Toward a Unified Approximate Analytical Representation for Spatially Running Spring-Loaded Inverted Pendulum Model

As a versatile template characterizing the center of mass movements in legged locomotion, the sagittal spring-loaded inverted pendulum (SLIP) model has been extensively explored in both biomechanics and robotics. Despite concise in mathematical formulation, the accurate analytical representation of the SLIP model is unaccessible due to its intrinsic nonlinearity. This article extends the traditional SLIP model from sagittal hopping into spatially running. A novel perturbation-based approach is proposed to obtain an analytical approximate solution for the 3-D-SLIP model, resulting in a straight-forward closed-form formulation wherein the numerical integration or iteration is avoided. The derived solution does not rely on the negligible gravity assumption as conventional simplification reported in the existing literature and offers satisfactory prediction performance in a wide range of model parameter combinations. The merits of the acquired approximate solution in both critical state of the apex return map and trajectory prediction with high accuracy have been demonstrated via performance evaluation, endowing this approximation the potential in motion planning and gait control for legged robots.