Modelling Human Gait using a Nonlinear Differential Equation

Abstract

We introduce an innovative method for the investigation of human gait, which is based on the visualisation of the vertical component of the movement of the centre of mass during walking or running, in the space of the coordinates position, velocity, and acceleration of the centre of mass. Collected data has been numerically approximated by the best fitting curve for a non-linear model. The resulting equation for the best fitting plane or curve in this space is a differential equation of second order. The model that we suggest is a Duffing equation with coefficients that depend on the height of a walker or runner and on the angular frequency of the oscillation. Statistics about the distribution of the Duffing stiffness depending on the speed is presented. 1 Author SummaryWe study the human gait modelled by the movement of the centre of mass of the test person. This is an example of a biological process which can be considered as a periodical dynamic system. Roughly, this movement behaves in a similar way to a vibrating mass suspended on a spring, but it is more complex. The vertical component of the movement during walking or running can be visualised as an oscillogram: a graph of the position as a function of time. We suggest a visualisation of the data in 3D space, where the coordinates describe position, velocity, and acceleration. Our new visualisation method allows us to model the movement of a persons centre of mass by a nonlinear differential equation. The resulting curve for an ideal spring-mass movement, without viscosity or external force, is an ellipse in the suggested 3D space. The shape of the data curve shows at which position an additional force was applied, or the movement slowed down. Some deviations are common for all test persons and others are different. In the future we plan to investigate the reasons for these deviations, such as different running techniques or the presence of injuries.