low speed gradient walking. Therefore, the model considers only potential energy. A mathematical parameter that depends on step length was introduced, showing that the optimal gradient is a function of that parameter. Consequently, the optimal negative gradient depends on the individual step length. The model explains why recent results do not suggest a single optimal gradient but rather a range around −10%.

#### 1. Introduction

Human walking requires energy for a variety of reasons. For instance, in level walking, alternate stages of braking and acceleration exist. Although there is a pendulum-like transfer between potential and kinetic energy of the body center of mass, this is only an energy-saving system. Since the transfer is not complete, additional energy must be incorporated into the system in each step (Cavagna et al. [1, 2]).

In gradient walking the situation changes depending on whether walking up- or downhill. In the former case (positive gradient) positive work is needed to provide gravitational potential energy. In downhill walking, the lost potential energy is absorbed by muscles compelled to stretch. Cavagna [3] showed that the lost energy is transformed into heat through negative or braking work. Direct experiments measuring oxygen uptake in subjects walking on different gradients showed that the minimum energetic cost is not accomplished on level groundbut on a negative gradient of about −10% (Margaria [4]). Further studies (Minetti et al. [5]) demonstrated that minimum energetic cost does not depend on speed and that the optimal path within a positive gradient, considering the vertical cost of transportation, is not always the straight one (Minetti [6]). Further studies (Kamon [7]) showed that oxygen uptake during descent can be about 30% of that required during ascent. Therefore we can deduce that the process of muscular braking, which involves negative work, is energetically different from positive work due to different efficiency factors [8–11]. A complete mechanical analysis must include both kinetic and potential energies but we can calculate each contribution to determine whether one of these (kinetic or potential) is more dominant or whether both energies contribute equally to the whole energetic cost.

Since walking implies low and rather constant velocity, kinetic energy does not vary greatly during the different walking phases. Supposing standard walking at a speed of 1.25 m·s^{−1}, the total kinetic energy involved in the movement is 0.78 J per unit mass. This energy is not supplied at every step since people do not come to a complete standstill between steps. During walking the center of mass moves at almost constant speed. Gottschall and Kram [12] quantified the variation in velocity of the center of mass during different step phases and for different gradients. This variation is about 0.09 m·s^{−1 }per step for level walking with a maximum of 0.18 m·s^{−1} per step in some downhill walking situations. During each step one brakes and accelerates about 0.09 m·s^{−1}, leading to a small variation in speed (from 1.20 m·s^{−1} to 1.30 m·s^{−1}). Calculation of the energy per unit mass taking this speed variation into account shows that the kinetic energy per unit mass needed is about 0.12 J·kg^{−1} per step for level walking and up to 0.20 J·kg^{−1 }per step for high negative gradients.

For potential energy, the vertical oscillation of the center of mass varies from 8 to 10 cm, depending on step length. This means a potential energy oscillation per unit mass from 0.78 J·kg^{−1} per step up to 0.98 J·kg^{−1} per step. It is clear that kinetic energy plays a lesser role in walking at low speeds, being from 5 to almost 10 times smaller than potential energy depending on a number of variables. Another factor to take into account is that the transfer of energy from one walking phase to another usually transforms the excess potential energy, achieved during the single support phase, to kinetic energy for the body. The kinetic energy of the center of mass is almost constant and the main loss of kinetic energy is due to the contact between the still feet on the ground and braking work to avoid acceleration. In the next step phase, the muscles perform positive work to raise the center of mass, thus gaining potential energy again. This is another reason for focusing the analysis on potential energy: the energy transfer involves transforming potential energy into kinetic energy in such a way that the calculation done above for kinetic energy could be overestimated. The significant effect of gravity on walking has been evaluated in previous studies [5, 13] and the incomplete energetic transfer between potential and kinetic energy during walking has been widely discussed [14–17]. As shown by Heglund and Schepens [18], this energy transfer also varies depending on age, with less recovery in children than in adults*.*

For the reasons stated above, this work focuses on the variation in potential energy during the walking process, as a simple and first approximation analysis. Further corrections such as kinetic energy components could be introduced if the model’s predictions are not sufficiently accurate. The main objective of the model is to prove that the vertical oscillation of the center of mass is ultimately responsible for the minimum energy spent at low negative gradient and that only with potential energy analysis will the model fit previous experimental results.

#### 2. Model

In human walking there are basically two stages. In the first stage, the feet are simultaneously on the ground (double support) and the center of mass is at its lowest point, at a distance from the ground. In the second stage, one foot is on the ground (single support), with the corresponding leg straight. The center of mass is at its highest position, at a maximum distance from the ground ().

Consider now a human with leg length . Usually the step length tends to be smaller than the leg length. Thus it can be modeled as , with the variable being an arbitrary parameter that differs for each individual, within a range.

As shown in Figure 1, the maximum height of the center of mass occurs during the single support phase and can be defined as follows: where is the vertical distance from the center of mass to the acetabular joint and is the distance from the acetabular joint to the ground. That means is the leg length.