A mathematical model describes the most efficient way of walking
Researchers at the University of Barcelona have developed a mathematical model that describes an important problem concerning human walking: minimum energy expenditure. In the early 20th century, experiments proved that minimum oxygen consumption do not occur in level walking, but in downhill walking, particularly on a negative gradient of about -10%. The study, which provides mathematical evidence supporting this assumption, has been developed by Gerard Saborit, researcher at the University of Barcelona, and Adrià Casinos, professor in the Department of Animal Biology of the University. Researchers have developed a model that allows predicting the optimal gradient for each person by linking it to leg and step lengths.
Researchers at the University of Barcelona have developed a mathematical model that describes an important problem concerning human walking: minimum energy expenditure. In the early 20th century, experiments proved that minimum oxygen consumption do not occur in level walking, but in downhill walking, particularly on a negative gradient of about -10%. The study, which provides mathematical evidence supporting this assumption, has been developed by Gerard Saborit, researcher at the University of Barcelona, and Adrià Casinos, professor in the Department of Animal Biology of the University. Researchers have developed a model that allows predicting the optimal gradient for each person by linking it to leg and step lengths.
The study, published in the journal Computational and Mathematical Methods in Medicine, could be applied to the rehabilitation of patients with heart diseases. Currently, researchers are using the model and fossil analysis to compare locomotion across human species.
Personalized minimum energy expenditure
We assume that walking downhill is easier than doing it on level ground. Rodolfo Margaria, researcher at the University of Milan, confirmed it experimentally in the early 1930s, when he measured oxygen uptake in subjects walking on different gradients. He showed that minimum energetic cost is not accomplished on level ground but on a negative gradient of about -10%, which equals to drop 10 metres for every 100 metres you walk on level ground. Several explanations were suggested so far, but this is the first time that researchers find a theoretical demonstration.
Researchers at the University of Barcelona focused on the centre of mass, which fluctuates up and down in every step we take and requires some potential energy —the energy stored in an object as a result of its position and the gravitational attraction of the Earth— to overcome gravity forces. When walking on a negative gradient of -10%, the centre of mass drops, so no extra potential energy is needed in order to take a step. Optimal conditions are maintained if gradient does not go beyond this limit; it marks the point from which extra energy is needed as the body has to use legs to brake.
“According to the model, the exact point that indicates minimum energy expenditure depends on the relation between step and leg lengths. This means that each person, according to his/her leg and step lengths, has a particular minimum energy expenditure”, says Gerard Saborit.
To minimize energy expenditure, whether walking up or downhill, step length should be adjusted to level differences. “This variation explains why some recent research studies contradict Margariaʼs work, which affirmed that minimum energy expenditure when walking was a certain value, a gradient of -10%. However, recent studies prove that minimum energy expenditure can expand within a limited range of negative gradients”, emphasizes Adrià Casinos.
Comparison with other human species
Now, researchers are collecting data from a group of people; they are measuring their leg and step lengths when they walk at different speeds. The goal is to obtain information about anatomy and biomechanics in order to compare it with other species. “We want to compare these data with leg and footprint fossils of different human species. Our aim is to observe locomotion differences across species”, points out Gerard Saborit.
Researchers emphasize that the mathematical model could be also applied to the rehabilitation of patients with heart diseases. “In the case of a person who cannot practise aerobic exercise but he/she must exercise maintaining a low heart rate, the model could allow him/her knowing what differences in level he/she should walk in order to spend as minimum energy as possible and make minimum cardiac effort”, concludes Gerard Saborit.
G. Saborit, A. Casions. «Parametric Modeling of Human Gradient Walking for Predicting Minimum Energy Expenditure». Computational and Mathematical Methods in Medicine, August 2015. Doi: 10.1155/2015/407156