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%.

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. [

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 [

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 [^{−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 [

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.

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

Consider now a human with leg length

As shown in Figure

Height variation of the human center of mass during successive step phases. Double support phase (left) and single support phase (right) are shown. Extracted and modified from Alexander [

On the other hand, the minimum height of the center of mass occurs during the double support phase and can be defined as

From Figure

The total oscillation of the center of mass

Using the values found in (

By definition, the variation in potential energy

Let us consider now the variation in potential energy due to a gradient. Conditions differ depending upon the sign of the gradient, but for mathematical simplicity a positive gradient will be considered. It should be noted that during gradient walking there is an adaptation of the step phases and they may not necessarily occur at the same point as in level walking. For instance, in downhill walking the rising of the center of mass is done faster than in level walking, and the later decline is done slower. In any case the height variation of the center of mass is not different than in level walking. So in terms of energy it is indifferent whether the elevation of the center of mass happens sooner or later, as at each step very similar variations of height occur. For simplicity of this first approximation model we suppose that the different timing of the walking phases between gradient and level walking does not affect significantly the energetic calculation.

Take the height attained (

Gradient that a human overcomes in one step during uphill walking. Extracted and modified from Alexander [

By definition, the gradient (

The total potential energy contribution to the whole energetic cost is the sum of two main factors: one for the oscillation of the center of mass due to the walking process and the other to overcome a given variable gradient, if any.

Thus, the total potential energy contribution (

It must be taken into account that if the total energy is negative, that is, the body receives energy from a negative gradient, it must brake to avoid acceleration. As mentioned previously, the negative work is about five times more efficient than positive work. To reflect this, an auxiliary function

Plotting

Total potential energy contribution to the energy expenditure as a function of gradient ^{−2}.

Figure

Taking the values found in (

It should be noted that for low gradients (

Optimum gradient as a function of the

The model, which is based on analysis of the variation in potential energy during walking, fits the experimental results on minimum energy expenditure obtained in previous studies and links this minimum to the step length of the subject. The hypothesis that kinetic energy plays a small role at low speeds and is not needed in a first approximation of the mechanical analysis of gradient walking appears to be correct.

The model proves mathematically that the minimum energy expenditure is due to potential energy exchange and is related to the subject’s step length. The reason for this is that in every step the center of mass is raised and the body must supply some potential energy, but with a low negative gradient, this energy can be supplied instead by the loss of potential energy. In these circumstances the body saves energy and can move in a more efficient way, requiring less oxygen uptake.

Our work demonstrates that the negative gradient that minimizes energy expenditure depends on the

Common walking strategies for minimizing the energetic cost of movement usually involve either changing step length or stride frequency [

In conclusion, this work presents a parametric model based on an analysis of the variation in potential energy during gradient walking, which explains the energetic mechanism behind the minimum energy spent experimentally in many previous studies, and links this optimum gradient with step length.

All authors declare that there is no conflict of interests regarding the publication of this paper.

Gerard Saborit and Adrià Casinos contributed equally to the work.

This work was funded by the Spanish Ministry of Economy and Competitiveness (CGL2011-23919) and the Generalitat de Catalunya, program AGAUR (2009SGR884). The authors thank Dr. Rémi Hackert (Muséum, Paris) for technical advice and suggestions.