Energy variations and periodic solutions in a switched inverted pendulum model of human running gaits

Abstract

In the paper, we investigate the dynamics of a switched spring loaded inverted pendulum (SLIP) model of running. We first show how switching between two sets of differential equations, corresponding to two phases of running namely to the support and flight phase, lead to discontinuities in the system’s overall vector field. These discontinuities allow for the existence of asymptotically stable solutions in an otherwise conservative system. We then present a modification to the standard SLIP model by introducing an additional parameter ξ, which can be thought of as a parameter encapsulating energy dissipation and energy generation during running. Conditions for the existence of periodic running in the model, under small angle approximation and a constraint on the touch-down angle, are then given. A one-dimensional reduced mapping, which captures the system’s dynamics, is then introduced. A linearisation of this mapping, which includes discontinuity mappings, is obtained, which, in turn, allows to determine the stability of periodic solutions. Finally, a one-parameter orbit diagram in ξ parameter depicts numerically found stable asymmetric periodic solutions which terminate in the fold bifurcation.