Seeking the Analytical Approximation of the Stance Dynamics of the 3D Spring-Loaded Inverted Pendulum Model By Using Perturbation Approach

The Spring-Loaded Inverted Pendulum (SLIP) has been widely exploited in both biomechanical and robotics research due to its simple form in mathematics and high accuracy in fitting experimental biology data. However the intrinsic nonlinearity of the SLIP dynamics makes accurate analytical representation unavailable. Traditional methods take advantage of numerical integration to handle this issue while several existing analytical approximations focusing on 2D-SLIP model. The 3D-SLIP suitable to physical reality is rarely investigated. This paper presents a novel perturbation-based approach to obtain the closed-form analytical approximations of the 3D-SLIP model in stance phase. In contrast to existing work ignoring the gravitational forces, the proposed approach just relies on assumptions of small leg compression and small leg swept angle. The performance of the derived approximations has been evaluated via comprehensive numerical analysis. The quality of accurate apex prediction promises the approximation as an advantageous and reliable tool for locomotion control of legged robots.