A simple model of human walking

Abstract

Aim. We investigate Alexander’s inverted pendulum model, the simplest mathematical model of human walking. Although it successfully explains some kinematic features of human walking, such as the velocity of the body's centre of mass, it does not account for others, like the vertical reaction force and the maximum walking speed. This paper aims to minimally extend Alexander’s model in such a way as to make it a viable and quantitative model of human walking for clinical biomechanics.Material and methods. In order to compare the predictions of Alexander’s model with experimental data on walking, we incorporate in it a robust phenomenological relation between stride frequency and stride length derived in the literature, and we introduce a step-angle dependent muscle force along the pendulum. We then analytically solve the pendulum's motion equation and find the corresponding analytical expression for the average walking speed.Results. The values of the average walking speed for different heights predicted by our model are in excellent agreement with the ones obtained in treadmill experiments. Moreover, it successfully predicts the observed walking-running transition speed, which occurs when the stride length equals the height of an individual. Finally, our extended model satisfactorily reproduces the experimentally observed ground reaction forces in the midstance and terminal stance phases. Consequently, the predicted value of the (height-dependent) maximum walking speed is in reasonable agreement with the one obtained in more sophisticated models of human walking.Conclusions. Augmented with our minimal extensions, Alexander’s model becomes an effective and realistic model of human walking applicable in clinical investigations of the human gate.