Spring-loaded inverted pendulum goes through two contraction-extension cycles during the single-support phase of walking.

Gabriel Antoniak,Tirthabir Biswas,Nelson Cortes,Vikas Bhandawat,Chanwoo Chun,Siddhartha Sikdar

doi:10.1242/bio.043695

Gabriel Antoniak, Tirthabir Biswas + Show 4 more

Open Access

https://doi.org/10.1242/bio.043695

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Abstract

ABSTRACTDespite the overall complexity of legged locomotion, the motion of the center of mass (COM) itself is relatively simple, and can be qualitatively described by simple mechanical models. In particular, walking can be qualitatively modeled by a simple model in which each leg is described by a spring-loaded inverted pendulum (SLIP). However, SLIP has many limitations and is unlikely to serve as a quantitative model. As a first step to obtaining a quantitative model for walking, we explored the ability of SLIP to model the single-support phase of walking, and found that SLIP has two limitations. First, it predicts larger horizontal ground reaction forces (GRFs) than empirically observed. A new model – angular and radial spring-loaded inverted pendulum (ARSLIP) – can overcome this deficit. Second, although the leg spring (surprisingly) goes through contraction-extension-contraction-extensions (CECEs) during the single-support phase of walking and can produce the characteristic M-shaped vertical GRFs, modeling the single-support phase requires active elements. Despite these limitations, SLIP as a model provides important insights. It shows that the CECE cycling lengthens the stance duration allowing the COM to travel passively for longer, and decreases the velocity redirection between the beginning and end of a step.

Highlights

Legged locomotion is complex; many approaches to legged locomotion focus on the motion of the animal’s center of mass (COM) rather than on the detailed dynamics of each joint (Full and Koditschek, 1999)
Non-dimensional analysis of spring-loaded inverted pendulum (SLIP) and ARSLIP shows that ARSLIP expands the parameter space over which biologically observed gaits are possible Successful models of locomotion have to be constrained by the average COM kinematics, and by the within-step fluctuations in the COM kinematics because these constraints are typically observed during locomotion
We refer to the parameter subspace within which these constraints are satisfied as the ‘gaitspace’. This analysis extends a previous analysis of SLIP (Biswas et al, 2018) by including a new model, ARSLIP, and by including ground reaction forces (GRFs) in our analysis

Summary

INTRODUCTION

Legged locomotion is complex; many approaches to legged locomotion focus on the motion of the animal’s center of mass (COM) rather than on the detailed dynamics of each joint (Full and Koditschek, 1999). These studies suggest that simple mechanical models can serve as conceptual models for locomotion. A successful model must produce realistic GRFs within the constraints of experimentally observed COM kinematics over experimentally observed stance duration These three constraints are rarely satisfied (Maus et al, 2010, 2014; Lipfert et al, 2012) simultaneously in most studies of locomotion. Fitting SLIP to empirical walking data shows that the stance leg goes through a contraction-expansion-contraction-expansion (CECE) cycle during single-support This cycling appears important for reducing the extent to which velocity vectors have to be redirected during the double-support phase, and increases the stance duration and makes it possible to travel passively for longer distances

METHODS AND RESULTS

10-4 VGRF convex at mid-stance

DISCUSSION

Inverted walking