Approximating the stance map of the SLIP runner based on perturbation approach

The Spring-Loaded Inverted Pendulum (SLIP), or monopedal runner, is widely used to depict running and hopping in mammalian and human locomotion, which is also serving as a template for running robot design. This classic model describes quite a simple mechanical system. Nevertheless issue of seeking the accurate analytic solution revealing the characteristics of the motion during stance remains unsettled due to the nonintegrable terms contained in the system equations. Moreover, several existing analytic approximations by simply ignoring or linearizing the gravitational force can not reveal the entire dynamical behavior of nonlinear system as well as can be breakdown rapidly when applied to a non-symmetric motion case. In this paper, a novel method with perturbation technique is proposed to obtain analytic approximate solutions to the SLIP dynamics in stance phase with considering the effect of gravity. The perturbation solution achieves higher accuracy in predicting the apex trajectory and stance locomotion by comparing with typical existing analytical approximations. Particularly, our solution is validated for non-symmetric case in a large angle range. Additionally, the prediction for stance trajectory is also verified through numerical evaluation.